82.19 Intersecting with an… to 82.32 Degrees of zero cycles

Section82.19 Intersecting with an invertible sheaf and push and pull#

Section 82.19 of the Stacks Project, Tag 0EQP. 4 tags here are carried by 16 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0EQQ, a lemma

Tag 0EQQ in the Stacks Project.

AlgebraicGeometry.Space.orderOfVanishingSection_eq_multiplicity_mul, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackOrderOfVanishing.lean, line 439. The attribute's comment: “coefficientwise at a prime divisor Z of X with generic point ξ lying over a prime divisor Z' of Y with generic point ξ': the order of vanishing of a meromorphic section pulls back with the multiplicity, ord_{Z,f^L}(f^s) = m · ord_{Z',L}(s), where m is the colength of the extended maximal ideal — the multiplicity with which Z occurs in f^{-1}(Z'), hence the coefficient the flat pullback of Stacks 0EP8 attaches to Z. PROVED, not assumed: that the order of vanishing multiplies by the multiplicity, from Stacks 02M1 (length_baseChange_eq_length_mul_length_quotient) applied to the cyclic modules O^h/a and O^h/b. For X integral this is the source's identity f^ div_L(s) = ∑ n_i div_{f^L|_{X_i}}(s_i) in Z_{n+r-1}(X) read at the coefficient of Z, the one component having multiplicity one; the statement is coefficientwise because the flat pullback of cycles is Stacks 0EP8, which this repository does not have, and for the same reason Stacks 0EPV is already carried coefficientwise here. NOT COVERED: for X with several irreducible components, the step from this identity to the cycle identity, namely the additivity over the components of the multiplicities n_i = m_{X_i,X} of Stacks 0EE2 and Stacks 0EE9; the statement applies to X and to each component X_i separately, with m read in each, but no additivity of those multiplicities is proved in this repository. RESTRICTED: carried as explicit named hypotheses are (i) the flat local homomorphism φ of the henselian local rings at the generic points, with hloc and hflat; (ii) the module N and the meromorphic section fs standing for f^L and f^s, this repository having no pullback of a meromorphic section along a morphism of algebraic spaces, together with hfs, that the ratio of fs at ξ against the pulled-back generator is the image under φ of the ratio of s at ξ' — which is the source's own step of writing s/s_{ξ'} = a/b and pulling the fraction back, the chartwise rule a/b ↦ f-sharp a / f-sharp b of Stacks 0ENC; (iii) the ratios of Stacks 0EPR at the two generic points as the parameters ratioY and ratioX, exactly as Stacks 0EPR and 0EPV carry them; and (iv) finiteness of the multiplicity, as hm naming the natural number m it equals. The source's reduction through the multiplicative torsor of Stacks 0EQG is not used: the length computation is the same one and is carried out directly on the ratio”.

Stacks 0EQQ, coefficientwise. The order of vanishing of a meromorphic section pulls back with multiplicities. Let f : X ⟶ Y be flat, let ℒ be an 𝒪_Y-module and 𝒩 an 𝒪_X-module standing for f^*ℒ, let Z' ⊆ Y and Z ⊆ X be prime divisors with Z over Z', and let m be the multiplicity of Z over Z' along the induced flat local homomorphism φ. For a nonzero meromorphic section s of ℒ whose ratio s / s_{ξ'} at ξ' is a / b, and for a meromorphic section fs of 𝒩 standing for f^*s, whose ratio at ξ against the pulled-back generator is therefore φ(a) / φ(b),

ord_{Z,𝒩}(fs) = m · ord_{Z',ℒ}(s).

The multiplicity m is the one with which Z occurs in f^{-1}(Z'), so for X integral this is the source's identity f^* div_ℒ(s) = ∑ n_i div_{f^*ℒ|_{X_i}}(s_i) read at the coefficient of Z; for X with several components it is each of the two readings of that identity, as the implementation notes explain. The source proves it by passing to the multiplicative torsor q : T ⟶ Y, where the section becomes a rational function, and then invoking Stacks 0EQA; here the two are the same length computation, read on the ratio parameter of Stacks 0EPR rather than on an embedding of the function field.

theorem orderOfVanishingSection_eq_multiplicity_mul
    (hloc : IsLocalHom φ) (hflat : φ.Flat) {m : ℕ}
    (hm : primeDivisorFlatMultiplicity B S hS hX hdecX ZX hY hdecY ZY φ = (m : ℕ∞))
    (L : Modules.{u, max u z} B S hS Y hY) (N : Modules.{u, max u z} B S hS X hX)
    (s : MeromorphicSection.{u, z} B S hS Y hY L)
    (fs : MeromorphicSection.{u, z} B S hS X hX N)
    (ratioY : LocalRatio.{u, z} B S hS hY hdecY ZY L)
    (ratioX : LocalRatio.{u, z} B S hS hX hdecX ZX N)
    (a b : nonZeroDivisors ↥(ZY.genericHenselianLocalRing B S hS hY hdecY))
    (hs : (ratioY s : ZY.genericTotalFractionRing B S hS hY hdecY) =
      IsLocalization.mk' (ZY.genericTotalFractionRing B S hS hY hdecY)
        (a : ↥(ZY.genericHenselianLocalRing B S hS hY hdecY)) b)
    (hfs : (ratioX fs : ZX.genericTotalFractionRing B S hS hX hdecX) =
      IsLocalization.mk' (ZX.genericTotalFractionRing B S hS hX hdecX)
        (φ (a : ↥(ZY.genericHenselianLocalRing B S hS hY hdecY)))
        (⟨φ (b : ↥(ZY.genericHenselianLocalRing B S hS hY hdecY)),
          Ring.map_mem_nonZeroDivisors_of_flatHom hflat b.2⟩ :
          nonZeroDivisors ↥(ZX.genericHenselianLocalRing B S hS hX hdecX))) :
    orderOfVanishingSection.{u, z} B S hS hX hdecX ZX N fs ratioX =
      (m : ℤ) * orderOfVanishingSection.{u, z} B S hS hY hdecY ZY L s ratioY

AlgebraicGeometry.Space.flatPullbackMap_sectionWeilDivisor_eq_componentDivisorSum, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackSectionWeilDivisor.lean, line 85.

In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be a morphism over B which is flat of relative dimension r, let ℒ be an invertible 𝒪_Y-module, assume Y is integral with n = dim_δ(Y), and let s be a nonzero meromorphic section of ℒ. Then

f^* div_ℒ(s) = ∑ n_i div_{f^*ℒ|_{X_i}}(s_i)

in Z_{n + r - 1}(X): the left-hand side is the flat pullback (Stacks 0EP8) of the Weil divisor div_ℒ(s) (Stacks 0EQE); on the right the sum runs over the irreducible components X_i ⊂ X of δ-dimension n + r, s_i = f|_{X_i}^*(s) is the pullback of s (Stacks 0ENC), a nonzero meromorphic section of the invertible module f^*ℒ|_{X_i} = f|_{X_i}^*ℒ, div_{f^*ℒ|_{X_i}}(s_i) is its Weil divisor on X_i pushed forward to X, and n_i = m_{X_i, X} is the multiplicity of X_i in X (Stacks 0EE9).

theorem flatPullbackMap_sectionWeilDivisor_eq_componentDivisorSum :
    flatPullbackMap D sX hX sY hY f hbase r hflat hr (n - 1) (n + r - 1) (by ring)
        (sectionWeilDivisor D sY hY hintY L hL hn s hs) =
      componentDivisorSum D sX hX sY hY f r hflat hintY (n := n) L hL s hs

Tag 0EQR, a lemma

Tag 0EQR in the Stacks Project.

AlgebraicGeometry.Space.flatCyclePullback_firstChernCapCycleMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackFirstChernCap.lean, line 783. The attribute's comment: “as an identity of cycles in Z_{k+r}(X) rather than of classes in CH, which is the finer statement and implies the source's once the class maps are known compatible — see flatPullback_firstChernClassCap, which states the source's own grading. The cap product is the cap product of Stacks 0EQJ as carried in this repository, firstChernCapCycleMap, so the component cycle c_1(L|_W) ∩ [W] remains an explicit parameter standing in for Stacks 0EQE, which does not exist for algebraic spaces here. PROVED, not assumed: the commutation itself. At a single integral closed subspace W both coefficients collapse to one term and those terms are equal by the pointwise Stacks 0EQQ (orderOfVanishingSection_eq_multiplicity_mul); the passage to a general cycle is the locally finite bookkeeping the source performs by appealing to Remark 0EQ8, and it is carried out here by exchanging two double sums. No additive map is assumed to commute with the cap product and no conjunct of the conclusion is a hypothesis. RESTRICTED: carried as explicit named hypotheses are (i) the flat pullback of cycles of Stacks 0EP8, which this repository does not construct, carried as FlatCyclePullback, that is TOGETHER WITH ITS DEFINING FORMULA: a multiplicity mult W T for each pair of integral closed subspaces, the maps Z_m(Y) → Z_n(X) for m + r = n, the statement that the coefficient of f^α at T is ∑ α_W · mult W T, and that f^[W] is carried by f^{-1}(W); (ii) the Weil divisor presentations of Stacks 0EQE for the two component rules, exactly as Stacks 0EQK carries them, with the sheaves L|_W and f^L|_{f^{-1}(W)} and the sections s and f^s as unconstrained parameters identified by their ratios at the generic points, as in the carried Stacks 0EQQ, this repository having no pullback of a sheaf of modules or of a meromorphic section along a morphism of algebraic spaces; (iii) at each prime divisor of f^{-1}(W) the flat local homomorphism of henselian local rings over the prime divisor of W beneath it, with its locality and flatness, as in the carried Stacks 0EQA and 0EQQ; and (iv) the identification of the multiplicity Stacks 0EP8 attaches to that prime divisor with the colength primeDivisorFlatMultiplicity, which is Stacks 0EE9 read along a flat morphism and is not proved here. The inverse image of each integral closed subspace occurring in α is assumed INTEGRAL, which is the case i = 1, n_1 = 1 of the source's sum over the irreducible components of f^{-1}(W); the step to several components is the additivity over them of the multiplicities of Stacks 0EE2 and Stacks 0EE9, which is NOT COVERED by Stacks 0EQA and 0EQQ as stated here, and it is NOT COVERED here either”.

Stacks 0EQR, as an identity of cycles. In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be flat of relative dimension r, let ℒ be an invertible O_Y-module and let α be a (k+1)-cycle on Y. Then

f^*(c₁(ℒ) ∩ α) = c₁(f^*ℒ) ∩ f^*α

in Z_{k+r}(X).

The cap product is the carried Stacks 0EQJ at the level of cycles, AlgebraicGeometry.Space.firstChernCapCycleMap, whose component rule is an explicit parameter; the flat pullback is the carried Stacks 0EP8, whose defining formula is the field AlgebraicGeometry.Space.FlatCyclePullback.coeff. The identity of cycles is finer than the source's identity of classes in CH_{k+r-1}(X), which it implies; see AlgebraicGeometry.Space.flatPullback_firstChernClassCap.

Nothing about either side of the equality is assumed. At a single integral closed subspace the identity is AlgebraicGeometry.Space.flatCyclePullback_mult_firstChernComponentCycle, whose content is the pointwise Stacks 0EQQ; the passage to a general cycle exchanges two locally finite double sums, which is the bookkeeping the source performs through Remark 0EQ8.

theorem flatCyclePullback_firstChernCapCycleMap
    (_hflat : IsFlat D.site D.base D.base_mem f)
    (_hr : HasRelativeDimension r D.site D.base D.base_mem f)
    (cmp : ∀ (W : IntegralClosedSubspace D.site D.base D.base_mem Y)
      (hW : deltaDimension D sY W = k + 1),
      FlatChernDivisorComparison D sX hX sY hY f r k pb setupY setupX chernY chernX W hW)
    (α : CycleGroup D sY (k + 1)) :
    pb.map k (k + (r : ℤ)) rfl (firstChernCapCycleMap D sY hY k chernY α) =
      firstChernCapCycleMap D sX hX (k + (r : ℤ)) chernX
        (pb.map (k + 1) (k + (r : ℤ) + 1) (by ring) α)

AlgebraicGeometry.Space.flatPullback_firstChernClassCap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackFirstChernCap.lean, line 902. The attribute's comment: “the source's own grading: f^(c_1(L) ∩ α) = c_1(f^L) ∩ f^α in the Chow groups, for the cap product of Stacks 0EQJ as carried here, so the component cycle c_1(L|_W) ∩ [W] and the Chow groups CH of Stacks 0EQ6 remain explicit parameters, standing in for Stacks 0EQE and Stacks 0EQ6 parts (2)-(4). PROVED, not assumed: the commutation, which is flatCyclePullback_firstChernCapCycleMap — the identity of cycles whose content is the pointwise Stacks 0EQQ — read through the two class maps. RESTRICTED: in addition to the hypotheses of flatCyclePullback_firstChernCapCycleMap, which are listed on it (the flat pullback of cycles of Stacks 0EP8 with its defining formula; the Weil divisor presentations of Stacks 0EQE; the flat local homomorphisms of henselian local rings; the identification of the multiplicity of Stacks 0EP8 with the colength; and the inverse image of each integral closed subspace occurring in α assumed integral), the flat pullback on the Chow groups is the parameter fstarCH together with hclass, that it is induced by the flat pullback of cycles — which is Stacks 0EQ9, that flat pullback respects rational equivalence, and says nothing about the cap product. The invertible sheaves L on Y and N on X, the latter standing for f^L, are unconstrained parameters as they are in the carried Stacks 0EQQ and as they are for firstChernClassCap itself, which does not consume them”.

Stacks 0EQR. In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be flat of relative dimension r, let ℒ be an invertible O_Y-module and let α be a (k+1)-cycle on Y. Then

f^*(c₁(ℒ) ∩ α) = c₁(f^*ℒ) ∩ f^*α

in CH_{k+r}(X), for the cap product of Stacks 0EQJ as carried in this repository.

This is AlgebraicGeometry.Space.flatCyclePullback_firstChernCapCycleMap composed with the class maps of the two Chow groups. The flat pullback on Chow groups is the parameter fstarCH, and all that is asked of it is that it be induced by the flat pullback of cycles, which is the content of Stacks 0EQ9 — that flat pullback respects rational equivalence — and is the hypothesis hclass. That hypothesis says nothing about the cap product.

theorem flatPullback_firstChernClassCap
    (_hflat : IsFlat D.site D.base D.base_mem f)
    (_hr : HasRelativeDimension r D.site D.base D.base_mem f)
    (cmp : ∀ (W : IntegralClosedSubspace D.site D.base D.base_mem Y)
      (hW : deltaDimension D sY W = k + 1),
      FlatChernDivisorComparison D sX hX sY hY f r k pb setupY setupX chernY chernX W hW)
    (L : Modules.{u, u} D.site D.base D.base_mem Y hY.1)
    (hL : IsInvertibleSheaf D sY hY L)
    (N : Modules.{u, u} D.site D.base D.base_mem X hX.1)
    (hN : IsInvertibleSheaf D sX hX N)
    {CHY : Type w} [AddCommGroup CHY] (chowY : ChowTarget.{u, w, v} D sY hY k CHY)
    {CHX : Type w} [AddCommGroup CHX]
    (chowX : ChowTarget.{u, w, v} D sX hX (k + (r : ℤ)) CHX)
    (fstarCH : CHY →+ CHX)
    (hclass : ∀ β : CycleGroup D sY k,
      chowX.cls (pb.map k (k + (r : ℤ)) rfl β) = fstarCH (chowY.cls β))
    (α : CycleGroup D sY (k + 1)) :
    fstarCH (firstChernClassCap D sY hY L k hL chernY chowY α) =
      firstChernClassCap D sX hX N (k + (r : ℤ)) hN chernX chowX
        (pb.map (k + 1) (k + (r : ℤ) + 1) (by ring) α)

AlgebraicGeometry.Space.flatPullbackChowMapCanonical_canonicalFirstChernCapCanonical, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackFirstChernCapCanonical.lean, line 222.

Stacks 0EQR. In Situation 0EDT let X, Y/B be good, let f : X ⟶ Y be flat of relative dimension r, let ℒ be an invertible 𝒪_Y-module and let α be a (k+1)-cycle on Y. Then

f^*(c₁(ℒ) ∩ α) = c₁(f^*ℒ) ∩ f^*α

in CH_{k+r}(X). Here c₁(ℒ) ∩ - : Z_{k+1}(Y) → CH_k(Y) is the cap product of Stacks 0EQJ (canonicalFirstChernCapCanonical) into the Chow group CH_k(Y) of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK, f^* : CH_k(Y) → CH_{k+r}(X) is the flat pullback on Chow groups of Stacks 0EQB (flatPullbackChowMapCanonical), f^*α is the flat pullback of cycles of Stacks 0EP8 (flatPullbackMap) and f^*ℒ is the inverse image of Stacks 03LU (modulePullback). The source's k-cycle and CH_{k+r-1} are this statement's (k+1)-cycle and CH_{k+r}.

theorem flatPullbackChowMapCanonical_canonicalFirstChernCapCanonical
    (α : CycleGroup D sY (k + 1)) :
    flatPullbackChowMapCanonical.{u + 1} D sX hX sY hY f hbase r hflat hr k (k + (r : ℤ)) rfl
        (canonicalFirstChernCapCanonical.{u + 1} D sY hY L hL k α) =
      canonicalFirstChernCapCanonical.{u + 1} D sX hX
        ((modulePullback D.site D.base D.base_mem
          (ObjectProperty.homMk f : (⟨X, hX.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
            ⟨Y, hY.1⟩)).obj L)
        (modulePullbackInvertible
          (ObjectProperty.homMk f : (⟨X, hX.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
            ⟨Y, hY.1⟩) L hL) (k + (r : ℤ))
        (flatPullbackMap D sX hX sY hY f hbase r hflat hr (k + 1) (k + (r : ℤ) + 1) (by ring)
          α)

Tag 0EQS, a lemma

Tag 0EQS in the Stacks Project.

AlgebraicGeometry.Space.properPushforward_divisorCycleOfSection_eq_iff, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardOfDivisor.lean, line 528. The attribute's comment: “first conclusion REDUCED, not proved: f_(div_{f^L}(f^s)) = [R(X):R(Y)] div_L(s) in Z_{n-1}(Y) IF AND ONLY IF it holds at every prime divisor of Y; proved away from them, using dim_δ(X) = dim_δ(Y); f^L and f^*s are the unconstrained parameters LX and σX”.

Stacks 0EQS, the identity in Z_{n-1}(Y), reduced to its coefficients at the prime divisors of Y. In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be dominant and quasi-compact over B, let X and Y be integral with dim_δ(X) = dim_δ(Y) = n, let ℒ be an invertible 𝒪_Y-module and let s be a nonzero meromorphic section of ℒ. Then

f_*(div_{f^*ℒ}(f^*s)) = [R(X) : R(Y)] div_ℒ(s) in Z_{n-1}(Y)

holds exactly when it holds coefficientwise at every prime divisor of Y, that is when

∑_Z ord_{Z, f^*ℒ}(f^*s) · deg(Z/Z') = [R(X) : R(Y)] · ord_{Z', ℒ}(s)

for every prime divisor Z' ⊆ Y, the sum being over the integral closed subspaces Z ⊆ X.

Both sides are coefficient families on the integral closed subspaces of Y, and away from the prime divisors of Y both vanish: the coefficient of div_ℒ(s) at an integral closed subspace whose point set does not have codimension one is zero by Stacks 0EQE, part (1), and the pushforward has no contribution there by AlgebraicGeometry.Space.properPushforwardCoeff_divisorCycleCoefficients_eq_zero, which is where the equality of the δ-dimensions is used. That is the whole content of the reduction; the coefficient identity at a prime divisor of Y is not proved here.

theorem properPushforward_divisorCycleOfSection_eq_iff (_hL : IsInvertibleSheaf D sY hY L)
    {f : X ⟶ Y} (hfB : f ≫ sY = sX) (hf : IsQuasiCompact D.site D.base D.base_mem f)
    (_hdom : IsDominant D.site D.base D.base_mem f)
    (hDvX : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdecX LX σX ratioX W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hDvY : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem Y =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hY.1 hdecY L σ ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem Y) :
    properCyclePushforward D sX hX sY hY hfB hf (n - 1)
          (divisorCycleOfSection D sX hX hdecX LX σX ratioX hDvX hintX dX hdX hnX) =
        (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
          divisorCycleOfSection D sY hY hdecY L σ ratio hDvY hintY dY hdY hnY ↔
      ∀ W : PrimeDivisor D.site D.base D.base_mem Y,
        (∑ᶠ Z : IntegralClosedSubspace D.site D.base D.base_mem X,
            divisorCycleCoefficients D sX hX hdecX LX σX ratioX Z *
              properCycleMult D sX hX sY hY f Z
                (W.toIntegralClosedSubspace D.site D.base D.base_mem)) =
          (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) *
            divisorCoefficients.{u, u} D.site D.base D.base_mem hY.1 hdecY L σ ratio W

AlgebraicGeometry.Space.properPushforward_divisorCycleOfSection, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardOfDivisor.lean, line 622. The attribute's comment: “first conclusion: f_(div_{f^L}(f^s)) = [R(X):R(Y)] div_L(s) in Z_{n-1}(Y), resting on the single carried input ProperPushforwardNormRule — Stacks 0EQ4 at the prime divisors of Y, which this repository does not have; f^L and f^*s are the parameters LX, σX”.

Stacks 0EQS, first conclusion. In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be dominant and quasi-compact over B, let X and Y be integral with dim_δ(X) = dim_δ(Y) = n, let ℒ be an invertible 𝒪_Y-module and let s be a nonzero meromorphic section of ℒ. Then

f_*(div_{f^*ℒ}(f^*s)) = [R(X) : R(Y)] div_ℒ(s) in Z_{n-1}(Y).

This is the plain equality the source asserts, resting on the single carried input AlgebraicGeometry.Space.ProperPushforwardNormRule, which is Stacks 0EQ4 at the prime divisors of Y. Everything else — that the two coefficient families agree away from the prime divisors of Y, which is where dim_δ(X) = dim_δ(Y) is used — is proved, in AlgebraicGeometry.Space.properPushforward_divisorCycleOfSection_eq_iff.

theorem properPushforward_divisorCycleOfSection (hL : IsInvertibleSheaf D sY hY L)
    {f : X ⟶ Y} (hfB : f ≫ sY = sX) (hf : IsQuasiCompact D.site D.base D.base_mem f)
    (hdom : IsDominant D.site D.base D.base_mem f)
    (hDvX : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdecX LX σX ratioX W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hDvY : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem Y =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hY.1 hdecY L σ ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem Y)
    (hnorm : ProperPushforwardNormRule D sX hX sY hY hdecX hdecY LX σX ratioX L σ ratio
      hintX f) :
    properCyclePushforward D sX hX sY hY hfB hf (n - 1)
        (divisorCycleOfSection D sX hX hdecX LX σX ratioX hDvX hintX dX hdX hnX) =
      (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
        divisorCycleOfSection D sY hY hdecY L σ ratio hDvY hintY dY hdY hnY

AlgebraicGeometry.Space.properPushforward_firstChernClassCapFundamental_of_normRule, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardOfDivisor.lean, line 653. The attribute's comment: “second conclusion: f_(c_1(f^L) ∩ [X]) = c_1(L) ∩ f_[X] in CH_{n-1}(Y), resting on the same carried ProperPushforwardNormRule; the right side is [R(X):R(Y)] · (c_1(L) ∩ [Y]) by the proved f_[X] = [R(X):R(Y)][Y], the left the class on Y of the pushforward cycle”.

Stacks 0EQS, second conclusion. In the situation above,

f_*(c₁(f^*ℒ) ∩ [X]) = c₁(ℒ) ∩ f_*[X] in CH_{n-1}(Y),

the right side being [R(X) : R(Y)] · (c₁(ℒ) ∩ [Y]) by AlgebraicGeometry.Space.properPushforward_integralCycle_integralClosedSubspaceUniv, which is f_*[X] = [R(X) : R(Y)] [Y].

This rests on the same single carried input as the first conclusion; it is not conditional on the first conclusion, which is proved from that input.

theorem properPushforward_firstChernClassCapFundamental_of_normRule
    (hL : IsInvertibleSheaf D sY hY L)
    {CH : Type w} [AddCommGroup CH] (chow : ChowTarget.{u, w, v} D sY hY (n - 1) CH)
    {f : X ⟶ Y} (hfB : f ≫ sY = sX) (hf : IsQuasiCompact D.site D.base D.base_mem f)
    (hdom : IsDominant D.site D.base D.base_mem f)
    (hDvX : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdecX LX σX ratioX W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hDvY : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem Y =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hY.1 hdecY L σ ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem Y)
    (hnorm : ProperPushforwardNormRule D sX hX sY hY hdecX hdecY LX σX ratioX L σ ratio
      hintX f) :
    chow.cls (properCyclePushforward D sX hX sY hY hfB hf (n - 1)
        (divisorCycleOfSection D sX hX hdecX LX σX ratioX hDvX hintX dX hdX hnX)) =
      (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
        firstChernClassCapFundamental D sY hY hdecY L chow σ ratio hDvY hintY dY hdY hnY

AlgebraicGeometry.Space.properPushforward_firstChernClassCapFundamental, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardOfDivisor.lean, line 697. The attribute's comment: “second conclusion, DERIVED from the first (the hypothesis heq): f_(c_1(f^L) ∩ [X]) = c_1(L) ∩ f_[X] in CH_{n-1}(Y), right side read as [R(X):R(Y)] · (c_1(L) ∩ [Y]) via f_[X] = [R(X):R(Y)][Y], left side as the class on Y of the pushforward CYCLE”.

Stacks 0EQS, the identity in CH_{n-1}(Y), from the identity in Z_{n-1}(Y). In the situation above, if

f_*(div_{f^*ℒ}(f^*s)) = [R(X) : R(Y)] div_ℒ(s) in Z_{n-1}(Y),

which is the first conclusion of the source and the hypothesis heq, then

f_*(c₁(f^*ℒ) ∩ [X]) = c₁(ℒ) ∩ f_*[X] in CH_{n-1}(Y),

the right side being [R(X) : R(Y)] · (c₁(ℒ) ∩ [Y]) by f_*[X] = [R(X) : R(Y)] [Y].

This is the source's own derivation of its second conclusion from its first. On the left, c₁(f^*ℒ) ∩ [X] is by Stacks 0EQE, part (2), the class of div_{f^*ℒ}(f^*s), and proper pushforward on Chow groups is not available here, only on cycles, so the left side is the class on Y of the pushforward cycle — the same reading the carried Stacks 0EQJ, case (1), uses for i_*. On the right, c₁(ℒ) ∩ [Y] is again Stacks 0EQE, part (2), AlgebraicGeometry.Space.firstChernClassCapFundamental, a class independent of the chosen nonzero meromorphic section by AlgebraicGeometry.Space.firstChernClassCapFundamental_eq, so the second identity is the image of the first under the class map of the Chow group.

theorem properPushforward_firstChernClassCapFundamental (_hL : IsInvertibleSheaf D sY hY L)
    {CH : Type w} [AddCommGroup CH] (chow : ChowTarget.{u, w, v} D sY hY (n - 1) CH)
    {f : X ⟶ Y} (hfB : f ≫ sY = sX) (hf : IsQuasiCompact D.site D.base D.base_mem f)
    (_hdom : IsDominant D.site D.base D.base_mem f)
    (hDvX : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdecX LX σX ratioX W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hDvY : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem Y =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hY.1 hdecY L σ ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem Y)
    (heq : properCyclePushforward D sX hX sY hY hfB hf (n - 1)
        (divisorCycleOfSection D sX hX hdecX LX σX ratioX hDvX hintX dX hdX hnX) =
      (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
        divisorCycleOfSection D sY hY hdecY L σ ratio hDvY hintY dY hdY hnY) :
    chow.cls (properCyclePushforward D sX hX sY hY hfB hf (n - 1)
        (divisorCycleOfSection D sX hX hdecX LX σX ratioX hDvX hintX dX hdX hnX)) =
      (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
        firstChernClassCapFundamental D sY hY hdecY L chow σ ratio hDvY hintY dY hdY hnY

AlgebraicGeometry.Space.properCyclePushforward_canonicalDivisorCycle_pullback, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardOfDivisorCanonical.lean, line 435. The attribute's comment: “first conclusion: f_(div_{f^L}(f^s)) = [R(X):R(Y)] div_L(s) in Z_{n-1}(Y), for f : X → Y proper and dominant of good integral spaces with dim_δ X = dim_δ Y = n, L invertible on Y, s a nonzero meromorphic section; f^L, f^s, div (0EQE (1) at the canonical generators of 0EPR), f_ and [R(X):R(Y)] all constructed; nothing carried”. References: EGA IV (21.10.19), “special case”.

In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be a proper morphism, let ℒ be an invertible 𝒪_Y-module, assume X and Y integral, f dominant and dim_δ(X) = dim_δ(Y) = n, and let s be a nonzero meromorphic section of ℒ. Then

f_*(div_{f^*ℒ}(f^*s)) = [R(X) : R(Y)] div_ℒ(s) in Z_{n-1}(Y).

Here f^*ℒ is the inverse image AlgebraicGeometry.Space.modulePullback (invertible by AlgebraicGeometry.Space.modulePullbackInvertible), f^*s is the pullback of meromorphic sections, defined because f carries the generic point of X to the generic point of Y (AlgebraicGeometry.Space.pullbacksMeromorphicSections_of_genericPoint, from dominance), and it is nonzero because f^* preserves regular meromorphic sections; div is the Weil divisor of Stacks 0EQE, part (1), at the canonical generators of Stacks 0EPR, f_* is proper pushforward of cycles (Stacks 0EP2) and [R(X) : R(Y)] is the degree of the extension of function fields (Stacks 0ENY).

theorem properCyclePushforward_canonicalDivisorCycle_pullback {f : X ⟶ Y} (hfB : f ≫ sY = sX)
    (hf : IsProper D.site D.base D.base_mem f) (hdom : IsDominant D.site D.base D.base_mem f)
    {n : ℤ} (hnX : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hnY : deltaDimensionClosed D sY hY Set.univ isClosed_univ = (n : EInt)) (hσ : σ ≠ 0) :
    properCyclePushforward D sX hX sY hY hfB hf.2.1.2 (n - 1)
        (canonicalDivisorCycle D sX hX hintX
          ((modulePullback D.site D.base D.base_mem (goodSpacesHom D sX hX sY hY f)).obj L)
          (modulePullbackInvertible (goodSpacesHom D sX hX sY hY f) L hL)
          (pullbackMeromorphicSection D.site D.base D.base_mem (goodSpacesHom D sX hX sY hY f) L
            (pullbacksMeromorphicSections_of_genericPoint D.site D.base D.base_mem (Z := ⟨X, hX.1⟩)
              (Y := ⟨Y, hY.1⟩) hintX hintY f
              (pointsMap_genericPoint_of_isDominant D.site D.base D.base_mem hX.1 hY.1 hintX hintY
                f hdom)) σ)
          (isRegularMeromorphicSection_pullbackMeromorphicSection D.site D.base D.base_mem
            (goodSpacesHom D sX hX sY hY f)
            (pullbacksMeromorphicSections_of_genericPoint D.site D.base D.base_mem (Z := ⟨X, hX.1⟩)
              (Y := ⟨Y, hY.1⟩) hintX hintY f
              (pointsMap_genericPoint_of_isDominant D.site D.base D.base_mem hX.1 hY.1 hintX hintY
                f hdom)) L
            (Stacks.SheafOfModules.IsInvertible.toAlgebraicGeometry _ _ _ hL) σ
            ((isRegularMeromorphicSection_iff_ne_zero D.site D.base D.base_mem hY.1 hintY
              (hasEtaleLocalProperty_isLocallyNoetherian_of_isGood D sY hY) L hL σ).2 hσ)) hnX) =
      (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
        canonicalDivisorCycle D sY hY hintY L hL σ
          ((isRegularMeromorphicSection_iff_ne_zero D.site D.base D.base_mem hY.1 hintY
            (hasEtaleLocalProperty_isLocallyNoetherian_of_isGood D sY hY) L hL σ).2 hσ) hnY

AlgebraicGeometry.Space.properCyclePushforward_sectionWeilDivisor_pullback, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardOfDivisorCanonical.lean, line 553. The attribute's comment: “both conclusions: f_(div_{f^L}(f^s)) = [R(X):R(Y)] div_L(s) in Z_{n-1}(Y), and f_(c_1(f^L) ∩ [X]) = c_1(L) ∩ f_[X] = [R(X):R(Y)] (c_1(L) ∩ [Y]) in CH_{n-1}(Y), for f : X → Y proper and dominant of good integral spaces with dim_δ X = dim_δ Y = n, L invertible on Y, s a nonzero meromorphic section; div and c_1(-) ∩ [-] are Stacks 0EQE (1) and (2), f^L, f^s, f_ and [R(X):R(Y)] constructed; the left side of the second identity is the class of the pushforward of the cycle div_{f^L}(f^s) representing c_1(f^L) ∩ [X]; nothing carried”.

In Situation 0EDT let X/B and Y/B be good, let f : X ⟶ Y be a proper morphism, let ℒ be an invertible 𝒪_Y-module, assume X and Y integral, f dominant and dim_δ(X) = dim_δ(Y) = n, and let s be a nonzero meromorphic section of ℒ. Then

f_*(div_{f^*ℒ}(f^*s)) = [R(X) : R(Y)] div_ℒ(s) as cycles on Y, in Z_{n-1}(Y),

and in particular

f_*(c₁(f^*ℒ) ∩ [X]) = c₁(ℒ) ∩ f_*[X] in CH_{n-1}(Y).

div is the Weil divisor AlgebraicGeometry.Space.sectionWeilDivisor of Stacks 0EQE, part (1), f^*ℒ the inverse image AlgebraicGeometry.Space.modulePullback, f^*s the pullback of meromorphic sections (defined by AlgebraicGeometry.Space.pullbacksMeromorphicSections_of_isDominant_of_isIntegralSpace and nonzero by AlgebraicGeometry.Space.pullbackMeromorphicSection_ne_zero_of_isDominant), f_* proper pushforward of cycles (Stacks 0EP2) and [R(X) : R(Y)] the degree of the extension of function fields (Stacks 0ENY).

In the second identity c₁(f^*ℒ) ∩ [X] is represented by the class of div_{f^*ℒ}(f^*s) in CH_{n-1}(X) (Stacks 0EQE, part (2)), and c₁(ℒ) ∩ f_*[X] is written as [R(X) : R(Y)] (c₁(ℒ) ∩ [Y]), since f_*[X] = [R(X) : R(Y)] [Y] (Stacks 0EP1), with c₁(ℒ) ∩ [Y] the class AlgebraicGeometry.Space.weilDivisorOfInvertibleSheaf.

theorem properCyclePushforward_sectionWeilDivisor_pullback (hL : IsInvertibleSheaf D sY hY L)
    {f : X ⟶ Y} (hfB : f ≫ sY = sX) (hf : IsProper D.site D.base D.base_mem f)
    (hdom : IsDominant D.site D.base D.base_mem f) {n : ℤ}
    (hnX : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hnY : deltaDimensionClosed D sY hY Set.univ isClosed_univ = (n : EInt)) (hσ : σ ≠ 0) :
    properCyclePushforward D sX hX sY hY hfB hf.2.1.2 (n - 1)
        (sectionWeilDivisor D sX hX hintX
          ((modulePullback D.site D.base D.base_mem (goodSpacesHom D sX hX sY hY f)).obj L)
          (Stacks.SheafOfModules.IsInvertible.toAlgebraicGeometry _ _ _
            (modulePullbackInvertible (goodSpacesHom D sX hX sY hY f) L
              (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _ hL))) hnX
          (pullbackMeromorphicSection D.site D.base D.base_mem (goodSpacesHom D sX hX sY hY f) L
            (pullbacksMeromorphicSections_of_isDominant_of_isIntegralSpace D sX hX sY hY hintX
              hintY hdom) σ)
          (pullbackMeromorphicSection_ne_zero_of_isDominant D sX hX sY hY hintX hintY L
            (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _ hL) σ hdom hσ)) =
        (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
          sectionWeilDivisor D sY hY hintY L hL hnY σ hσ ∧
      chowClass.{v} D sY hY (n - 1) (properCyclePushforward D sX hX sY hY hfB hf.2.1.2 (n - 1)
        (sectionWeilDivisor D sX hX hintX
          ((modulePullback D.site D.base D.base_mem (goodSpacesHom D sX hX sY hY f)).obj L)
          (Stacks.SheafOfModules.IsInvertible.toAlgebraicGeometry _ _ _
            (modulePullbackInvertible (goodSpacesHom D sX hX sY hY f) L
              (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _ hL))) hnX
          (pullbackMeromorphicSection D.site D.base D.base_mem (goodSpacesHom D sX hX sY hY f) L
            (pullbacksMeromorphicSections_of_isDominant_of_isIntegralSpace D sX hX sY hY hintX
              hintY hdom) σ)
          (pullbackMeromorphicSection_ne_zero_of_isDominant D sX hX sY hY hintX hintY L
            (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _ hL) σ hdom hσ))) =
        (functionFieldDegreeAtGenericPoint D sX hX sY hY hintX f : ℤ) •
          weilDivisorOfInvertibleSheaf.{v} D sY hY hintY L hL hnY

Tag 0EQT, a lemma

Tag 0EQT in the Stacks Project.

AlgebraicGeometry.Space.properPushforward_firstChernCapCycleMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardFirstChernCap.lean, line 636. The attribute's comment: “at the level of cycles: p_(c_1(p^L) ∩ α) = c_1(L) ∩ p_α in Z_k(Y) for α in Z_{k+1}(X), with the cap product of Stacks 0EQJ and the pushforward of Stacks 0EP1; on the two carried inputs of ProperPushforwardChernComparison — Stacks 0EP2 for the closed-immersion square of the source's reduction, and Stacks 0EQS along the component morphism p|_W of Stacks 0EEG, which is proper because p is; the component rule of p^L is the parameter chernX, the repository having no pullback of sheaves of modules; the reduction to a single subspace is PROVED”.

Stacks 0EQT, at the level of cycles. In Situation 0EDT let X/B and Y/B be good, let p : X ⟶ Y be a proper morphism over B, let ℒ be an invertible 𝒪_Y-module and let α ∈ Z_{k+1}(X). Then

p_*(c₁(p^*ℒ) ∩ α) = c₁(ℒ) ∩ p_*α in Z_k(Y),

an identity of k-cycles, hence also of classes in CH_k(Y).

The cap products are the cycle-level cap product of Stacks 0EQJ, with the component rule of p^*ℒ on X and of ℒ on Y, and the pushforwards are those of Stacks 0EP1, taken along the quasi-compactness that properness of p supplies. The two carried inputs are the fields of AlgebraicGeometry.Space.ProperPushforwardChernComparison, both of them statements about the pushforward along the component morphism p|_W : W ⟶ W' of Stacks 0EEG, which is proper because p is: Stacks 0EP2 for the square of the source's reduction, and Stacks 0EQS along p|_W, which is the source's final step. The passage from a single integral closed subspace to an arbitrary cycle is proved.

theorem properPushforward_firstChernCapCycleMap
    (cmp : ProperPushforwardChernComparison D hX hY hpB hp k chernX chernY)
    (α : CycleGroup D sX (k + 1)) :
    properCyclePushforward D sX hX sY hY hpB hp.2.1.2 k
        (firstChernCapCycleMap D sX hX k chernX α) =
      firstChernCapCycleMap D sY hY k chernY
        (properCyclePushforward D sX hX sY hY hpB hp.2.1.2 (k + 1) α)

AlgebraicGeometry.Space.properPushforward_firstChernClassCap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardFirstChernCap.lean, line 664. The attribute's comment: “p_(c_1(p^L) ∩ α) = c_1(L) ∩ p_α in CH_k(Y) for α in Z_{k+1}(X), the right side being the cap product firstChernClassCap of Stacks 0EQJ at the invertible L, the left the class on Y of the pushforward CYCLE, proper pushforward of Chow classes being unavailable here; on the two carried inputs of ProperPushforwardChernComparison — Stacks 0EP2 for the closed-immersion square of the source's reduction and Stacks 0EQS along the component morphism of Stacks 0EEG, proper because p is; the component rule of p^L is the parameter chernX”.

Stacks 0EQT. In Situation 0EDT let X/B and Y/B be good, let p : X ⟶ Y be a proper morphism over B, let α ∈ Z_{k+1}(X) and let ℒ be an invertible sheaf on Y. Then

p_*(c₁(p^*ℒ) ∩ α) = c₁(ℒ) ∩ p_*α in CH_k(Y).

The right side is the cap product of Stacks 0EQJ on Y, AlgebraicGeometry.Space.firstChernClassCap at ℒ, applied to the pushforward of cycles p_*α. Proper pushforward on Chow groups is not available in this repository, only on cycles, so the left side is the class on Y of the pushforward of the cycle representing c₁(p^*ℒ) ∩ α, which is the reading Stacks 0EQS and Stacks 0EQJ, case (1), already use. The identity holds already at the level of cycles, which is AlgebraicGeometry.Space.properPushforward_firstChernCapCycleMap.

theorem properPushforward_firstChernClassCap
    (L : Modules.{u, u} D.site D.base D.base_mem Y hY.1) (hL : IsInvertibleSheaf D sY hY L)
    {CH : Type w} [AddCommGroup CH] (chow : ChowTarget.{u, w, v} D sY hY k CH)
    (cmp : ProperPushforwardChernComparison D hX hY hpB hp k chernX chernY)
    (α : CycleGroup D sX (k + 1)) :
    chow.cls (properCyclePushforward D sX hX sY hY hpB hp.2.1.2 k
        (firstChernCapCycleMap D sX hX k chernX α)) =
      firstChernClassCap D sY hY L k hL chernY chow
        (properCyclePushforward D sX hX sY hY hpB hp.2.1.2 (k + 1) α)

AlgebraicGeometry.Space.chowGroupCanonicalProperPushforward_canonicalFirstChernCapCanonical, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardFirstChernCapChowGroup.lean, line 100.

Stacks 0EQT. In Situation 0EDT let X/B and Y/B be good, let p : X ⟶ Y be a proper morphism over B, let α ∈ Z_{k+1}(X) and let ℒ be an invertible 𝒪_Y-module. Then

p_*(c₁(p^*ℒ) ∩ α) = c₁(ℒ) ∩ p_*α in CH_k(Y).

Every object is the constructed one, nothing is a parameter:

• p^*ℒ is the inverse image (modulePullback p).obj ℒ of Stacks 03LU, invertible by Stacks 0B8P (AlgebraicGeometry.Space.modulePullbackInvertible);
• c₁(-) ∩ - is the cap product of Stacks 0EQJ, Z_{k+1} → CH_k, at the canonical component rule (AlgebraicGeometry.Space.canonicalFirstChernCapCanonical), on X for p^*ℒ and on Y for ℒ;
• on the left p_* is proper pushforward on Chow groups, CH_k(X) → CH_k(Y), of Stacks 0EQC (AlgebraicGeometry.Space.chowGroupCanonicalProperPushforward); on the right it is proper pushforward of cycles Z_{k+1}(X) → Z_{k+1}(Y) of Stacks 0EP1;
• CH_k is the Chow group of Stacks 0EQ6 (AlgebraicGeometry.Space.chowGroupCanonical).

The proof is the source's: the formula is reduced to a single integral closed subspace W and its image W', where it is Stacks 0EQS for W ⟶ W' when the δ-dimension is preserved; the case where it drops, which the source passes over, is properCyclePushforward_canonicalDivisorCycleAt_pullback_eq_zero_of_deltaValue_drop.

theorem chowGroupCanonicalProperPushforward_canonicalFirstChernCapCanonical
    (α : CycleGroup D sX (k + 1)) :
    chowGroupCanonicalProperPushforward.{u + 1} D sX hX sY hY hqB hq k
        (canonicalFirstChernCapCanonical.{u + 1} D sX hX
          ((modulePullback D.site D.base D.base_mem q).obj L) (modulePullbackInvertible q L hL) k
          α) =
      canonicalFirstChernCapCanonical.{u + 1} D sY hY L hL k
        (properCyclePushforward D sX hX sY hY hqB hq.2.1.2 (k + 1) α)

AlgebraicGeometry.Space.properPushforward_firstChernCapCycleMap_iff_component, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardFirstChernCapReduction.lean, line 309. The attribute's comment: “REDUCED, not proved: p_(c_1(p^L) ∩ α) = c_1(L) ∩ p_*α in Z_k(Y) for every α in Z_{k+1}(X) IF AND ONLY IF it holds on the fundamental cycle [W] of every integral closed subspace W of X. BOTH directions are proved and nothing is assumed, with the cap product of Stacks 0EQJ and the pushforward of Stacks 0EP1. The remaining clause, at a single W, is Stacks 0EQS along the component morphism p|_W of Stacks 0EEG, which rests on Stacks 0EQ4 and is open in this repository”.

Stacks 0EQT, reduced to the source's final step, in both directions. In Situation 0EDT let X/B and Y/B be good, let p : X ⟶ Y be a proper morphism over B, let ℒ be an invertible 𝒪_Y-module and let α range over Z_{k+1}(X). Then the projection formula

p_*(c₁(p^*ℒ) ∩ α) = c₁(ℒ) ∩ p_*α in Z_k(Y)

holds for every α exactly when it holds on the fundamental cycle [W] of every integral closed subspace W ⊆ X, that is when

p_*(c₁(p^*ℒ) ∩ [W]) = deg(W/W') · (c₁(ℒ) ∩ [W'])

for W' the closure of the image of |W| and deg(W/W') the degree of Stacks 0EP1, part (1).

Both directions are proved and nothing is assumed: the forward direction is AlgebraicGeometry.Space.properPushforward_firstChernComponentCycle_of_firstChernCapCycleMap, the reverse the source's steps (2)–(4), AlgebraicGeometry.Space.properPushforward_firstChernCapCycleMap_of_component.

The cap products are the cycle-level cap product of Stacks 0EQJ and the pushforwards those of Stacks 0EP1, taken along the quasi-compactness that properness of p supplies. The component rules of p^*ℒ and of ℒ are the parameters chernX and chernY; the instance at the canonical rules, where c₁(p^*ℒ) is the first Chern class of the genuine inverse image p^*ℒ of Stacks 03LU, is AlgebraicGeometry.Space.properPushforward_canonicalFirstChernCapCycleMap_iff_component.

The right-hand clause is Stacks 0EQS along the component morphism p|_W : W ⟶ W' of Stacks 0EEG, and it is open here; see the implementation notes.

theorem properPushforward_firstChernCapCycleMap_iff_component :
    (∀ α : CycleGroup D sX (k + 1),
        properCyclePushforward D sX hX sY hY hpB hp.2.1.2 k
            (firstChernCapCycleMap D sX hX k chernX α) =
          firstChernCapCycleMap D sY hY k chernY
            (properCyclePushforward D sX hX sY hY hpB hp.2.1.2 (k + 1) α)) ↔
      ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
        properCyclePushforward D sX hX sY hY hpB hp.2.1.2 k
            (firstChernComponentCycle D sX hX k chernX W) =
          properCycleDegree D sX hX sY hY p W •
            firstChernComponentCycle D sY hY k chernY
              (IntegralClosedSubspace.closureImage D.site D.base D.base_mem hY.1
                (D.isDecentSpace_of_isGood sY hY) p W)

AlgebraicGeometry.Space.properPushforward_canonicalFirstChernCapCycleMap_iff_component, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProperPushforwardFirstChernCapReduction.lean, line 376. The attribute's comment: “REDUCED, not proved, with BOTH component rules tied to L rather than parametric: p_(c_1(p^L) ∩ α) = c_1(L) ∩ p_α in Z_k(Y) for every α in Z_{k+1}(X) IF AND ONLY IF it holds on the fundamental cycle [W] of every integral closed subspace W of X. BOTH directions are proved and nothing is assumed. c_1(p^L) is the canonical rule of Stacks 0EQJ at the genuine inverse image p^*L of Stacks 03LU, invertible by Stacks 0B8P, and c_1(L) the canonical rule at L. The remaining clause, at a single W, is Stacks 0EQS along the component morphism p|_W of Stacks 0EEG, which rests on Stacks 0EQ4 and is open in this repository”.

Stacks 0EQT, reduced to the source's final step, with both component rules tied to ℒ. In Situation 0EDT let X/B and Y/B be good, let p : X ⟶ Y be a proper morphism over B, let ℒ be an invertible 𝒪_Y-module and let α range over Z_{k+1}(X). Then

p_*(c₁(p^*ℒ) ∩ α) = c₁(ℒ) ∩ p_*α in Z_k(Y)

holds for every α exactly when it holds on the fundamental cycle [W] of every integral closed subspace W ⊆ X. Both directions are proved and nothing is assumed; this is AlgebraicGeometry.Space.properPushforward_firstChernCapCycleMap_iff_component at the canonical component rules of Stacks 0EQJ.

Here c₁(p^*ℒ) is the first Chern class of the genuine inverse image p^*ℒ of Stacks 03LU, through AlgebraicGeometry.Space.canonicalPullbackFirstChernRule, and c₁(ℒ) that of ℒ, through AlgebraicGeometry.Space.canonicalFirstChernRule; neither is a parameter, and both are the rule W ↦ div(t_W) of Stacks 0EQJ at the canonical ratios of Stacks 0EPR.

The remaining clause, at a single W, is Stacks 0EQS along the component morphism p|_W : W ⟶ W' of Stacks 0EEG, and is open here: Stacks 0EQS rests on the coefficient identity of Stacks 0EQ4, which this repository does not prove. See the implementation notes of StacksFrontier.AlgebraicGeometry.Spaces.Chow.ProperPushforwardOfDivisor.

theorem properPushforward_canonicalFirstChernCapCycleMap_iff_component (hqB : q.hom ≫ sY = sX)
    (hq : IsProper D.site D.base D.base_mem q.hom) (k : ℤ) :
    (∀ α : CycleGroup D sX (k + 1),
        properCyclePushforward D sX hX sY hY hqB hq.2.1.2 k
            (firstChernCapCycleMap D sX hX k
              (canonicalPullbackFirstChernRule D hX hY q L hL k) α) =
          firstChernCapCycleMap D sY hY k (canonicalFirstChernRule D sY hY L hL k)
            (properCyclePushforward D sX hX sY hY hqB hq.2.1.2 (k + 1) α)) ↔
      ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
        properCyclePushforward D sX hX sY hY hqB hq.2.1.2 k
            (firstChernComponentCycle D sX hX k
              (canonicalPullbackFirstChernRule D hX hY q L hL k) W) =
          properCycleDegree D sX hX sY hY q.hom W •
            firstChernComponentCycle D sY hY k (canonicalFirstChernRule D sY hY L hL k)
              (IntegralClosedSubspace.closureImage D.site D.base D.base_mem hY.1
                (D.isDecentSpace_of_isGood sY hY) q.hom W)

Sources

• Section 82.19, Tag 0EQP
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.orderOfVanishingSection_eq_multiplicity_mul · AlgebraicGeometry.Space.flatPullbackMap_sectionWeilDivisor_eq_componentDivisorSum · AlgebraicGeometry.Space.flatCyclePullback_firstChernCapCycleMap · AlgebraicGeometry.Space.flatPullback_firstChernClassCap · AlgebraicGeometry.Space.flatPullbackChowMapCanonical_canonicalFirstChernCapCanonical · AlgebraicGeometry.Space.properPushforward_divisorCycleOfSection_eq_iff · AlgebraicGeometry.Space.properPushforward_divisorCycleOfSection · AlgebraicGeometry.Space.properPushforward_firstChernClassCapFundamental_of_normRule

Section82.20 The key formula#

Section 82.20 of the Stacks Project, Tag 0EQU. 2 tags here are carried by 11 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0EQU, the section

Tag 0EQU in the Stacks Project.

AlgebraicGeometry.Space.PrimeDivisor.genericResidueFieldEquivFunctionField, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormula/GenericResidueField.lean, line 188. The attribute's comment: “the identification κ(ξ) = R(Z) of the residue field at the generic point of a prime divisor with its function field, at the residue field of the chosen chart”.

The canonical isomorphism θ_Z : κ(u) ≃+* R(Z) of Section 0EQU between the residue field of the local ring of the chart at the generic point of a prime divisor and the function field of the prime divisor. Its inverse is evaluation at ρ_Z.

def PrimeDivisor.genericResidueFieldEquivFunctionField :
    Z.genericResidueField B S hS hX hdec ≃+* rationalFunctions B S hS Z.closed.1

AlgebraicGeometry.Space.PrimeDivisor.henselianResidueFieldEquivFunctionField, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormula/GenericResidueField.lean, line 306. The attribute's comment: “the residue field of the henselian local ring at the generic point of a prime divisor is its function field”.

The composite κ(𝒪^h_{X, ξ}) ≃+* R(Z): the residue field of the henselian local ring at the generic point of a prime divisor is the function field of the prime divisor, through the residue field κ(u) of the chosen chart. A unit v of 𝒪^h_{X, ξ} thus has a residue θ(v̄) ∈ R(Z), which is how the key formula of Section 0EQU reads a change of generator on Z.

def PrimeDivisor.henselianResidueFieldEquivFunctionField :
    letI := henselianLocalRing_isLocalRing B S hS X hX (Z.genericPoint B S hS X) hdec
    IsLocalRing.ResidueField ↥(Z.genericHenselianLocalRing B S hS hX hdec) ≃+*
      rationalFunctions B S hS Z.closed.1

AlgebraicGeometry.Space.PrimeDivisor.genericRestriction, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormula/GenericRestriction.lean, line 268. The attribute's comment: “the image s_ξ|_Z of a generator s_ξ of ℒ_ξ in the fibre at the generic point, as a meromorphic section of ℒ|_Z”.

σ_Z = s_ξ|_Z, the restriction to a prime divisor of the generator at its generic point: the unique meromorphic section of ℒ|_Z whose pullback along ρ_Z : Spec κ(u) → Z is the image of the generator s_ξ of ℒ at the generic point.

def PrimeDivisor.genericRestriction :
    MeromorphicSection.{u, u} B S hS Z.subspace.toFunctor Z.closed.1
      (Z.restrictModule B S hS hX L)

AlgebraicGeometry.Space.firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaComponentDivisor.lean, line 420. The attribute's comment: “either cycle identity of the section, c_1(N) ∩ div_L(s) = ∑ ord_{B_i}(f_i) (Z_i → X)_* div_{N|Z_i}(t_i|Z_i) in Z_{n-2}(X), with the component cycle no longer a parameter: it is the Weil divisor div_{N|W}(t_W) of Stacks 0EQE part (1) on W, for the restriction N|W of Stacks 03LU. N is arbitrary here, and what is then carried on W is what Stacks 0EQE carries on X: the meromorphic section t W, the Stacks 0EPR ratio family, the Stacks 0EPS local finiteness and the Stacks 0EDN dimension function. For N invertible, which is the section's own hypothesis, t W and the dimension function are produced and not carried, see firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum_of_isInvertible. hdim is Stacks 0EPZ. The key formula 0EQV IS proved, unconditionally and axiom-clean, as keyFormula_of_ne_zero of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, and 0EQX follows there as canonicalFirstChernCap_divisor_comm; what is still open is only the quantified predicate KeyFormula of KeyFormulaCycleIdentities, which ranges over the component rules of 0EQJ and the tame-symbol divisors”.

Stacks 0EQU: either cycle identity of the section preceding the key formula, with the component cycle div_{𝒩|Z_i}(t_i|Z_i) constructed and not assumed. In Situation 0EDT let X/B be good and integral with dim_δ(X) = n, let ℒ and 𝒩 be 𝒪_X-modules, let s be a nonzero meromorphic section of ℒ, and let {Z_i}, i ∈ I, be an injective locally finite family of prime divisors of X containing every prime divisor along which s has nonzero order of vanishing. Then

c₁(𝒩) ∩ div_ℒ(s) = ∑_i ord_{Z_i,ℒ}(s) · (Z_i → X)_* div_{𝒩|Z_i}(t_i|Z_i)

in Z_{n-2}(X), the component cycle c₁(𝒩|_{Z_i}) ∩ [Z_i] being the Weil divisor div_{𝒩|Z_i}(t_i|Z_i) of Stacks 0EQE part (1) on Z_i, that is AlgebraicGeometry.Space.componentDivisorCycleRule. The two identities displayed in Section 0EQU before the key formula are the two instances of this statement, exchanging (ℒ, s) with (𝒩, t) and 𝒩 with ℒ.

The orders ord_{Z_i,ℒ}(s) are those of Stacks 0EPR, which the source identifies with its ord_{B_i}(f_i); the δ-dimension hypothesis hdim is Stacks 0EPZ, proved as AlgebraicGeometry.Space.deltaDimension_toIntegralClosedSubspace.

This statement takes, as inputs on each W, exactly what Stacks 0EQE takes on X: the meromorphic section t W of 𝒩|_W, which at W = Z_i is the source's reduction t_i|Z_i (AlgebraicGeometry.Space.PrimeDivisor.genericFibreReduction, read through Stacks 0EN0); the ratio family ratioN of Stacks 0EPR; the local finiteness hfinN of Stacks 0EPS; and the dimension function dimN of Stacks 0EDN. The same four, on X, are ratio, hfin, d and hd here. Two of those four are no longer needed once 𝒩 is invertible, which is Section 0EQU's own hypothesis on 𝒩: firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum_of_isInvertible is this identity with t produced by Stacks 0EPQ on W and with dimN derived from d. This version is kept because it needs no hypothesis on 𝒩 at all.

The key formula itself, Stacks 0EQV, is proved, unconditionally and axiom-clean: AlgebraicGeometry.Space.keyFormula_of_ne_zero of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, at the canonical choices and for the constructed tame-symbol sum, from which Stacks 0EQX follows there as AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm. What is still open is only the predicate AlgebraicGeometry.Space.KeyFormula of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaCycleIdentities, which quantifies the component rules of Stacks 0EQJ and the principal divisors of the tame symbols; at that generality it is not a theorem of this library.

theorem firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum
    {k : ℤ} {I : Type v}
    (t : ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
      MeromorphicSection.{u, u} D.site D.base D.base_mem W.subspace.toFunctor
        (setup.isSpace W) (integralClosedSubspaceRestrict D sX hX N W))
    (ratioN : ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
      LocalRatioFamily.{u, u} D.site D.base D.base_mem (setup.isSpace W) (setup.isDecent W)
        (integralClosedSubspaceRestrict D sX hX N W))
    (dimN : ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
      DimensionFunction (points D.site D.base D.base_mem W.subspace.toFunctor))
    (hdimN : ∀ (W : IntegralClosedSubspace D.site D.base D.base_mem X)
      (x : points D.site D.base D.base_mem W.subspace.toFunctor),
      dimN W x = D.deltaValue (W.subspace.ι ≫ sX) x)
    (hfinN : ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
      LocallyFinite fun P : PrimeDivisor D.site D.base D.base_mem W.subspace.toFunctor =>
        if setup.sectionOrder W (integralClosedSubspaceRestrict D sX hX N W) (t W) (ratioN W) P = 0
          then ∅ else P.support D.site D.base D.base_mem W.subspace.toFunctor)
    (s : MeromorphicSection.{u, z} D.site D.base D.base_mem X hX.1 L)
    (ratio : LocalRatioFamily.{u, z} D.site D.base D.base_mem hX.1 hdec L)
    (hfin : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hint : IsIntegralSpace D.site D.base D.base_mem X)
    (d : DimensionFunction (points D.site D.base D.base_mem X))
    (hd : ∀ x, d x = D.deltaValue sX x) {n : ℤ}
    (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hk : k + 1 = n - 1)
    (Z : I → PrimeDivisor D.site D.base D.base_mem X)
    (hZinj : Function.Injective Z)
    (hZfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X (Z i).subspace)
    (hdim : ∀ i, deltaDimension D sX
      ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) = k + 1)
    (hcover : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratio W ≠ 0 →
        ∃ i, Z i = W) :
    firstChernCapCycleMap D sX hX k
        (componentDivisorCycleRule sX hX setup N k t ratioN dimN hdimN hfinN)
        (divisorCycleAt.{u, z} D sX hX hdec L s ratio hfin hint d hd hn hk) =
      cycleFamilySum D sX hX
        (fun i => (Z i).toIntegralClosedSubspace D.site D.base D.base_mem) hZfin k
        (fun i => orderOfVanishingSection.{u, z} D.site D.base D.base_mem hX.1 hdec (Z i) L s
          (ratio (Z i)) •
            componentDivisorCycleRule sX hX setup N k t ratioN dimN hdimN hfinN
              ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) (hdim i))

AlgebraicGeometry.Space.firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum_of_isInvertible, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaComponentDivisor.lean, line 597. The attribute's comment: “either cycle identity of the section for an invertible N, with the component cycle div_{N|W}(t_W) of Stacks 0EQE part (1) on W and its meromorphic section t_W no longer a parameter: t_W is a regular meromorphic section of N|W produced by Stacks 0EPQ on W, and the Stacks 0EDN dimension function on W is the one on X pulled back along the closed embedding. What is carried on W is the Stacks 0EPR ratio family and the Stacks 0EPS local finiteness of the prime divisors of nonzero order, both blocked by LocalRatio being an unconstrained function type. hdim is Stacks 0EPZ. The key formula 0EQV IS proved, unconditionally and axiom-clean, as keyFormula_of_ne_zero of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, and 0EQX follows there as canonicalFirstChernCap_divisor_comm; what is still open is only the quantified predicate KeyFormula of KeyFormulaCycleIdentities, which ranges over the component rules of 0EQJ and the tame-symbol divisors”.

Stacks 0EQU for an invertible 𝒩, with the component cycle's meromorphic section and dimension function produced and not assumed. Section 0EQU's cycle identity

c₁(𝒩) ∩ div_ℒ(s) = ∑_i ord_{Z_i,ℒ}(s) · (Z_i → X)_* div_{𝒩|Z_i}(t_i|Z_i)

in Z_{n-2}(X), where the component cycle at an integral closed subspace W ⊆ X is the Weil divisor div_{𝒩|_W}(t_W) of Stacks 0EQE part (1) on W of a meromorphic section t_W this statement constructs: AlgebraicGeometry.Space.componentMeromorphicSection, a regular meromorphic section of 𝒩|_W produced by Stacks 0EPQ on W, whose hypotheses (a), (b), (c) of Stacks 0ENA hold because W is integral and locally Noetherian. The source's t_i|Z_i — the reduction of a chosen generator of 𝒩_{ξ_i}, AlgebraicGeometry.Space.PrimeDivisor.genericFibreReduction — is one such section, and the identity holds for any of them: the component cycle is a parameter of Stacks 0EQJ occurring on both of its sides.

The dimension function of Stacks 0EDN on each W is likewise no longer carried: it is AlgebraicGeometry.Space.subspaceDeltaDimensionFunction, the dimension function d on |X| pulled back along the closed embedding |W| → |X|, which is a dimension function because an intermediate point of a specialization starting in a closed set lies in that closed set.

Invertibility of 𝒩 is Section 0EQU's own hypothesis on 𝒩. What is still carried on each W is the pair Stacks 0EQE carries on X: the ratio family ratioN of Stacks 0EPR and the local finiteness hfinN of the prime divisors of W along which t_W has nonzero order, Stacks 0EPS. Neither can be discharged while AlgebraicGeometry.Space.LocalRatio is an unconstrained function type — a ratio family not required to be s ↦ s / s_ξ for a generator s_ξ makes ord arbitrary, so the set of prime divisors of nonzero order is not controlled; the first set of Stacks 0EPS, which does not mention the ratio, is proved and applies to t_W because t_W is regular, by AlgebraicGeometry.Space.locallyFinite_primeDivisor_not_memGenericModuleStalk_of_isIntegral.

The key formula itself, Stacks 0EQV, is proved, unconditionally and axiom-clean: AlgebraicGeometry.Space.keyFormula_of_ne_zero of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, at the canonical choices and for the constructed tame-symbol sum, from which Stacks 0EQX follows there as AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm. What is still open is only the predicate AlgebraicGeometry.Space.KeyFormula of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaCycleIdentities, which quantifies the component rules of Stacks 0EQJ and the principal divisors of the tame symbols; at that generality it is not a theorem of this library.

theorem firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum_of_isInvertible
    {k : ℤ} {I : Type v}
    (hN : IsInvertibleModule (⟨X, hX.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) N)
    (ratioN : ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
      LocalRatioFamily.{u, u} D.site D.base D.base_mem (setup.isSpace W) (setup.isDecent W)
        (integralClosedSubspaceRestrict D sX hX N W))
    (d : DimensionFunction (points D.site D.base D.base_mem X))
    (hd : ∀ x, d x = D.deltaValue sX x)
    (hfinN : ∀ W : IntegralClosedSubspace D.site D.base D.base_mem X,
      LocallyFinite fun P : PrimeDivisor D.site D.base D.base_mem W.subspace.toFunctor =>
        if setup.sectionOrder W (integralClosedSubspaceRestrict D sX hX N W)
            (componentMeromorphicSection sX hX setup N hN W) (ratioN W) P = 0
          then ∅ else P.support D.site D.base D.base_mem W.subspace.toFunctor)
    (s : MeromorphicSection.{u, z} D.site D.base D.base_mem X hX.1 L)
    (ratio : LocalRatioFamily.{u, z} D.site D.base D.base_mem hX.1 hdec L)
    (hfin : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hint : IsIntegralSpace D.site D.base D.base_mem X) {n : ℤ}
    (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hk : k + 1 = n - 1)
    (Z : I → PrimeDivisor D.site D.base D.base_mem X)
    (hZinj : Function.Injective Z)
    (hZfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X (Z i).subspace)
    (hdim : ∀ i, deltaDimension D sX
      ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) = k + 1)
    (hcover : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratio W ≠ 0 →
        ∃ i, Z i = W) :
    firstChernCapCycleMap D sX hX k
        (componentDivisorCycleRule sX hX setup N k
          (componentMeromorphicSection sX hX setup N hN) ratioN
          (fun W => subspaceDeltaDimensionFunction D sX hX d W)
          (fun W => subspaceDeltaDimensionFunction_eq_deltaValue D sX hX d hd W) hfinN)
        (divisorCycleAt.{u, z} D sX hX hdec L s ratio hfin hint d hd hn hk) =
      cycleFamilySum D sX hX
        (fun i => (Z i).toIntegralClosedSubspace D.site D.base D.base_mem) hZfin k
        (fun i => orderOfVanishingSection.{u, z} D.site D.base D.base_mem hX.1 hdec (Z i) L s
          (ratio (Z i)) •
            componentDivisorCycleRule sX hX setup N k
              (componentMeromorphicSection sX hX setup N hN) ratioN
              (fun W => subspaceDeltaDimensionFunction D sX hX d W)
              (fun W => subspaceDeltaDimensionFunction_eq_deltaValue D sX hX d hd W) hfinN
              ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) (hdim i))

AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaProof/Headline.lean, line 137. The attribute's comment: “the key formula of the section, c_1(N) ∩ div_L(s) = c_1(L) ∩ div_N(t) in CH_{n-2}(X), at the section's hypotheses; Lemma 0EQV is keyFormula_of_ne_zero”.

Section 0EQU, the key formula: c₁(𝒩) ∩ div_ℒ(s) = c₁(ℒ) ∩ div_𝒩(t) in CH_{n-2}(X). In Situation 0EDT let X/B be good, X integral with dim_δ(X) = n, let ℒ and 𝒩 be invertible 𝒪_X-modules, and let s, t be nonzero meromorphic sections of ℒ, 𝒩. Then c₁(𝒩) ∩ div_ℒ(s) = c₁(ℒ) ∩ div_𝒩(t) in the Chow group CH_{n-2}(X) of Stacks 0EQ6, the cap products being those of Stacks 0EQJ and the divisors those of Stacks 0EQE, all at the canonical choices.

The proof is the section's: the family {Z_i} of prime divisors where s or t fails to generate is locally finite (Stacks 0EPS), the two cap products are the sums over it of ord_{B_i}(f_i) (Z_i → X)_* div(t_i|_{Z_i}) and ord_{B_i}(g_i) (Z_i → X)_* div(s_i|_{Z_i}), and the key formula (Stacks 0EQV) makes their difference a locally finite sum of principal divisors of tame symbols on the Z_i.

theorem canonicalFirstChernCap_divisor_comm
    (s : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 L) (hs : s ≠ 0)
    (t : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 N) (ht : t ≠ 0)
    {n : ℤ} (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt)) :
    canonicalFirstChernCap.{u, u + 1} D sX hX N hN (n - 2)
        (canonicalDivisorCycleAt D sX hX hint L hL s
          (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint L hL hs)
          (deltaDimensionClosed_eq_sub_two_add_one_add_one D sX hX hn)) =
      canonicalFirstChernCap.{u, u + 1} D sX hX L hL (n - 2)
        (canonicalDivisorCycleAt D sX hX hint N hN t
          (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint N hN ht)
          (deltaDimensionClosed_eq_sub_two_add_one_add_one D sX hX hn))

AlgebraicGeometry.Space.canonicalFirstChernCap_canonicalDivisorCycleAt, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaSection.lean, line 309. The attribute's comment: “the cycle identities of the section, in CH_{n-2}(X) as the source states them: c_1(N) ∩ div_L(s) = ∑ ord_{B_i}(f_i) (Z_i → X)_* div_{N|Z_i}(t_i|Z_i), with everything at the canonical choices (the cap product, the divisors and the ratios s = f_i s_i are constructed, not carried) and for an arbitrary regular meromorphic section of N|Z_i in place of t_i|Z_i; the second identity is the same statement with (L, s) and (N, t) exchanged. Claims (1)-(2) of the section are canonicalDivisorCycle_apply_eq_zero_of_notMem, canonicalDivisorCycle_apply_primeDivisor and orderOfVanishingSection_canonical_eq_ord”.

Section 0EQU, claim (3). In Situation 0EDT let X/B be good, X integral with dim_δ(X) = n = k + 2, let ℒ, 𝒩 be invertible with s a regular meromorphic section of ℒ, and let {Z_i} be a locally finite family of prime divisors such that s generates ℒ_ξ at the generic point of every prime divisor outside it. For generators s_i = v_i s_{ξ_i} of ℒ_{ξ_i} write s = f_i s_i, f_i ∈ Q(B_i)^*. Then for any regular meromorphic sections τ_i of 𝒩|_{Z_i} — in particular for the source's t_i|_{Z_i} —

c₁(𝒩) ∩ div_ℒ(s) = ∑ ord_{B_i}(f_i) (Z_i → X)_* div_{𝒩|_{Z_i}}(τ_i) in CH_{n-2}(X).

The cap product is the one of Stacks 0EQJ at the canonical choices, AlgebraicGeometry.Space.canonicalFirstChernCap, the divisors are those of Stacks 0EQE at the canonical ratios, and CH_{n-2}(X) is the Chow group of Stacks 0EQ6. The second identity of the section is this one with (ℒ, s) and (𝒩, t) exchanged.

theorem canonicalFirstChernCap_canonicalDivisorCycleAt
    (N : Modules.{u, u} D.site D.base D.base_mem X hX.1)
    (hN : IsInvertibleModule (⟨X, hX.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) N)
    {I : Type v} (Zf : I → PrimeDivisor D.site D.base D.base_mem X)
    (hinj : Function.Injective Zf)
    (hfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X
        ((Zf i).toIntegralClosedSubspace D.site D.base D.base_mem).subspace)
    (hcov : ∀ W : PrimeDivisor D.site D.base D.base_mem X, (∀ i, Zf i ≠ W) →
      W.SectionGenerates D sX hX hint L hL s)
    {k : ℤ}
    (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = ((k + 1 + 1 : ℤ) : EInt))
    (v : ∀ i, ↥((Zf i).genericHenselianLocalRing D.site D.base D.base_mem hX.1
      (isDecentSpace_of_isGood D sX hX)))
    (hv : ∀ i, IsUnit (v i))
    (f : ∀ i, ((Zf i).genericTotalFractionRing D.site D.base D.base_mem hX.1
      (isDecentSpace_of_isGood D sX hX))ˣ)
    (hf : ∀ i, (f i : (Zf i).genericTotalFractionRing D.site D.base D.base_mem hX.1
        (isDecentSpace_of_isGood D sX hX)) * algebraMap _ _ (v i) =
      (Zf i).canonicalGenericRatio D.site D.base D.base_mem hX.1
        (isDecentSpace_of_isGood D sX hX) L hL hint
        (hasEtaleLocalProperty_isLocallyNoetherian_of_isGood D sX hX) s)
    (τ : ∀ i, MeromorphicSection.{u, u} D.site D.base D.base_mem
      ((Zf i).toIntegralClosedSubspace D.site D.base D.base_mem).subspace.toFunctor
      (isGood_integralClosedSubspace D sX hX _).1
      (integralClosedSubspaceRestrict D sX hX N
        ((Zf i).toIntegralClosedSubspace D.site D.base D.base_mem)))
    (hτ : ∀ i, IsRegularMeromorphicSection.{u, u} D.site D.base D.base_mem _
      (isGood_integralClosedSubspace D sX hX _).1 _ (τ i)) :
    haveI := fun W : PrimeDivisor D.site D.base D.base_mem X =>
      W.nontrivial_genericHenselianLocalRing D.site D.base D.base_mem hX.1
        (isDecentSpace_of_isGood D sX hX) hint
        (hasEtaleLocalProperty_isLocallyNoetherian_of_isGood D sX hX)
    haveI := fun W : PrimeDivisor D.site D.base D.base_mem X =>
      W.isNoetherianRing_genericHenselianLocalRing D.site D.base D.base_mem hX.1
        (isDecentSpace_of_isGood D sX hX) hint
        (hasEtaleLocalProperty_isLocallyNoetherian_of_isGood D sX hX)
    haveI := fun W : PrimeDivisor D.site D.base D.base_mem X =>
      W.krullDimLE_genericHenselianLocalRing D.site D.base D.base_mem hX.1
        (isDecentSpace_of_isGood D sX hX) hint
        (hasEtaleLocalProperty_isLocallyNoetherian_of_isGood D sX hX)
    canonicalFirstChernCap.{u, v} D sX hX N hN k
        (canonicalDivisorCycleAt D sX hX hint L hL s hs hn) =
      chowClass.{v} D sX hX k (cycleFamilySum D sX hX
        (fun i => (Zf i).toIntegralClosedSubspace D.site D.base D.base_mem) hfin k
        (fun i => (Zf i).ord D.site D.base D.base_mem hX.1 (isDecentSpace_of_isGood D sX hX)
            (f i : (Zf i).genericTotalFractionRing D.site D.base D.base_mem hX.1
              (isDecentSpace_of_isGood D sX hX)) (f i).isUnit •
          canonicalDivisorCycleAt D _ (isGood_integralClosedSubspace D sX hX _)
            ((Zf i).toIntegralClosedSubspace D.site D.base D.base_mem).isIntegralSpace
            (integralClosedSubspaceRestrict D sX hX N
              ((Zf i).toIntegralClosedSubspace D.site D.base D.base_mem))
            (isInvertibleModule_integralClosedSubspaceRestrict D sX hX N hN _) (τ i) (hτ i)
            (deltaDimensionClosed_univ_toIntegralClosedSubspace D sX hX hint hn (Zf i))))

AlgebraicGeometry.Space.locallyFinite_primeDivisor_not_sectionGenerates, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaSection.lean, line 436. The attribute's comment: “existence of the family {Z_i}: the prime divisors at whose generic point s or t fails to generate form a locally finite family (by 0EPS)”.

Section 0EQU: the family exists ("Such a set exists by Spaces over Fields, Lemma 0EPS"). The prime divisors at whose generic point s fails to generate ℒ_ξ or t fails to generate 𝒩_ξ form a locally finite family; it is the smallest family the section may be taken over.

theorem locallyFinite_primeDivisor_not_sectionGenerates :
    LocallyFinite fun W : {W : PrimeDivisor D.site D.base D.base_mem X //
        ¬ (W.SectionGenerates D sX hX hint L hL s ∧ W.SectionGenerates D sX hX hint N hN t)} =>
      W.1.support D.site D.base D.base_mem X

Tag 0EQV, a lemma

Tag 0EQV in the Stacks Project.

AlgebraicGeometry.Space.firstChernCapCycleMap_divisorCycleAt_eq_cycleFamilySum, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaCycleIdentities.lean, line 345. The attribute's comment: “the situation above, not the lemma: either cycle identity of Section 0EQU, c_1(N) ∩ div_L(s) = ∑ ord_{B_i}(f_i) (Z_i → X)_* div_{N|Z_i}(t_i|Z_i) in Z_{n-2}(X), for an arbitrary component rule chern of Stacks 0EQJ, of which it uses no property; hdim is Stacks 0EPZ. DISCHARGED ELSEWHERE: the instance at the registered cap product — the component cycle canonicalFirstChernRule at the canonical choices, in CH_{n-2}(X) — is canonicalFirstChernCap_canonicalDivisorCycleAt, and the key formula itself is proved unconditionally as keyFormula_of_ne_zero; neither carries a free component rule”.

Either cycle identity of the section preceding the key formula, for an arbitrary component rule. In Situation 0EDT let X/B be good and integral with dim_δ(X) = n, let ℒ be an 𝒪_X-module with a nonzero meromorphic section s, and let {Z_i}, i ∈ I, be an injective locally finite family of prime divisors of X containing every prime divisor along which s has nonzero order of vanishing. Then

c₁(𝒩) ∩ div_ℒ(s) = ∑_i ord_{Z_i,ℒ}(s) · (Z_i → X)_*(c₁(𝒩|_{Z_i}) ∩ [Z_i])

in Z_{n-2}(X). The two identities displayed in Section 0EQU before the key formula are the two instances of this statement: the first is the one above, and the second is obtained by exchanging (ℒ, s) with (𝒩, t) and 𝒩 with ℒ, that is by using the component rule of ℒ and the divisor of t.

The orders ord_{Z_i,ℒ}(s) are those of Stacks 0EPR, which the source identifies with its ord_{B_i}(f_i), and they depend on the ratio parameter of Stacks 0EPR only through its value at the generic points, not at all by AlgebraicGeometry.Space.orderOfVanishingSection_unit_smul. The δ-dimension hypothesis hdim is Stacks 0EPZ, proved as AlgebraicGeometry.Space.deltaDimension_toIntegralClosedSubspace.

The component cycle c₁(𝒩|_{Z_i}) ∩ [Z_i], which the source writes div_{𝒩|Z_i}(t_i|Z_i), is here the arbitrary component rule chern of Stacks 0EQJ: nothing below uses any property of it. AlgebraicGeometry.Space.firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaComponentDivisor is the instance of this statement in which that cycle is the Weil divisor div_{𝒩|Z_i}(t_i|Z_i) of Stacks 0EQE part (1) on Z_i; that is the form Section 0EQU displays, and it carries the tag.

The key formula itself, Stacks 0EQV, is proved, at the canonical choices and for the constructed tame-symbol sum: AlgebraicGeometry.Space.keyFormula_of_ne_zero of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, from which Section 0EQU's c₁(𝒩) ∩ div_ℒ(s) = c₁(ℒ) ∩ div_𝒩(t) in CH_{n-2}(X) follows as AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm. What is left open is only the predicate AlgebraicGeometry.Space.KeyFormula below, which quantifies the component rules and the tame-symbol divisors and so cannot be discharged at that generality.

theorem firstChernCapCycleMap_divisorCycleAt_eq_cycleFamilySum
    {k : ℤ} (chern : FirstChernComponentCycleRule D sX k) {I : Type v}
    (s : MeromorphicSection.{u, z} D.site D.base D.base_mem X hX.1 L)
    (ratio : LocalRatioFamily.{u, z} D.site D.base D.base_mem hX.1 hdec L)
    (hfin : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratio W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hint : IsIntegralSpace D.site D.base D.base_mem X)
    (d : DimensionFunction (points D.site D.base D.base_mem X))
    (hd : ∀ x, d x = D.deltaValue sX x) {n : ℤ}
    (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hk : k + 1 = n - 1)
    (Z : I → PrimeDivisor D.site D.base D.base_mem X)
    (hZinj : Function.Injective Z)
    (hZfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X (Z i).subspace)
    (hdim : ∀ i, deltaDimension D sX
      ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) = k + 1)
    (hcover : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratio W ≠ 0 →
        ∃ i, Z i = W) :
    firstChernCapCycleMap D sX hX k chern
        (divisorCycleAt.{u, z} D sX hX hdec L s ratio hfin hint d hd hn hk) =
      cycleFamilySum D sX hX
        (fun i => (Z i).toIntegralClosedSubspace D.site D.base D.base_mem) hZfin k
        (fun i => orderOfVanishingSection.{u, z} D.site D.base D.base_mem hX.1 hdec (Z i) L s
          (ratio (Z i)) •
            chern ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) (hdim i))

AlgebraicGeometry.Space.firstChernCap_sub_firstChernCap_eq_cycleFamilySum, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaCycleIdentities.lean, line 528. The attribute's comment: “the situation above, not the lemma: the FIRST cycle of the key formula equals c_1(N) ∩ div_L(s) - c_1(L) ∩ div_N(t), from the two cycle identities of Section 0EQU, for arbitrary component rules chernN and chernL of Stacks 0EQJ, of which no property is used; hdim is Stacks 0EPZ. DISCHARGED ELSEWHERE: the key formula itself is proved unconditionally, at the canonical choices and for the constructed tame-symbol sum, as keyFormula_of_ne_zero, and its Section 0EQU consequence in CH_{n-2}(X) as canonicalFirstChernCap_divisor_comm”.

The first cycle of the key formula is the difference of the two cap products of Section 0EQU: with the family {Z_i} containing every prime divisor along which s or t has nonzero order of vanishing,

c₁(𝒩) ∩ div\_ℒ(s) - c₁(ℒ) ∩ div\_𝒩(t) = ∑\_i (Z\_i → X)\_*(ord\_\{B\_i}(f\_i) div\_\{𝒩|Z\_i}(t\_i|Z\_i) - ord\_\{B\_i}(g\_i) div\_\{ℒ|Z\_i}(s\_i|Z\_i))

in Z_{n-2}(X). It is the two cycle identities of Section 0EQU together with additivity of the locally finite sum of Stacks 0EQ6 part (1).

The two restricted divisors div_{𝒩|Z_i}(t_i|Z_i) and div_{ℒ|Z_i}(s_i|Z_i) are the component cycles carried as the parameters chernN and chernL of Stacks 0EQJ, standing in for Stacks 0EQE on Z_i; the two orders are those of Stacks 0EPR, which the source identifies with ord_{B_i}(f_i) and ord_{B_i}(g_i); the δ-dimension hypothesis hdim is Stacks 0EPZ, proved as AlgebraicGeometry.Space.deltaDimension_toIntegralClosedSubspace.

The key formula itself — the equality of this cycle with ∑_i (Z_i → X)_* div(∂_{B_i}(f_i, g_i)) — is proved at the canonical choices, as AlgebraicGeometry.Space.keyFormula_of_ne_zero; only the predicate AlgebraicGeometry.Space.KeyFormula, which quantifies the component rules and the tame-symbol divisors, is left open.

theorem firstChernCap_sub_firstChernCap_eq_cycleFamilySum
    (s : MeromorphicSection.{u, z} D.site D.base D.base_mem X hX.1 L)
    (t : MeromorphicSection.{u, z} D.site D.base D.base_mem X hX.1 N)
    (ratioL : LocalRatioFamily.{u, z} D.site D.base D.base_mem hX.1 hdec L)
    (ratioN : LocalRatioFamily.{u, z} D.site D.base D.base_mem hX.1 hdec N)
    (hfinL : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratioL W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hfinN : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec N t ratioN W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hint : IsIntegralSpace D.site D.base D.base_mem X)
    (dim : DimensionFunction (points D.site D.base D.base_mem X))
    (hd : ∀ x, dim x = D.deltaValue sX x) {n : ℤ}
    (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hk : k + 1 = n - 1)
    (hZinj : Function.Injective Z)
    (hZfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X (Z i).subspace)
    (hdim : ∀ i, deltaDimension D sX
      ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) = k + 1)
    (hcoverL : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec L s ratioL W ≠ 0 →
        ∃ i, Z i = W)
    (hcoverN : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, z} D.site D.base D.base_mem hX.1 hdec N t ratioN W ≠ 0 →
        ∃ i, Z i = W) :
    firstChernCapCycleMap D sX hX k chernN
        (divisorCycleAt.{u, z} D sX hX hdec L s ratioL hfinL hint dim hd hn hk) -
      firstChernCapCycleMap D sX hX k chernL
        (divisorCycleAt.{u, z} D sX hX hdec N t ratioN hfinN hint dim hd hn hk) =
      cycleFamilySum D sX hX
        (fun i => (Z i).toIntegralClosedSubspace D.site D.base D.base_mem) hZfin k
        (keyFormulaDivisorTerm.{u, z} D sX hX hdec L N chernL chernN Z s t ratioL ratioN hdim)

AlgebraicGeometry.Space.keyFormula_of_ne_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaProof/Headline.lean, line 93.

Stacks 0EQV, the key formula, for nonzero meromorphic sections. In Situation 0EDT let X/B be good, X integral with dim_δ(X) = n, let ℒ, 𝒩 be invertible with nonzero meromorphic sections s, t, let {Z_i} be an injective locally finite family of prime divisors outside of which s generates ℒ_ξ and t generates 𝒩_ξ, and let s_i = v_i s_{ξ_i}, t_i = w_i t_{ξ_i} be generators of ℒ_{ξ_i}, 𝒩_{ξ_i} (v_i, w_i units of B_i = 𝒪^h_{X,ξ_i}; every generator is of this form for a unique unit), so that s = f_i s_i, t = g_i t_i with f_i, g_i ∈ Q(B_i)^*. Then

∑ (Z\_i → X)\_*(ord\_\{B\_i}(f\_i) div\_\{𝒩|\_\{Z\_i}}(t\_i|\_\{Z\_i}) - ord\_\{B\_i}(g\_i) div\_\{ℒ|\_\{Z\_i}}(s\_i|\_\{Z\_i})) = ∑ (Z\_i → X)\_* div(∂\_\{B\_i}(f\_i, g\_i))

in Z_{n-2}(X), with t_i|_{Z_i} the image of t_i in the fibre 𝒩_{ξ_i} ⊗ κ(ξ_i) read as a meromorphic section of 𝒩|_{Z_i}, and ∂_{B_i} the tame symbol of Stacks 0EAH read in κ(ξ_i)^* = R(Z_i)^*.

theorem keyFormula_of_ne_zero
    (s : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 L) (hs : s ≠ 0)
    (t : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 N) (ht : t ≠ 0)
    {n : ℤ} (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    {I : Type*} (Zf : I → PrimeDivisor D.site D.base D.base_mem X)
    (hfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X
        ((Zf i).toIntegralClosedSubspace D.site D.base D.base_mem).subspace)
    (hinj : Function.Injective Zf)
    (hcov : ∀ W : PrimeDivisor D.site D.base D.base_mem X, (∀ i, Zf i ≠ W) →
      W.SectionGenerates D sX hX hint L hL s ∧ W.SectionGenerates D sX hX hint N hN t)
    (v w : ∀ i, (↥((Zf i).genericHenselianLocalRing D.site D.base D.base_mem hX.1
      (isDecentSpace_of_isGood D sX hX)))ˣ) :
    keyFormulaDivisorSum D sX hX hint L hL s
        (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint L hL hs) N hN t
        (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint N hN ht)
        (deltaDimensionClosed_eq_sub_two_add_one_add_one D sX hX hn) Zf hfin v w =
      keyFormulaTameSum D sX hX hint L hL s
        (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint L hL hs) N hN t
        (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint N hN ht)
        (deltaDimensionClosed_eq_sub_two_add_one_add_one D sX hX hn) Zf hfin v w

Sources

• Section 82.20, Tag 0EQU
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.PrimeDivisor.genericResidueFieldEquivFunctionField · AlgebraicGeometry.Space.PrimeDivisor.henselianResidueFieldEquivFunctionField · AlgebraicGeometry.Space.PrimeDivisor.genericRestriction · AlgebraicGeometry.Space.firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum · AlgebraicGeometry.Space.firstChernCapCycleMap_divisorCycleAt_eq_componentDivisorSum_of_isInvertible · AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm · AlgebraicGeometry.Space.canonicalFirstChernCap_canonicalDivisorCycleAt · AlgebraicGeometry.Space.locallyFinite_primeDivisor_not_sectionGenerates

Section82.21 Intersecting with an invertible sheaf and rational equivalence#

Section 82.21 of the Stacks Project, Tag 0EQW. 3 tags here are carried by 6 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0EQX, a lemma

Tag 0EQX in the Stacks Project.

AlgebraicGeometry.Space.firstChernClassCap_comm_iff_isRationallyEquivalentToZero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernClassCapCommutativity.lean, line 188. The attribute's comment: “not the lemma itself but its equivalence form: the equality c_1(N) ∩ div_L(s) = c_1(L) ∩ div_N(t) in CH_{n-2}(X) is shown to hold if and only if the first cycle of the key formula 0EQV is rationally equivalent to zero, for an arbitrary component rule chern of 0EQJ. The lemma's own statement is PROVED ELSEWHERE, unconditionally and at the hypotheses of 0EQX: canonicalFirstChernCap_divisor_comm of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, at the canonical component rule and the canonical divisors, registered under 0EQX as well. The equivalence is kept for its generality in chern, at which the key formula is the open predicate KeyFormula”.

The conclusion of Stacks 0EQX, reduced to the key formula of Stacks 0EQV, for an arbitrary component rule. In Situation 0EDT let X/B be good and integral with dim_δ(X) = n, let ℒ and 𝒩 be invertible with nonzero meromorphic sections s and t, and set α = div_ℒ(s), β = div_𝒩(t). Then

c₁(𝒩) ∩ α = c₁(ℒ) ∩ β in CH_{n-2}(X)

holds if and only if the first cycle of the key formula of Section 0EQU is rationally equivalent to zero, where that cycle is ∑_i (Z_i → X)_*(ord_{B_i}(f_i) div_{𝒩|Z_i}(t_i|Z_i) - ord_{B_i}(g_i) div_{ℒ|Z_i}(s_i|Z_i)).

This is an equivalence, so it does not by itself prove the equality; what proves it is the key formula, which gives the right hand side. Stacks 0EQX itself is proved, unconditionally and at its own hypotheses, by AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm of StacksFrontier.AlgebraicGeometry.Spaces.Chow.KeyFormulaProof.Headline, which is where the tag's mathematical content lives. What this statement adds is the generality: the component cycle c₁(ℒ|_W) ∩ [W] of Stacks 0EQE is the explicit parameter chern of Stacks 0EQJ and the equivalence holds for every value of it, so it is the form that converts the open predicate AlgebraicGeometry.Space.KeyFormula — the key formula quantified over component rules and tame-symbol divisors — into the equality at that generality. CH_{n-2}(X) is the Chow group of Stacks 0EQ6 constructed in AlgebraicGeometry.Space.chowGroup.

theorem firstChernClassCap_comm_iff_isRationallyEquivalentToZero
    (hL : IsInvertibleSheaf D sX hX L) (hN : IsInvertibleSheaf D sX hX N)
    (s : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 L)
    (t : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 N)
    (ratioL : LocalRatioFamily.{u, u} D.site D.base D.base_mem hX.1 hdec L)
    (ratioN : LocalRatioFamily.{u, u} D.site D.base D.base_mem hX.1 hdec N)
    (hfinL : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdec L s ratioL W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hfinN : LocallyFinite fun W : PrimeDivisor D.site D.base D.base_mem X =>
      if divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdec N t ratioN W = 0 then ∅
      else W.support D.site D.base D.base_mem X)
    (hint : IsIntegralSpace D.site D.base D.base_mem X)
    (dim : DimensionFunction (points D.site D.base D.base_mem X))
    (hd : ∀ x, dim x = D.deltaValue sX x) {n : ℤ}
    (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt))
    (hk : k + 1 = n - 1)
    (hZinj : Function.Injective Z)
    (hZfin : LocallyFinite fun i : I =>
      subspacePointSet D.site D.base D.base_mem X (Z i).subspace)
    (hdim : ∀ i, deltaDimension D sX
      ((Z i).toIntegralClosedSubspace D.site D.base D.base_mem) = k + 1)
    (hcoverL : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdec L s ratioL W ≠ 0 →
        ∃ i, Z i = W)
    (hcoverN : ∀ W : PrimeDivisor D.site D.base D.base_mem X,
      divisorCoefficients.{u, u} D.site D.base D.base_mem hX.1 hdec N t ratioN W ≠ 0 →
        ∃ i, Z i = W) :
    firstChernClassCap D sX hX N k hN chernN (chowTarget.{v} D sX hX k)
          (divisorCycleAt.{u, u} D sX hX hdec L s ratioL hfinL hint dim hd hn hk) =
        firstChernClassCap D sX hX L k hL chernL (chowTarget.{v} D sX hX k)
          (divisorCycleAt.{u, u} D sX hX hdec N t ratioN hfinN hint dim hd hn hk) ↔
      IsRationallyEquivalentToZero.{v} D sX hX k
        (cycleFamilySum D sX hX
          (fun i => (Z i).toIntegralClosedSubspace D.site D.base D.base_mem) hZfin k
          (keyFormulaDivisorTerm.{u, u} D sX hX hdec L N chernL chernN Z s t ratioL ratioN
            hdim))

AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/KeyFormulaProof/Headline.lean, line 137. The attribute's comment: “Lemma 82.21.1 itself: for X good and integral with δ-dimension n, L and N invertible and s, t nonzero meromorphic sections, c_1(N) ∩ div_L(s) = c_1(L) ∩ div_N(t) in CH_{n-2}(X). Unconditional at the source's hypotheses: integrality and goodness are the section variables, regularity of s and t is derived from s ≠ 0, t ≠ 0 rather than assumed, and the cap product of 0EQJ, the divisors of 0EQE and the Chow group of 0EQ6 are the constructed ones, not carried”.

Section 0EQU, the key formula: c₁(𝒩) ∩ div_ℒ(s) = c₁(ℒ) ∩ div_𝒩(t) in CH_{n-2}(X). In Situation 0EDT let X/B be good, X integral with dim_δ(X) = n, let ℒ and 𝒩 be invertible 𝒪_X-modules, and let s, t be nonzero meromorphic sections of ℒ, 𝒩. Then c₁(𝒩) ∩ div_ℒ(s) = c₁(ℒ) ∩ div_𝒩(t) in the Chow group CH_{n-2}(X) of Stacks 0EQ6, the cap products being those of Stacks 0EQJ and the divisors those of Stacks 0EQE, all at the canonical choices.

The proof is the section's: the family {Z_i} of prime divisors where s or t fails to generate is locally finite (Stacks 0EPS), the two cap products are the sums over it of ord_{B_i}(f_i) (Z_i → X)_* div(t_i|_{Z_i}) and ord_{B_i}(g_i) (Z_i → X)_* div(s_i|_{Z_i}), and the key formula (Stacks 0EQV) makes their difference a locally finite sum of principal divisors of tame symbols on the Z_i.

theorem canonicalFirstChernCap_divisor_comm
    (s : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 L) (hs : s ≠ 0)
    (t : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 N) (ht : t ≠ 0)
    {n : ℤ} (hn : deltaDimensionClosed D sX hX Set.univ isClosed_univ = (n : EInt)) :
    canonicalFirstChernCap.{u, u + 1} D sX hX N hN (n - 2)
        (canonicalDivisorCycleAt D sX hX hint L hL s
          (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint L hL hs)
          (deltaDimensionClosed_eq_sub_two_add_one_add_one D sX hX hn)) =
      canonicalFirstChernCap.{u, u + 1} D sX hX L hL (n - 2)
        (canonicalDivisorCycleAt D sX hX hint N hN t
          (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint N hN ht)
          (deltaDimensionClosed_eq_sub_two_add_one_add_one D sX hX hn))

Tag 0EQY, a lemma

Tag 0EQY in the Stacks Project.

AlgebraicGeometry.Space.isRationallyEquivalentToZeroCanonical_firstChernCapCycleMap_canonicalFirstChernRule, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernCapRationalEquivalenceCanonicalSummand.lean, line 567. The attribute's comment: “the cap product with the first Chern class is compatible with rational equivalence, at the level of cycles: it carries a cycle rationally equivalent to zero to a cycle rationally equivalent to zero, both read at the canonical embeddings of Stacks 0ENK”.

Stacks 0EQY at the level of cycles. For X/B good and 𝒩 an invertible 𝒪_X-module, the cap product c₁(𝒩) ∩ - : Z_{k+1}(X) ⟶ Z_k(X) of Stacks 0EQJ carries a (k+1)-cycle rationally equivalent to zero to a k-cycle rationally equivalent to zero, both in the sense of part (2) of Stacks 0EQ6 read at the canonical embeddings R(W) → Q(𝒪^h_{W,ξ}) of Stacks 0ENK.

Nothing is assumed. The reassembly over the presenting family of the rational equivalence is isRationallyEquivalentToZeroCanonical_firstChernCapCycleMap_of_pushforwardDivisor, and the summand statement it asks for is the previous theorem.

theorem isRationallyEquivalentToZeroCanonical_firstChernCapCycleMap_canonicalFirstChernRule
    {α : CycleGroup D sX (k + 1)}
    (hα : IsRationallyEquivalentToZeroCanonical.{u + 1} D sX hX (k + 1) α) :
    IsRationallyEquivalentToZeroCanonical.{u + 1} D sX hX k
      (firstChernCapCycleMap D sX hX k (canonicalFirstChernRule D sX hX N hN k) α)

AlgebraicGeometry.Space.canonicalFirstChernCapCanonicalChowMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernCapRationalEquivalenceCanonicalSummand.lean, line 589. The attribute's comment: “the descended operation CH_{k+1}(X) -> CH_k(X), the Chow groups being those of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK on both sides”.

Stacks 0EQY. The cap product of Stacks 0EQJ descends to

c₁(𝒩) ∩ - : CH_{k+1}(X) ⟶ CH_k(X),

with CH_*(X) the Chow group AlgebraicGeometry.Space.chowGroupCanonical of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK on both sides.

The descent is AlgebraicGeometry.Space.chowGroupCanonicalLift and the hypothesis it needs is the previous theorem.

def canonicalFirstChernCapCanonicalChowMap :
    chowGroupCanonical.{u + 1} D sX hX (k + 1) →+ chowGroupCanonical.{u + 1} D sX hX k

AlgebraicGeometry.Space.canonicalFirstChernCapCanonicalChowMap_chowClassCanonical, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernCapRationalEquivalenceCanonicalSummand.lean, line 601. The attribute's comment: “the descended operation agrees with the cap product of Stacks 0EQJ on the class of a cycle, which is the factorization through rational equivalence”.

Stacks 0EQY. The descended operation agrees with the cap product of Stacks 0EQJ on the class of a cycle, which is what it means for the cap product to factor through rational equivalence.

theorem canonicalFirstChernCapCanonicalChowMap_chowClassCanonical
    (α : CycleGroup D sX (k + 1)) :
    canonicalFirstChernCapCanonicalChowMap D sX hX k N hN
        (chowClassCanonical.{u + 1} D sX hX (k + 1) α) =
      canonicalFirstChernCapCanonical.{u + 1} D sX hX N hN k α

Tag 0EQZ, a lemma

Tag 0EQZ in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_firstChernCapCycleMap_firstChernCapCycleMap_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/IteratedFirstChernCapCommutativity.lean, line 766. The attribute's comment: “c₁(ℒ) ∩ c₁(𝒩) ∩ α = c₁(𝒩) ∩ c₁(ℒ) ∩ α in CH_k(X), for every (k+2)-cycle α on X representing a class of CH_{k+2}(X); both cap products are the cycle-level operations of Stacks 0EQJ at the canonical choices, the classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical); nothing carried”.

Stacks 0EQZ: the two iterated cap products with first Chern classes commute. In Situation 0EDT let X/B be good and let ℒ, 𝒩 be invertible 𝒪_X-modules. For every (k + 2)-cycle α on X,

c₁(ℒ) ∩ c₁(𝒩) ∩ α = c₁(𝒩) ∩ c₁(ℒ) ∩ α in CH_k(X).

The cap products are those of Stacks 0EQJ at the canonical choices (AlgebraicGeometry.Space.canonicalFirstChernRule): the inner one is the cycle-level operation Z_{k+2}(X) → Z_{k+1}(X) defining c₁(-) ∩ - on representatives, and so is the outer one Z_{k+1}(X) → Z_k(X), the class being taken afterwards in the Chow group CH_k(X) of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical, whose rational equivalences are sums of principal divisors at the canonical embeddings of Stacks 0ENK. As every class of CH_{k+2}(X) is the class of a cycle, this is the identity of the source on representatives.

theorem chowClassCanonical_firstChernCapCycleMap_firstChernCapCycleMap_comm (k : ℤ)
    (α : CycleGroup D sX (k + 1 + 1)) :
    chowClassCanonical.{max (u + 1) v} D sX hX k
        (firstChernCapCycleMap D sX hX k (canonicalFirstChernRule D sX hX L hL k)
          (firstChernCapCycleMap D sX hX (k + 1) (canonicalFirstChernRule D sX hX N hN (k + 1))
            α)) =
      chowClassCanonical.{max (u + 1) v} D sX hX k
        (firstChernCapCycleMap D sX hX k (canonicalFirstChernRule D sX hX N hN k)
          (firstChernCapCycleMap D sX hX (k + 1) (canonicalFirstChernRule D sX hX L hL (k + 1))
            α))

Sources

• Section 82.21, Tag 0EQW
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.firstChernClassCap_comm_iff_isRationallyEquivalentToZero · AlgebraicGeometry.Space.canonicalFirstChernCap_divisor_comm · AlgebraicGeometry.Space.isRationallyEquivalentToZeroCanonical_firstChernCapCycleMap_canonicalFirstChernRule · AlgebraicGeometry.Space.canonicalFirstChernCapCanonicalChowMap · AlgebraicGeometry.Space.canonicalFirstChernCapCanonicalChowMap_chowClassCanonical · AlgebraicGeometry.Space.chowClassCanonical_firstChernCapCycleMap_firstChernCapCycleMap_comm

Section82.22 Intersecting with effective Cartier divisors#

Section 82.22 of the Stacks Project, Tag 0ER0. 7 tags here are carried by 23 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ER1, a definition

Tag 0ER1 in the Stacks Project.

AlgebraicGeometry.Space.refinedGysin, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierIntersection.lean, line 592. The attribute's comment: “the refined Gysin homomorphism i^* : Z_{k+1}(X) → CH_k(D) of the vanishing locus D = Z(s), parts (1)(a), (1)(b) and (2); the Chow group of D and the two component cycles [D ∩ W]_k and c_1(L|_W) ∩ [W] are explicit parameters, instantiated by Stacks 0EQ6 parts (2)-(4) (chowTarget), Stacks 0EE9 (closedSubspaceCycle) and Stacks 0EQJ (canonicalFirstChernRule)”.

Stacks 0ER1, parts (1) and (2). The refined Gysin homomorphism i^* : Z_{k+1}(X) ⟶ CH_k(E) of the vanishing locus E = Z(s) of a global section s of an invertible sheaf ℒ on a good algebraic space X/B, with i : E → X the closed immersion.

On the cycle of an integral closed subspace W ⊆ X of δ-dimension k + 1 it is the class of [E ∩ W]_k when W ⊄ E, which is part (1)(a), and the class of i'_*(c₁(ℒ|_W) ∩ [W]) when W ⊆ E, which is part (1)(b); see AlgebraicGeometry.Space.refinedGysin_integralCycle_of_not_le and AlgebraicGeometry.Space.refinedGysin_integralCycle_of_le. Part (2), the extension of that rule over an arbitrary (k+1)-cycle, is AlgebraicGeometry.Space.gysinCycleMap, which is proved to land in Z_k(E) and to be additive.

The Chow group CH_k(E) and the two component cycles [E ∩ W]_k and c₁(ℒ|_W) ∩ [W] are explicit parameters, standing in for Stacks 0EQ6 parts (2)–(4), Stacks 0EE9 and Stacks 0EQJ, none of which is available for algebraic spaces in this repository; see the implementation notes of StacksFrontier.AlgebraicGeometry.Spaces.Chow.EffectiveCartierIntersection.

def refinedGysin (_hL : IsInvertibleSheaf D p hp L)
    (rule : GysinComponentRule D p (vanishingLocus D p hp L s) k)
    {CH : Type w} [AddCommGroup CH]
    (chow : ChowTarget.{u, w, v} D ((vanishingLocus D p hp L s).1.ι ≫ p) hE k CH) :
    CycleGroup D p (k + 1) →+ CH

AlgebraicGeometry.Space.effectiveCartierIntersection, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierIntersection.lean, line 646. The attribute's comment: “part (3): the operation D · α = i_ i^ α of an effective Cartier divisor D = Z(s), with the Chow group of X and the component cycles [D ∩ W]_k and c_1(L|_W) ∩ [W] taken as explicit parameters, instantiated by Stacks 0EQ6 parts (2)-(4) (chowTarget), Stacks 0EE9 (closedSubspaceCycle) and Stacks 0EQJ (canonicalFirstChernRule); the pushforward i_* is applied to the cycle-level Gysin map, the source applying it to the class”.

Stacks 0ER1, part (3). Intersecting with an effective Cartier divisor: when E = Z(s) is an effective Cartier divisor on X one writes E · α = i_* i^* α for the pushforward along the closed immersion i : E → X of the refined Gysin class of α.

The pushforward is the honest pushforward of cycles along a closed immersion, AlgebraicGeometry.Space.cycleGroupClosedPushforward, applied to the cycle-level Gysin map; the class map of a Chow group for X is applied afterwards. The source applies i_* to a Chow class, which needs proper pushforward to respect rational equivalence; the form below is that statement at the level of cycles, and is the same once Stacks 0EQ6 parts (2)–(4) land.

def effectiveCartierIntersection (_hL : IsInvertibleSheaf D p hp L)
    (_hCartier : IsEffectiveCartierDivisor D.site D.base D.base_mem ⟨X, hp.1⟩
      (vanishingLocus D p hp L s))
    (rule : GysinComponentRule D p (vanishingLocus D p hp L s) k)
    {CH : Type w} [AddCommGroup CH]
    (chow : ChowTarget.{u, w, v} D p hp k CH) :
    CycleGroup D p (k + 1) →+ CH

AlgebraicGeometry.Space.sectionRefinedGysin, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RefinedGysinOfSection.lean, line 249. The attribute's comment: “parts (1)(a), (1)(b) and (2): the refined Gysin homomorphism i^* : Z_{k+1}(X) → CH_k(D) of the vanishing locus D = Z(s) of a global section of an invertible sheaf, with values in CH_k(D) = Z_k(D)/∼_rat of Stacks 0EQ6, clause (1)(a) the cycle [D ∩ W]_k of Stacks 0EE9 of the scheme-theoretic intersection and clause (1)(b) the pushforward of div_{L|_W}(t_W) for the regular meromorphic section of Stacks 0EPQ, every ingredient constructed”.

The refined Gysin homomorphism i^* : Z_{k+1}(X) ⟶ CH_k(E) of the vanishing locus E = Z(s) of a global section s of an invertible sheaf ℒ on a good algebraic space X/B, i : E → X the closed immersion, with values in the Chow group CH_k(E) = Z_k(E)/∼_rat.

On the cycle [W] of an integral closed subspace W ⊆ X of δ-dimension k + 1 it is the class of [E ∩ W]_k when W ⊄ E (AlgebraicGeometry.Space.sectionRefinedGysin_integralCycle_of_not_le) and the class of i'_*(c₁(ℒ|_W) ∩ [W]) when W ⊆ E (AlgebraicGeometry.Space.sectionRefinedGysin_integralCycle_of_le); on a general cycle α = ∑ n_W [W] it is the class of ∑ n_W i^*[W] (AlgebraicGeometry.Space.sectionRefinedGysin_apply).

def sectionRefinedGysin (hL : IsInvertibleSheaf D p hp L) :
    CycleGroup D p (k + 1) →+
      chowGroup.{v} D ((vanishingLocus D p hp L s).1.ι ≫ p)
        (isGood_vanishingLocus D p hp L s) k

AlgebraicGeometry.Space.sectionCartierIntersection, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RefinedGysinOfSection.lean, line 313. The attribute's comment: “part (3): the operation D · α = i_ i^ α of an effective Cartier divisor D = Z(s), with values in CH_k(X) = Z_k(X)/∼_rat of Stacks 0EQ6, i^ the refined Gysin homomorphism of parts (1) and (2) with every ingredient constructed, and i_ the pushforward along the closed immersion i : D → X of the cycle representing i^* α”.

Intersecting with an effective Cartier divisor: when E = Z(s) is an effective Cartier divisor on X, the operation E · α = i_* i^* α with values in the Chow group CH_k(X), i_* being the pushforward along the closed immersion i : E → X of the cycle ∑ n_W i^*[W] representing i^* α.

def sectionCartierIntersection (hL : IsInvertibleSheaf D p hp L)
    (hCartier : IsEffectiveCartierDivisor D.site D.base D.base_mem ⟨X, hp.1⟩
      (vanishingLocus D p hp L s)) :
    CycleGroup D p (k + 1) →+ chowGroup.{v} D p hp k

Tag 0ER2, a remark

Tag 0ER2 in the Stacks Project.

AlgebraicGeometry.Space.canonicalTrivialRestrictionGysin, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/TrivialRestrictionGysin.lean, line 483. The attribute's comment: “the canonical map i^ : Z_{k+1}(X) → Z_k(D) on cycles of a pair (L, s) of Definition 0ER1 with D = Z(s), constructed with clause (1)(a) of Stacks 0ER1 given by the cycle [D ∩ W]_k of Stacks 0EE9 of the scheme-theoretic intersection D ∩ W = i^{-1}(W) and clause (1)(b) by the zero cycle, which is the requirement i^[W] = 0 for W ⊆ D of this remark; both values are then proved of the map, in canonicalTrivialRestrictionGysin_integralCycle_of_not_le and canonicalTrivialRestrictionGysin_integralCycle_of_le, and determine it, in gysinCycleMap_eq_canonicalTrivialRestrictionGysin”.

For a pair (ℒ, s) as in Stacks 0ER1, with E = Z(s) the vanishing locus of s and i : E → X its closed immersion, the canonical map i^* : Z_{k+1}(X) ⟶ Z_k(E) on cycles: the additive extension of the rule which sends the cycle of an integral closed subspace W ⊆ X of δ-dimension k + 1 to the cycle [E ∩ W]_k of the scheme-theoretic intersection when W ⊄ E, as Stacks 0ER1(1)(a) prescribes, and to 0 when W ⊆ E, which is the requirement of Stacks 0ER2.

def canonicalTrivialRestrictionGysin :
    CycleGroup D p (k + 1) →+ CycleGroup D ((vanishingLocus D p hp L s).1.ι ≫ p) k

AlgebraicGeometry.Space.canonicalTrivialRestrictionGysin_integralCycle_of_le, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/TrivialRestrictionGysin.lean, line 493. The attribute's comment: “the defining requirement i^*[W] = 0 for W ⊆ D, proved of the canonical map on cycles of the pair of Definition 0ER1”.

The defining requirement for the pair of Stacks 0ER1: the canonical map on cycles vanishes on the cycle of an integral closed subspace contained in E.

theorem canonicalTrivialRestrictionGysin_integralCycle_of_le
    {W : IntegralClosedSubspace D.site D.base D.base_mem X}
    (hW : deltaDimension D p W = k + 1)
    (hWE : W.subspace ≤ (vanishingLocus D p hp L s).1) :
    canonicalTrivialRestrictionGysin D p hp L s k (integralCycle D p (k + 1) W hW) = 0

AlgebraicGeometry.Space.canonicalTrivialRestrictionGysin_integralCycle_of_not_le, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/TrivialRestrictionGysin.lean, line 507. The attribute's comment: “the remaining value i^*[W] = [D ∩ W]_k of Stacks 0ER1(1)(a), proved of the canonical map on cycles, the right-hand side being the cycle of Stacks 0EE9 of the scheme-theoretic intersection”.

On the cycle of an integral closed subspace not contained in E the canonical map is the cycle [E ∩ W]_k of Stacks 0EE9 of the scheme-theoretic intersection, as Stacks 0ER1(1)(a) prescribes.

theorem canonicalTrivialRestrictionGysin_integralCycle_of_not_le
    {W : IntegralClosedSubspace D.site D.base D.base_mem X}
    (hW : deltaDimension D p W = k + 1)
    (hWE : ¬ W.subspace ≤ (vanishingLocus D p hp L s).1) :
    canonicalTrivialRestrictionGysin D p hp L s k (integralCycle D p (k + 1) W hW) =
      gysinIntersectionCycle D p hp (vanishingLocus D p hp L s)
        (isGood_vanishingLocus D p hp L s) k W hW hWE

AlgebraicGeometry.Space.gysinCycleMap_eq_canonicalTrivialRestrictionGysin, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/TrivialRestrictionGysin.lean, line 523. The attribute's comment: “canonicity of the canonical map on cycles of the pair of Definition 0ER1: the requirement i^*[W] = 0 for W ⊆ D, together with the value [D ∩ W]_k of Stacks 0ER1(1)(a) on the remaining integral closed subspaces, determines the map on cycles”.

Canonicity for the pair of Stacks 0ER1: a component rule whose cycle map takes the two prescribed values on the cycle of every integral closed subspace has the canonical map of Stacks 0ER2 as its cycle map.

theorem gysinCycleMap_eq_canonicalTrivialRestrictionGysin
    (rule' : GysinComponentRule D p (vanishingLocus D p hp L s) k)
    (hle : ∀ (W : IntegralClosedSubspace D.site D.base D.base_mem X)
      (hW : deltaDimension D p W = k + 1), W.subspace ≤ (vanishingLocus D p hp L s).1 →
      gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k rule'
        (integralCycle D p (k + 1) W hW) = 0)
    (hnotle : ∀ (W : IntegralClosedSubspace D.site D.base D.base_mem X)
      (hW : deltaDimension D p W = k + 1)
      (hWE : ¬ W.subspace ≤ (vanishingLocus D p hp L s).1),
      gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k rule'
          (integralCycle D p (k + 1) W hW) =
        gysinIntersectionCycle D p hp (vanishingLocus D p hp L s)
          (isGood_vanishingLocus D p hp L s) k W hW hWE) :
    gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k rule' =
      canonicalTrivialRestrictionGysin D p hp L s k

Tag 0ER3, a remark

Tag 0ER3 in the Stacks Project.

AlgebraicGeometry.Space.pullbackSheaf, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierPullback.lean, line 77. The attribute's comment: “the sheaf L' = f^*L of the construction, as the inverse image of Stacks 03LU”.

The inverse image ℒ' = f^*ℒ along f of a sheaf of modules ℒ on X.

abbrev pullbackSheaf : Modules.{u, u} D.site D.base D.base_mem X' hp'.1

AlgebraicGeometry.Space.pullbackGlobalSection, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierPullback.lean, line 83. The attribute's comment: “the section s' = f^*s of the construction, as the inverse image of a global section along f”.

The inverse image s' = f^*s along f of a global section s of ℒ.

abbrev pullbackGlobalSection : (pullbackSheaf D p hp p' hp' f L).sections

AlgebraicGeometry.Space.pullbackDivisor, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierPullback.lean, line 105. The attribute's comment: “the closed subspace D' = X' ×_X D of X', taken as the inverse image of D along f in the sense of Stacks 083Q; its identification with Z(s') is AlgebraicGeometry.Space.pullbackDivisor_eq_vanishingLocus”.

The closed subspace E' = X' ×_X E of X': the inverse image along f of the vanishing locus E = Z(s).

def pullbackDivisor : ClosedSubspaces D.site D.base D.base_mem X'

AlgebraicGeometry.Space.pullbackDivisor_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierPullback.lean, line 119. The attribute's comment: “commutativity of the diagram, g ≫ i = i' ≫ f”.

The diagram commutes: g ≫ i = i' ≫ f, for i' : E' → X' the inclusion of the closed subspace E' and i : E → X the inclusion of the vanishing locus.

theorem pullbackDivisor_comm :
    pullbackDivisorProjection D p hp p' hp' f L s ≫ (vanishingLocus D p hp L s).1.ι =
      (pullbackDivisor D p hp p' hp' f L s).1.ι ≫ f.hom

AlgebraicGeometry.Space.pullbackDivisor_isPullback, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierPullback.lean, line 128. The attribute's comment: “the identification D' = X' ×_X D, as the cartesian square with legs g : D' → D and i' : D' → X'”.

The square with legs g : E' → E and i' : E' → X' is cartesian, which is the identification E' = X' ×_X E.

theorem pullbackDivisor_isPullback :
    IsPullback (pullbackDivisorProjection D p hp p' hp' f L s)
      (pullbackDivisor D p hp p' hp' f L s).1.ι (vanishingLocus D p hp L s).1.ι f.hom

AlgebraicGeometry.Space.pullbackDivisor_eq_vanishingLocus, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/EffectiveCartierPullback.lean, line 136. The attribute's comment: “the identification D' = Z(s'), the vanishing locus of the pulled-back section”.

The identification E' = Z(s'): for an invertible ℒ, the inverse image along f of the vanishing locus of s is the vanishing locus of the pulled-back section s' = f^*s.

theorem pullbackDivisor_eq_vanishingLocus (hL : IsInvertibleSheaf D p hp L) :
    pullbackDivisor D p hp p' hp' f L s =
      vanishingLocus D p' hp' (pullbackSheaf D p hp p' hp' f L)
        (pullbackGlobalSection D p hp p' hp' f L s)

Tag 0ER4, a lemma

Tag 0ER4 in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_closedSubspaceCyclePushforward_gysinCycleMap_sectionGysinRule, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RefinedGysinPushforwardCap.lean, line 312. The attribute's comment: “i_ i^ α = c₁(L) ∩ α in CH_k(X) for every (k+1)-cycle α, for X/B good, L invertible, s a global section and i : D = Z(s) → X: i^ α is the cycle of the refined Gysin map of Stacks 0ER1 (gysinCycleMap at sectionGysinRule), i_ the pushforward of cycles along the closed immersion, c₁(L) ∩ α the cycle of Stacks 0EQJ at the canonical choices, both classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical); nothing carried”.

Stacks 0ER4. In Situation 0EDT let X/B be good, let ℒ be an invertible 𝒪_X-module with a global section s, and let i : E → X be the closed immersion of the vanishing locus E = Z(s). For every (k+1)-cycle α on X

i_* i^* α = c₁(ℒ) ∩ α in CH_k(X).

Here i^* α is represented by the cycle ∑ n_W i^*[W] of the refined Gysin map of Stacks 0ER1, AlgebraicGeometry.Space.gysinCycleMap at the rule AlgebraicGeometry.Space.sectionGysinRule; i_* is the pushforward of cycles along the closed immersion; c₁(ℒ) ∩ α is represented by the cycle of Stacks 0EQJ, AlgebraicGeometry.Space.firstChernCapCycleMap at the canonical component rule; and both classes are taken in CH_k(X) of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical, whose rational equivalences are sums of principal divisors at the canonical embeddings of Stacks 0ENK. As every class of CH_{k+1}(X) is the class of a cycle, this is the identity of the source on representatives.

The difference of the two sides is, by AlgebraicGeometry.Space.closedSubspaceCyclePushforward_gysinCycleMap_sub_firstChernCapCycleMap and AlgebraicGeometry.Space.firstChernCapCycleMap_eq_cycleFamilySum, the locally finite sum of the pushforwards of the cycles n_W ([W ×_X Z(s)]_k - c₁(ℒ|_W) ∩ [W]) for W ⊄ Z(s) and n_W (c₁(ℒ|_W) ∩ [W] - c₁(ℒ|_W) ∩' [W]) for all W, the two cap products being formed with the two regular meromorphic sections of ℒ|_W the two constructions choose. By Stacks 0EQF on W (AlgebraicGeometry.Space.closedSubspaceCycle_preimage_vanishingLocus_eq_canonicalDivisorCycleAt) each of these is an integer multiple of the difference of the Weil divisors of two regular meromorphic sections of ℒ|_W, hence a principal divisor at the canonical embeddings (AlgebraicGeometry.Space.isPrincipalDivisorCycleOfGood_zsmul_canonicalDivisorCycleAt_sub).

theorem chowClassCanonical_closedSubspaceCyclePushforward_gysinCycleMap_sectionGysinRule (k : ℤ)
    (α : CycleGroup D p (k + 1)) :
    chowClassCanonical.{max (u + 1) v} D p hp k (closedSubspaceCyclePushforward D p hp
        (vanishingLocus D p hp L s) k (gysinCycleMap D p (vanishingLocus D p hp L s)
          (isGood_vanishingLocus D p hp L s) k (sectionGysinRule D p hp L s k hL) α)) =
      chowClassCanonical.{max (u + 1) v} D p hp k (firstChernCapCycleMap D p hp k
        (canonicalFirstChernRule D p hp L
          (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _ hL) k) α)

AlgebraicGeometry.Space.chowClassCanonical_closedSubspaceCyclePushforward_gysinCycleMap_canonicalSection, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RefinedGysinPushforwardCap.lean, line 459. The attribute's comment: “in particular: D · α = c₁(O_X(D)) ∩ α in CH_k(X) for an effective Cartier divisor D ⊆ X, with D · α = i_ i^ α of Stacks 0ER1 part (3) formed on cycles for the pair (O_X(D), 1_D) of Stacks 0847, whose vanishing locus is D, c₁(O_X(D)) ∩ α the cycle of Stacks 0EQJ at the canonical choices, and the classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical)”.

Stacks 0ER4, in particular. In Situation 0EDT let X/B be good and let D ⊆ X be an effective Cartier divisor, with invertible sheaf 𝒪_X(D) and canonical section 1_D (Stacks 0841), whose vanishing locus Z(1_D) is D (Stacks 0847). For every (k+1)-cycle α on X

D · α = c₁(𝒪_X(D)) ∩ α in CH_k(X),

where D · α = i_* i^* α is the operation of Stacks 0ER1, part (3), for the pair (𝒪_X(D), 1_D): i^* α is represented by the cycle of the refined Gysin map, AlgebraicGeometry.Space.gysinCycleMap at AlgebraicGeometry.Space.sectionGysinRule, and i_* is the pushforward of cycles along i : Z(1_D) → X. The cap product c₁(𝒪_X(D)) ∩ α is represented by the cycle of Stacks 0EQJ at the canonical choices, and both classes are taken in CH_k(X) of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical. This is chowClassCanonical_closedSubspaceCyclePushforward_gysinCycleMap_sectionGysinRule for that pair.

theorem chowClassCanonical_closedSubspaceCyclePushforward_gysinCycleMap_canonicalSection
    (Dv : ClosedSubspaces D.site D.base D.base_mem X)
    (hDv : IsEffectiveCartierDivisor D.site D.base D.base_mem ⟨X, hp.1⟩ Dv) (k : ℤ)
    (α : CycleGroup D p (k + 1)) :
    chowClassCanonical.{max (u + 1) v} D p hp k (closedSubspaceCyclePushforward D p hp
        (vanishingLocus D p hp
          (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv)
          (canonicalSection D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv)) k
        (gysinCycleMap D p (vanishingLocus D p hp
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv)
            (canonicalSection D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv))
          (isGood_vanishingLocus D p hp _ _) k
          (sectionGysinRule D p hp
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv)
            (canonicalSection D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv) k
            (associatedInvertibleSheaf_isInvertible D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv))
          α)) =
      chowClassCanonical.{max (u + 1) v} D p hp k (firstChernCapCycleMap D p hp k
        (canonicalFirstChernRule D p hp
          (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv)
          (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _
            (associatedInvertibleSheaf_isInvertible D.site D.base D.base_mem ⟨X, hp.1⟩ Dv hDv))
          k) α)

Tag 0ER5, a lemma

Tag 0ER5 in the Stacks Project.

AlgebraicGeometry.Space.gysinCycleMap_properPushforward_eq_of_comparison, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinCompatibility.lean, line 592. The attribute's comment: “i^f_α' = g_(i')^α' at the level of cycles, on the single carried input GysinProperPushforwardComparison: the source's prime-cycle case analysis read coefficientwise at the two component parameters of Definition 0ER1”.

Stacks 0ER5. In Situation 0EDT let f : X' ⟶ X be a proper morphism of good algebraic spaces over B, let (ℒ, s, i : E → X) be as in Definition 0ER1, and form the diagram of Remark 0ER3, with E' = E ×_X X', g : E' ⟶ E and i' : E' ⟶ X'. Then for every (k+1)-cycle α' on X'

i^* f_* α' = g_* (i')^* α',

an identity of k-cycles on E, hence of classes in CH_k(E).

hsq is the cartesian square of Remark 0ER3, which makes E' the scheme-theoretic base change E ×_X X'. Only quasi-compactness of f and of g is used, through Stacks 0ENZ; properness, which the source assumes and from which it gets g proper by base change, is not needed for the pushforwards themselves.

The single hypothesis beyond the setup is cmp, the comparison datum AlgebraicGeometry.Space.GysinProperPushforwardComparison, whose docstring explains why it does not follow from the other data. Everything else — in particular the passage from the prime cycles of the source's case analysis to an arbitrary cycle, which is not formal because a cycle is an infinite formal sum — is proved.

theorem gysinCycleMap_properPushforward_eq_of_comparison
    (cmp : GysinProperPushforwardComparison D p hp p' hp' f E hE E' hE' g k rule rule')
    (hsq : IsPullback g E'.1.ι E.1.ι f) (α' : CycleGroup D p' (k + 1)) :
    gysinCycleMap D p E hE k rule (properCyclePushforward D p' hp' p hp hfbase hfqc (k + 1) α') =
      properCyclePushforward D (E'.1.ι ≫ p') hE' (E.1.ι ≫ p) hE hgbase hgqc k
        (gysinCycleMap D p' E' hE' k rule' α')

AlgebraicGeometry.Space.sectionGysin_properPushforward_eq_chowGroupCanonicalProperPushforward, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinProperPushforwardCompatibility.lean, line 124. The attribute's comment: “i^ f_ α' = g_ (i')^ α' in CH_k(D) for f : X' → X proper between good algebraic spaces over B, (L, s, i : D → X) as in 0ER1 and the diagram of 0ER3 (L' = f^L, s' = f^s, D' = Z(s') = X' ×_X D): both Gysin maps are 0ER1's at the constructed rules sectionGysinRule, CH_k = chowGroupCanonical of 0EQ6, f_ the proper pushforward of cycles of 0EP1 and g_ the proper pushforward on Chow groups of 0EQC; unconditional”.

In Situation 0EDT let f : X' ⟶ X be a proper morphism of good algebraic spaces over B. Let (ℒ, s, i : D → X) be as in Definition 0ER1: ℒ an invertible 𝒪_X-module, s ∈ Γ(X, ℒ), D = Z(s) its vanishing locus (here E, the letter D being the dimension situation) and i the closed immersion. Form the diagram of Remark 0ER3

E' --i'--> X'
|          |
g          f
|          |
v          v
E  --i-->  X

with ℒ' = f^*ℒ, s' = f^*s and E' = Z(s') = X' ×_X E (AlgebraicGeometry.Space.pullbackDivisor_eq_vanishingLocus). Then for every (k+1)-cycle α' on X'

i^* f_* α' = g_* (i')^* α' in CH_k(E),

where i^* and (i')^* are the refined Gysin homomorphisms Z_{k+1}(·) → CH_k(·) of Stacks 0ER1 of the two triples, at their constructed component rules (AlgebraicGeometry.Space.sectionGysinRule: clause (1)(a) the cycle [E ∩ W]_k of Stacks 0EE9, clause (1)(b) c₁(ℒ|_W) ∩ [W]), f_* is proper pushforward of cycles (Stacks 0EP1), the Chow groups are those of Stacks 0EQ6 (AlgebraicGeometry.Space.chowGroupCanonical, div(f) read at the canonical embeddings of Stacks 0ENK) and g_* is proper pushforward on Chow groups, Stacks 0EQC (AlgebraicGeometry.Space.chowGroupCanonicalProperPushforward).

theorem sectionGysin_properPushforward_eq_chowGroupCanonicalProperPushforward
    (α' : CycleGroup D p' (k + 1)) :
    chowClassCanonical.{u + 1} D ((vanishingLocus D p hp L s).1.ι ≫ p)
        (isGood_vanishingLocus D p hp L s) k
        (gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k
          (sectionGysinRule D p hp L s k hL)
          (properCyclePushforward D p' hp' p hp hfbase hf.2.1.2 (k + 1) α')) =
      chowGroupCanonicalProperPushforward.{u + 1} D
        ((pullbackSectionVanishingLocus D p hp p' hp' f L s).1.ι ≫ p')
        (isGood_vanishingLocus D p' hp' _ _)
        ((vanishingLocus D p hp L s).1.ι ≫ p) (isGood_vanishingLocus D p hp L s)
        (pullbackVanishingLocusProjection_comp_structure D p hp p' hp' f L s hL hfbase)
        (isProper_pullbackVanishingLocusProjection D p hp p' hp' f L s hL hf) k
        (chowClassCanonical.{u + 1} D
          ((pullbackSectionVanishingLocus D p hp p' hp' f L s).1.ι ≫ p')
          (isGood_vanishingLocus D p' hp' _ _) k
          (gysinCycleMap D p' (pullbackSectionVanishingLocus D p hp p' hp' f L s)
            (isGood_vanishingLocus D p' hp' _ _) k
            (sectionGysinRule D p' hp' (pullbackSheaf D p hp p' hp' f L)
              (pullbackGlobalSection D p hp p' hp' f L s) k
              (isInvertibleSheaf_pullbackSheaf D p hp p' hp' f L hL)) α'))

AlgebraicGeometry.Space.gysinCycleMap_properPushforward_eq_of_genericPointData, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinProperPushforwardGenericPoints.lean, line 532. The attribute's comment: “i^f_α' = g_(i')^α' in Z_k(E) at the canonical component rule, with the W' ⊄ E' branch PROVED from the generic-point dictionary (the degree count, from Stacks 02M0); only the W' ⊆ E' branch, the chern comparison of Stacks 0EQS(2), is carried”.

Stacks 0ER5, with the non-contained branch of the comparison proved rather than carried. In Situation 0EDT let f : X' ⟶ X be a quasi-compact morphism of good algebraic spaces over B, let E ⊆ X be a closed subspace with closed immersion i : E → X, and form the diagram of Remark 0ER3, with E' a closed subspace of X' whose point set is |f|^{-1}|E|, sitting in a cartesian square with g : E' ⟶ E and i' : E' ⟶ X'. Then for every (k+1)-cycle α' on X'

i^* f_* α' = g_* (i')^* α',

an identity of k-cycles on E, hence of classes in CH_k(E), for the refined Gysin maps of Stacks 0ER1 built on the canonical component rule of AlgebraicGeometry.Space.canonicalGysinComponentRule.

Two things are carried. hdata, the local dictionary at the generic points of the pairs with W' ⊄ E' — from which the degree count of the source's first case is proved, by AlgebraicGeometry.Space.properCycleDegree_mul_gysinIntersectionMult_eq_finsum. And hchern, the source's second case, which compares the cap products of Stacks 0EQJ on X and on X': clause (1)(b) of Stacks 0ER1 is the free parameter AlgebraicGeometry.Space.GysinComponentRule.chernCycle, so chern and chern' are unrelated data and no relation between them is derivable here. See the implementation notes of StacksFrontier.AlgebraicGeometry.Spaces.Chow.GysinProperPushforwardGenericPoints.

theorem gysinCycleMap_properPushforward_eq_of_genericPointData
    (hfbase : f ≫ p = p') (hfqc : IsQuasiCompact D.site D.base D.base_mem f)
    (hgbase : g ≫ (E.1.ι ≫ p) = E'.1.ι ≫ p')
    (hgqc : IsQuasiCompact D.site D.base D.base_mem g)
    (hsq : IsPullback g E'.1.ι E.1.ι f)
    (hpt : subspacePointSet D.site D.base D.base_mem X' E'.1 =
      pointsMap D.site D.base D.base_mem X' f ⁻¹'
        subspacePointSet D.site D.base D.base_mem X E.1)
    (chern : FirstChernComponentCycleRule D p k)
    (chern' : FirstChernComponentCycleRule D p' k)
    (hdata : ∀ (W' : IntegralClosedSubspace D.site D.base D.base_mem X')
        (T : IntegralClosedSubspace D.site D.base D.base_mem E.1.toFunctor),
      ¬ W'.subspace ≤ E'.1 →
        Nonempty (GysinProperPushforwardGenericPointDatum.{u, v} D p hp p' hp' f E hE E' hE' g k
            W' T))
    (hchern : ∀ (W' : IntegralClosedSubspace D.site D.base D.base_mem X'),
      deltaDimension D p' W' = k + 1 → W'.subspace ≤ E'.1 →
      ∀ T : IntegralClosedSubspace D.site D.base D.base_mem E.1.toFunctor,
        (∑ᶠ W : IntegralClosedSubspace D.site D.base D.base_mem X,
            properCycleMult D p' hp' p hp f W' W *
              (gysinComponentCycle D p E hE k
                (canonicalGysinComponentRule D p hp E hE k chern) W).1 T) =
          ∑ᶠ T' : IntegralClosedSubspace D.site D.base D.base_mem E'.1.toFunctor,
            (gysinComponentCycle D p' E' hE' k
                (canonicalGysinComponentRule D p' hp' E' hE' k chern') W').1 T' *
              properCycleMult D (E'.1.ι ≫ p') hE' (E.1.ι ≫ p) hE g T' T)
    (α' : CycleGroup D p' (k + 1)) :
    gysinCycleMap D p E hE k (canonicalGysinComponentRule D p hp E hE k chern)
        (properCyclePushforward D p' hp' p hp hfbase hfqc (k + 1) α') =
      properCyclePushforward D (E'.1.ι ≫ p') hE' (E.1.ι ≫ p) hE hgbase hgqc k
        (gysinCycleMap D p' E' hE' k
          (canonicalGysinComponentRule D p' hp' E' hE' k chern') α')

Tag 0ER6, a lemma

Tag 0ER6 in the Stacks Project.

AlgebraicGeometry.Space.gysinCycleMap_flatPullbackMap_eq_of_comparison, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinCompatibility.lean, line 950. The attribute's comment: “(i')^f^α = g^i^α at the level of cycles, on the single carried input GysinFlatPullbackComparison: the source's prime-cycle case analysis read coefficientwise at the two component parameters of Definition 0ER1”.

Stacks 0ER6. In Situation 0EDT let f : X' ⟶ X be a flat morphism of relative dimension r of good algebraic spaces over B, let (ℒ, s, i : E → X) be as in Definition 0ER1, and form the diagram of Remark 0ER3, with E' = E ×_X X', g : E' ⟶ E and i' : E' ⟶ X'. Then for every (k+1)-cycle α on X

(i')^* f^* α = g^* i^* α,

an identity of (k+r)-cycles on E', hence of classes in CH_{k+r}(E').

hsq is the cartesian square of Remark 0ER3, which makes E' the scheme-theoretic base change E ×_X X'. Flatness of g and its relative dimension r are hypotheses: the source gets them from f by base change along i. hkr is the degree bookkeeping (k+1) + r = (k+r) + 1, which is the convention of AlgebraicGeometry.Space.flatPullbackMap.

The single hypothesis beyond the setup is cmp, the comparison datum AlgebraicGeometry.Space.GysinFlatPullbackComparison, whose docstring explains why it does not follow from the other data. Everything else — in particular the passage from the prime cycles of the source's case analysis to an arbitrary cycle, which is not formal because a cycle is an infinite formal sum — is proved.

theorem gysinCycleMap_flatPullbackMap_eq_of_comparison
    (hkr : k + 1 + (r : ℤ) = k + (r : ℤ) + 1)
    (cmp : GysinFlatPullbackComparison D p hp p' hp' f hfbase r hflat hr E hE E' hE' g hgbase
      hgflat hgr k rule rule')
    (hsq : IsPullback g E'.1.ι E.1.ι f) (α : CycleGroup D p (k + 1)) :
    gysinCycleMap D p' E' hE' (k + (r : ℤ)) rule'
        (flatPullbackMap D p' hp' p hp f hfbase r hflat hr (k + 1) (k + (r : ℤ) + 1)
          hkr α) =
      flatPullbackMap D (E'.1.ι ≫ p') hE' (E.1.ι ≫ p) hE g hgbase r hgflat hgr k
        (k + (r : ℤ)) rfl (gysinCycleMap D p E hE k rule α)

AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_flatPullbackMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinFlatPullbackCanonical.lean, line 130. The attribute's comment: “(i')^ f^ α = g^ i^ α in CH_{k+r}(D') for f : X' → X flat of relative dimension r between good spaces over B, (L, s, i : D → X) as in 0ER1 and D' = Z(f^*s) = X' ×_X D as in 0ER3 (g : D' → D flat of relative dimension r by base change); the Gysin maps of 0ER1 at their constructed rules (sectionGysinRule), the flat pullbacks of cycles of 0EP8, CH = chowGroupCanonical of 0EQ6. Unconditional: the case W ⊄ D is 0EQQ for f^{-1}(W) → W (an identity of cycles), the case W ⊆ D is 0EQR at one integral closed subspace, reassembled by 0EQ8”.

Stacks 0ER6. In Situation 0EDT let f : X' ⟶ X be a morphism over B of good algebraic spaces which is flat of relative dimension r, let (ℒ, s, i : E → X) be as in Definition 0ER1 (ℒ invertible, s ∈ Γ(X, ℒ), E = Z(s)), and form the diagram of Remark 0ER3: ℒ' = f^*ℒ, s' = f^*s, E' = Z(s') = X' ×_X E with i' : E' → X' and the projection g : E' ⟶ E, which is flat of relative dimension r. Then for every (k + 1)-cycle α on X

(i')^* f^* α = g^* i^* α in CH_{k+r}(E').

Here i^*, (i')^* are the Gysin maps on cycles of Stacks 0ER1 at their constructed component rules, f^*, g^* the flat pullbacks of cycles of Stacks 0EP8, and CH_{k+r}(E') the Chow group of Stacks 0EQ6 (div(f_j) at the canonical embeddings of Stacks 0ENK).

theorem chowClassCanonical_gysinCycleMap_flatPullbackMap (α : CycleGroup D p (k + 1)) :
    chowClassCanonical.{u + 1} D ((pullbackSectionVanishingLocus D p hp p' hp' f L s).1.ι ≫ p')
        (isGood_vanishingLocus D p' hp' _ _) (k + (r : ℤ))
        (gysinCycleMap D p' (pullbackSectionVanishingLocus D p hp p' hp' f L s)
          (isGood_vanishingLocus D p' hp' _ _) (k + (r : ℤ))
          (sectionGysinRule D p' hp' (pullbackSheaf D p hp p' hp' f L)
            (pullbackGlobalSection D p hp p' hp' f L s) (k + (r : ℤ))
            (isInvertibleSheaf_pullbackSheaf D p hp p' hp' f L hL))
          (flatPullbackMap D p' hp' p hp f.hom hfbase r hflat hr (k + 1) (k + (r : ℤ) + 1)
            (by ring) α)) =
      chowClassCanonical.{u + 1} D ((pullbackSectionVanishingLocus D p hp p' hp' f L s).1.ι ≫ p')
        (isGood_vanishingLocus D p' hp' _ _) (k + (r : ℤ))
        (flatPullbackMap D ((pullbackSectionVanishingLocus D p hp p' hp' f L s).1.ι ≫ p')
          (isGood_vanishingLocus D p' hp' _ _) ((vanishingLocus D p hp L s).1.ι ≫ p)
          (isGood_vanishingLocus D p hp L s)
          (pullbackVanishingLocusProjection D p hp p' hp' f L s hL)
          (pullbackVanishingLocusProjection_comp_structure D p hp p' hp' f L s hL hfbase) r
          (isFlat_and_hasRelativeDimension_pullbackVanishingLocusProjection D p hp p' hp' f L s
            hfbase r hflat hr hL).1
          (isFlat_and_hasRelativeDimension_pullbackVanishingLocusProjection D p hp p' hp' f L s
            hfbase r hflat hr hL).2 k (k + (r : ℤ)) rfl
          (gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k
            (sectionGysinRule D p hp L s k hL) α))

Tag 0ER7, a lemma

Tag 0ER7 in the Stacks Project.

AlgebraicGeometry.Space.gysinCycleMap_closedSubspaceCycle_coe_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinClosedSubspaceIntersectionAdditivity.lean, line 661. The attribute's comment: “i^*([Z]_{k+1}) = [D ∩ Z]_k for a closed subspace Z with dim_δ(Z) ≤ k+1 whose intersection with D is an effective Cartier divisor on Z, stated coefficientwise on the integral closed subspaces of D so that no δ-dimension bound on D ∩ Z is needed; the Gysin map is the one of Stacks 0ER1 at canonicalGysinComponentRule, so clause (1)(a) is the cycle [D ∩ W]_k of Stacks 0EE9 and only clause (1)(b), the cap product of Stacks 0EQJ, is the free parameter chern that Definition 0ER1 takes it to be here; Stacks 0EQL for (Z, s|_Z) and the non-containment of the components of Z in D are proved, not assumed”.

Stacks 0ER7. In Situation 0EDT let X/B be good and let (ℒ, s, i : E → X) be a triple as in Definition 0ER1: ℒ invertible, s a global section of it and E = Z(s) its vanishing locus. Let Z ⊆ X be a closed subspace with dim_δ(Z) ≤ k + 1 whose intersection E ∩ Z is an effective Cartier divisor on Z. Then

i^*([Z]_{k + 1}) = [E ∩ Z]_k.

The left-hand side is the refined Gysin map of Stacks 0ER1 at the component rule AlgebraicGeometry.Space.canonicalGysinComponentRule, so clause (1)(a) is the cycle [E ∩ W]_k of Stacks 0EE9 that the source means and only clause (1)(b) — the cap product c₁(ℒ|_W) ∩ [W] of Stacks 0EQJ — is the free parameter chern that Definition 0ER1 takes it to be in this library. The cycle [Z]_{k+1} is AlgebraicGeometry.Space.closedSubspaceCycle, which is Stacks 0EE9 as an element of Z_{k+1}(X), and the identity is stated coefficientwise so that [E ∩ Z]_k needs no δ-dimension bound of its own: both sides are coefficient families on the integral closed subspaces of E.

Nothing of the source's proof is assumed. The first two sentences — that the hypothesis is the regularity of s|_Z and that therefore E ∩ Z = Z(s|_Z) — are AlgebraicGeometry.Space.isEffectiveCartierDivisor_interOnClosedSubspace_iff_isRegularSection and AlgebraicGeometry.Space.interOnClosedSubspace_eq_zeroScheme; Lemma 0EQL for (Z, ℒ|_Z, s|_Z) is AlgebraicGeometry.Space.cycleCoeffOfClosedSubspace_zeroScheme_eq_componentZeroCycleSumCoeff, transported to the ambient X by AlgebraicGeometry.Space.cycleCoeffOfClosedSubspace_interOnVanishingLocus_eq_finsum; the non-containment Z_i ⊄ E is AlgebraicGeometry.Space.not_le_vanishingLocus_of_cycleCoeffOfClosedSubspace_ne_zero; and the final matching against the definition of the Gysin map is AlgebraicGeometry.Space.gysinComponentCycle_of_not_le.

theorem gysinCycleMap_closedSubspaceCycle_coe_eq
    (hL : IsInvertibleSheaf D p hp L)
    (chern : FirstChernComponentCycleRule D p k)
    (hZdim : deltaDimensionSubspace D p hp Z.1 Z.2 ≤ ((k + 1 : ℤ) : EInt))
    (hCartier : IsEffectiveCartierDivisor D.site D.base D.base_mem ⟨Z.1.toFunctor, Z.2.1⟩
      (interOnClosedSubspace D p hp L s Z)) :
    (gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k
        (canonicalGysinComponentRule D p hp (vanishingLocus D p hp L s)
          (isGood_vanishingLocus D p hp L s) k chern)
        (closedSubspaceCycle D p hp Z.1 Z.2 (k + 1) hZdim)).1 =
      cycleCoeffOfClosedSubspace D ((vanishingLocus D p hp L s).1.ι ≫ p)
        (isGood_vanishingLocus D p hp L s)
        (interOnVanishingLocus D p hp L s Z).1 (interOnVanishingLocus D p hp L s Z).2 k
        (canonicalGeometricPoint D.site D.base D.base_mem
          (interOnVanishingLocus D p hp L s Z).1.toFunctor)

AlgebraicGeometry.Space.gysinCycleMap_canonical_eq_of_cycleCoeffOfClosedSubspace, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinClosedSubspaceIntersectionAdditivity.lean, line 715. The attribute's comment: “i^*([Z]_{k+1}) = [D ∩ Z]_k as an equality in Z_k(D), for a closed subspace Z with dim_δ(Z) ≤ k+1 whose intersection with D is an effective Cartier divisor on Z; the Gysin map is the one of Stacks 0ER1 at canonicalGysinComponentRule, so clause (1)(a) is the cycle [D ∩ W]_k of Stacks 0EE9 and only clause (1)(b), the cap product of Stacks 0EQJ, is the free parameter chern that Definition 0ER1 takes it to be here; the two cycles are elements α and β of the cycle groups whose coefficient families hα and hβ pin to Stacks 0EE9, which replaces the δ-dimension bound on D ∩ Z; Stacks 0EQL for (Z, s|_Z) and the non-containment of the components of Z in D are proved, not assumed”.

Stacks 0ER7, as an equality of cycles. In Situation 0EDT let X/B be good and let (ℒ, s, i : E → X) be a triple as in Definition 0ER1. Let Z ⊆ X be a closed subspace with dim_δ(Z) ≤ k + 1 whose intersection E ∩ Z is an effective Cartier divisor on Z. Then

i^*([Z]_{k + 1}) = [E ∩ Z]_k

in Z_k(E), for the refined Gysin map of Stacks 0ER1 at the component rule AlgebraicGeometry.Space.canonicalGysinComponentRule.

This is AlgebraicGeometry.Space.gysinCycleMap_closedSubspaceCycle_coe_eq read as an equality of cycles rather than of coefficient families. The two cycles are the elements α of Z_{k+1}(X) and β of Z_k(E) whose coefficient families hα and hβ pin to those of Stacks 0EE9 — on the left [Z]_{k+1}, on the right [E ∩ Z]_k for the intersection read inside E. For the left one that is no restriction, α being forced to be AlgebraicGeometry.Space.closedSubspaceCycle; supplying the right one takes the place of the bound dim_δ(E ∩ Z) ≤ k, which the source gets from the effective Cartier hypothesis and which has no declaration in this library. Nothing weakens: both are the coefficient families the source's two cycles have.

This is the shape in which AlgebraicGeometry.Space.gysinCycleMap_eq_of_cycleCoeffOfClosedSubspace states the identity at an arbitrary component rule, where clause 0ER1(1)(a) is free data; here the rule is the canonical one of Stacks 0ER1, so that clause is the cycle [E ∩ W]_k of Stacks 0EE9 and the two inputs that lemma takes for it — Stacks 0EQL for (Z, ℒ|_Z, s|_Z) and the non-containment Z_i ⊄ E — are AlgebraicGeometry.Space.cycleCoeffOfClosedSubspace_interOnVanishingLocus_eq_finsum and AlgebraicGeometry.Space.not_le_vanishingLocus_of_cycleCoeffOfClosedSubspace_ne_zero.

theorem gysinCycleMap_canonical_eq_of_cycleCoeffOfClosedSubspace
    (hL : IsInvertibleSheaf D p hp L)
    (chern : FirstChernComponentCycleRule D p k)
    (hZdim : deltaDimensionSubspace D p hp Z.1 Z.2 ≤ ((k + 1 : ℤ) : EInt))
    (hCartier : IsEffectiveCartierDivisor D.site D.base D.base_mem ⟨Z.1.toFunctor, Z.2.1⟩
      (interOnClosedSubspace D p hp L s Z))
    (α : CycleGroup D p (k + 1))
    (hα : α.1 = cycleCoeffOfClosedSubspace D p hp Z.1 Z.2 (k + 1)
      (canonicalGeometricPoint D.site D.base D.base_mem Z.1.toFunctor))
    (β : CycleGroup D ((vanishingLocus D p hp L s).1.ι ≫ p) k)
    (hβ : β.1 = cycleCoeffOfClosedSubspace D ((vanishingLocus D p hp L s).1.ι ≫ p)
      (isGood_vanishingLocus D p hp L s)
      (interOnVanishingLocus D p hp L s Z).1 (interOnVanishingLocus D p hp L s Z).2 k
      (canonicalGeometricPoint D.site D.base D.base_mem
        (interOnVanishingLocus D p hp L s Z).1.toFunctor)) :
    gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s) k
        (canonicalGysinComponentRule D p hp (vanishingLocus D p hp L s)
          (isGood_vanishingLocus D p hp L s) k chern) α = β

Sources

• Section 82.22, Tag 0ER0
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.refinedGysin · AlgebraicGeometry.Space.effectiveCartierIntersection · AlgebraicGeometry.Space.sectionRefinedGysin · AlgebraicGeometry.Space.sectionCartierIntersection · AlgebraicGeometry.Space.canonicalTrivialRestrictionGysin · AlgebraicGeometry.Space.canonicalTrivialRestrictionGysin_integralCycle_of_le · AlgebraicGeometry.Space.canonicalTrivialRestrictionGysin_integralCycle_of_not_le · AlgebraicGeometry.Space.gysinCycleMap_eq_canonicalTrivialRestrictionGysin

Section82.23 Gysin homomorphisms#

Section 82.23 of the Stacks Project, Tag 0ER8. 4 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ER9, a lemma

Tag 0ER9 in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_canonicalDivisorCycleAt, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinDivisorKeyFormula.lean, line 457. The attribute's comment: “i^*div_N(t) = c_1(N) ∩ [D]_{n-1} in CH_{n-2}(D), at the hypotheses of 0ER9 itself: X good and integral of δ-dimension n, i : D → X the effective Cartier divisor presented by an invertible L with a nonzero global section σ (so D = Z(σ), as in the proof's own 'Set L = O_X(D)'), N invertible and t a nonzero meromorphic section of N. The Gysin map is 0ER1's, the cap product 0EQJ's formed on D, [D]_{n-1} is 0EE9's and div_N(t) is 0EQE's at the canonical ratios of 0EPR; CH_{n-2}(D) is chowGroupCanonical, the quotient by the rational equivalences of 0EQ6 read at the canonical 0ENK embeddings. Unconditional: the family {Z_i} is constructed by 0EPS, the choice s_i = σ off D by 0845, and the rational equivalence is the key formula 0EQV carried inside |D| and descended along i”.

Stacks 0ER9 (Lemma 82.23.1). In Situation 0EDT let X/B be good, X integral with n = dim_δ(X) = k + 2, let ℒ be an invertible 𝒪_X-module with a nonzero global section σ and let i : D = Z(σ) → X be the closed immersion of its vanishing locus (Stacks 0845) — the effective Cartier divisor of the source, presented by the pair (𝒪_X(D), s) the source immediately passes to. Let 𝒩 be an invertible 𝒪_X-module and let t be a nonzero meromorphic section of 𝒩. Then

i^*div_𝒩(t) = c₁(𝒩) ∩ [D]_{n-1} in CH_{n-2}(D),

the Chow group being AlgebraicGeometry.Space.chowGroupCanonical, which is the source's CH_{n-2}(D): the embeddings R(W) → Q(𝒪^h_{W,ξ}) of Stacks 0ENK presenting the rational equivalences are the canonical ones.

The refined Gysin map is the one of Stacks 0ER1, AlgebraicGeometry.Space.gysinCycleMap at the component rule AlgebraicGeometry.Space.canonicalGysinComponentRule, whose free clause (1)(b) is here the generator rule of the source's own choice s_i = σ off D; the cap product is the one of Stacks 0EQJ formed on D at the generator rule of an arbitrary family of generators t_W of the 𝒩_{ξ_W}, which is the choice Section 0EQU leaves free; [D]_{n-1} is the cycle of Stacks 0EE9; and div_𝒩(t) is the Weil divisor of Stacks 0EQE at the canonical ratios of Stacks 0EPR. Regularity of t is derived from t ≠ 0 rather than assumed.

theorem chowClassCanonical_gysinCycleMap_canonicalDivisorCycleAt
    (t : MeromorphicSection.{u, u} D.site D.base D.base_mem X hX.1 N) (ht : t ≠ 0)
    (w : GenericUnitFamily D sX hX) :
    chowClassCanonical.{u + 1} D ((vanishingLocus D sX hX L σ).1.ι ≫ sX)
        (isGood_vanishingLocus D sX hX L σ) k
        (gysinCycleMap D sX (vanishingLocus D sX hX L σ) (isGood_vanishingLocus D sX hX L σ) k
          (canonicalGysinComponentRule D sX hX (vanishingLocus D sX hX L σ)
            (isGood_vanishingLocus D sX hX L σ) k
            (generatorChernRule D sX hX hint L hL hn (sectionGeneratorUnit D sX hX hint L hL σ)))
          (canonicalDivisorCycleAt D sX hX hint N hN t
            (isRegularMeromorphicSection_of_ne_zero_of_isGood D sX hX hint N hN ht) hn)) =
      chowClassCanonical.{u + 1} D ((vanishingLocus D sX hX L σ).1.ι ≫ sX)
        (isGood_vanishingLocus D sX hX L σ) k
        (firstChernCapCycleMap D ((vanishingLocus D sX hX L σ).1.ι ≫ sX)
          (isGood_vanishingLocus D sX hX L σ) k
          (closedSubspaceChernRule D sX hX (vanishingLocus D sX hX L σ).1
            (vanishingLocus D sX hX L σ).2 k (generatorChernRule D sX hX hint N hN hn w))
          (vanishingLocusTopCycleAt D sX hX hint L hL hσ hn))

Tag 0ERA, a lemma

Tag 0ERA in the Stacks Project.

AlgebraicGeometry.Space.exists_sectionGysinChowGroupMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinChowGroupMapOfSection.lean, line 255. The attribute's comment: “i^* : CH_{k+1}(X) → CH_k(D) for X/B good and (L, s, i : D → X) as in 0ER1 (L invertible, s a global section, D = Z(s)): the Gysin cycle map of 0ER1 at its own constructed rule sectionGysinRule descends through the Chow groups chowGroupCanonical of 0EQ6 (div(f_j) at the canonical 0ENK embeddings). Unconditional: the member cases are 0EQY on V ⊆ D and the key formula 0EQV (the content of 0ER9 at N = O_V) on V ⊄ D, both carried inside D; 0ER5 and 0EQC are replaced by the carrier-respecting reassembly”.

Stacks 0ERA. In Situation 0EDT let X/B be good and let (ℒ, σ, i : E → X) be as in Definition 0ER1: ℒ an invertible 𝒪_X-module, σ ∈ Γ(X, ℒ) and i : E = Z(σ) → X the closed immersion of its vanishing locus. The Gysin homomorphism i^* : Z_{k+1}(X) → CH_k(E) of Definition 0ER1 factors through rational equivalence: there is a homomorphism

i^* : CH_{k+1}(X) ⟶ CH_k(E)

sending the class of every (k+1)-cycle α to i^*α, i.e. to the class of the cycle ∑ n_W i^*[W] of Stacks 0ER1(2).

The Gysin map on cycles is AlgebraicGeometry.Space.gysinCycleMap at AlgebraicGeometry.Space.sectionGysinRule, the component rule of Stacks 0ER1(1) with every ingredient constructed, which is the rule of AlgebraicGeometry.Space.sectionRefinedGysin; the Chow groups are AlgebraicGeometry.Space.chowGroupCanonical, Stacks 0EQ6 with div(f_j) read at the canonical embeddings of Stacks 0ENK. The map is AlgebraicGeometry.Space.sectionGysinChowGroupMap; nothing is assumed.

theorem exists_sectionGysinChowGroupMap :
    ∃ F : chowGroupCanonical.{u + 1} D p hp (k + 1) →+
        chowGroupCanonical.{u + 1} D ((vanishingLocus D p hp L σ).1.ι ≫ p)
          (isGood_vanishingLocus D p hp L σ) k,
      ∀ α : CycleGroup D p (k + 1),
        F (chowClassCanonical.{u + 1} D p hp (k + 1) α) =
          chowClassCanonical.{u + 1} D ((vanishingLocus D p hp L σ).1.ι ≫ p)
            (isGood_vanishingLocus D p hp L σ) k
            (gysinCycleMap D p (vanishingLocus D p hp L σ) (isGood_vanishingLocus D p hp L σ) k
              (sectionGysinRule D p hp L σ k hL) α)

Tag 0ERB, a lemma

Tag 0ERB in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_firstChernCapCycleMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RefinedGysinFirstChernCap.lean, line 1319. The attribute's comment: “i^(c₁(N) ∩ α) = c₁(i^N) ∩ i^α in CH_k(E), for every (k+2)-cycle α on X representing a class of CH_{k+2}(X); i^ is the refined Gysin map of Stacks 0ER1 on cycles (gysinCycleMap at sectionGysinRule), both cap products the cycle-level operations of Stacks 0EQJ at the canonical choices, the classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical); nothing carried”.

Stacks 0ERB. In Situation 0EDT let X/B be good, let (ℒ, s, i : E → X) be a triple as in Stacks 0ER1 and let 𝒩 be an invertible 𝒪_X-module. For every (k+2)-cycle α on X,

i^*(c₁(𝒩) ∩ α) = c₁(i^*𝒩) ∩ i^*α in CH_k(E).

Here i^* is the refined Gysin map of Stacks 0ER1 on cycles, AlgebraicGeometry.Space.gysinCycleMap at the rule AlgebraicGeometry.Space.sectionGysinRule; both cap products are the cycle-level operations of Stacks 0EQJ at the canonical choices, AlgebraicGeometry.Space.firstChernCapCycleMap at AlgebraicGeometry.Space.canonicalFirstChernRule; and the classes are taken in CH_k(E) of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical, whose rational equivalences are sums of principal divisors at the canonical embeddings of Stacks 0ENK. As every class of CH_{k+2}(X) is the class of a cycle, this is the identity of the source on representatives.

The two composites are cycle maps of kernels, and on each [W] their difference is a sum of canonical principal divisors lying over |W| (isRationallyEquivalentToZeroWithin_gysinCycleMap_firstChernComponentCycle_sub); the general case is Stacks 0EQ8 (AlgebraicGeometry.Space.isCycleFamilySumWithin_sub_of_isCycleMapOfKernel).

theorem chowClassCanonical_gysinCycleMap_firstChernCapCycleMap (k : ℤ)
    (α : CycleGroup D sX (k + 1 + 1)) :
    chowClassCanonical.{max (u + 1) v} D ((vanishingLocus D sX hX L s).1.ι ≫ sX)
        (isGood_vanishingLocus D sX hX L s) k
        (gysinCycleMap D sX (vanishingLocus D sX hX L s) (isGood_vanishingLocus D sX hX L s) k
          (sectionGysinRule D sX hX L s k hL)
          (firstChernCapCycleMap D sX hX (k + 1) (canonicalFirstChernRule D sX hX N hN (k + 1))
            α)) =
      chowClassCanonical.{max (u + 1) v} D ((vanishingLocus D sX hX L s).1.ι ≫ sX)
        (isGood_vanishingLocus D sX hX L s) k
        (firstChernCapCycleMap D ((vanishingLocus D sX hX L s).1.ι ≫ sX)
          (isGood_vanishingLocus D sX hX L s) k
          (canonicalFirstChernRule D _ (isGood_vanishingLocus D sX hX L s)
            ((modulePullback D.site D.base D.base_mem (vanishingLocusInclusion D sX hX L s)).obj N)
            (modulePullbackInvertible (vanishingLocusInclusion D sX hX L s) N hN) k)
          (gysinCycleMap D sX (vanishingLocus D sX hX L s) (isGood_vanishingLocus D sX hX L s)
            (k + 1) (sectionGysinRule D sX hX L s (k + 1) hL) α))

Tag 0ERC, a lemma

Tag 0ERC in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_gysinCycleMap_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RefinedGysinCommutativity.lean, line 536. The attribute's comment: “j^ i^ α = (j')^ (i')^ α in CH_k(D ∩ D') for every (k+2)-cycle α on X representing a class of CH_{k+2}(X), all four maps the refined Gysin maps of Stacks 0ER1 on cycles at the canonical component cycles, D ∩ D' being Z(s'|_D) identified with Z(s|_{D'}), the classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical); nothing carried”.

Stacks 0ERC: the two refined Gysin maps of a pair of sections commute. In Situation 0EDT let X/B be good and let (ℒ, s, i : E → X), (ℒ', s', i' : E' → X) be two triples as in Stacks 0ER1. Let j : Z(s'|_E) → E and j' : Z(s|_{E'}) → E' be the closed immersions of the triples (ℒ'|_E, s'|_E) on E and (ℒ|_{E'}, s|_{E'}) on E', both vanishing loci being E ∩ E' (AlgebraicGeometry.Space.vanishingLocusIntersectionComparison). For every (k + 2)-cycle α on X,

j^* (i^* α) = (j')^* ((i')^* α) in CH_k(E ∩ E').

All four Gysin maps are the cycle-level maps of Stacks 0ER1, parts (1) and (2), at the canonical component cycles (AlgebraicGeometry.Space.gysinCycleMap at AlgebraicGeometry.Space.sectionGysinRule); the left side is carried to Z(s|_{E'}) by the identification of the two vanishing loci, and the classes are taken in the Chow group of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical, whose rational equivalences are sums of principal divisors at the canonical embeddings of Stacks 0ENK. As every class of CH_{k+2}(X) is the class of a cycle, this is the identity of the source on representatives.

The proof is the source's: the case α = [W] is the key formula on W (isRationallyEquivalentToZeroWithin_comparisonPushforward_gysinCycleMap_sub_integralCycle), a sum of canonical principal divisors lying over |W|, and the general case follows by Stacks 0EQ8 (AlgebraicGeometry.Space.isCycleFamilySumWithin_closedPushforward_gysin_gysin_sub).

theorem chowClassCanonical_gysinCycleMap_gysinCycleMap_comm (k : ℤ)
    (α : CycleGroup D p (k + 1 + 1)) :
    chowClassCanonical.{max (u + 1) v} D
        ((closedSubspaceVanishingLocus D p hp L s (vanishingLocus D p hp L' s')).1.ι ≫
          (vanishingLocus D p hp L' s').1.ι ≫ p)
        (isGood_vanishingLocus D _ (isGood_closedSubspace D p hp (vanishingLocus D p hp L' s')) _ _)
        k
        (comparisonPushforward D p (vanishingLocus D p hp L s) (vanishingLocus D p hp L' s') k
          (closedSubspaceVanishingLocus D p hp L' s' (vanishingLocus D p hp L s))
          (closedSubspaceVanishingLocus D p hp L s (vanishingLocus D p hp L' s'))
          (isGood_vanishingLocus D _ (isGood_closedSubspace D p hp (vanishingLocus D p hp L s)) _ _)
          (isGood_vanishingLocus D _ (isGood_closedSubspace D p hp (vanishingLocus D p hp L' s'))
            _ _)
          (isClosedImmersion_vanishingLocusIntersectionComparison D p hp s s' hL hL')
          (vanishingLocusIntersectionComparison_comp_ι D p hp s s' hL hL')
          (gysinCycleMap D _ (closedSubspaceVanishingLocus D p hp L' s' (vanishingLocus D p hp L s))
            (isGood_vanishingLocus D _
              (isGood_closedSubspace D p hp (vanishingLocus D p hp L s)) _ _)
            k (sectionGysinRule D _ (isGood_closedSubspace D p hp (vanishingLocus D p hp L s)) _
              (closedSubspaceModuleRestrictSection D p hp L' s' (vanishingLocus D p hp L s)) k
              (isInvertibleSheaf_closedSubspaceModuleRestrict D p hp (vanishingLocus D p hp L s)
                hL'))
            (gysinCycleMap D p (vanishingLocus D p hp L s) (isGood_vanishingLocus D p hp L s)
              (k + 1) (sectionGysinRule D p hp L s (k + 1) hL) α))) =
      chowClassCanonical.{max (u + 1) v} D
        ((closedSubspaceVanishingLocus D p hp L s (vanishingLocus D p hp L' s')).1.ι ≫
          (vanishingLocus D p hp L' s').1.ι ≫ p)
        (isGood_vanishingLocus D _ (isGood_closedSubspace D p hp (vanishingLocus D p hp L' s')) _ _)
        k
        (gysinCycleMap D _ (closedSubspaceVanishingLocus D p hp L s (vanishingLocus D p hp L' s'))
          (isGood_vanishingLocus D _
            (isGood_closedSubspace D p hp (vanishingLocus D p hp L' s')) _ _)
          k (sectionGysinRule D _ (isGood_closedSubspace D p hp (vanishingLocus D p hp L' s')) _
            (closedSubspaceModuleRestrictSection D p hp L s (vanishingLocus D p hp L' s')) k
            (isInvertibleSheaf_closedSubspaceModuleRestrict D p hp (vanishingLocus D p hp L' s')
              hL))
          (gysinCycleMap D p (vanishingLocus D p hp L' s') (isGood_vanishingLocus D p hp L' s')
            (k + 1) (sectionGysinRule D p hp L' s' (k + 1) hL') α))

Sources

• Section 82.23, Tag 0ER8
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_canonicalDivisorCycleAt · AlgebraicGeometry.Space.exists_sectionGysinChowGroupMap · AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_firstChernCapCycleMap · AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_gysinCycleMap_comm

Section82.24 Relative effective Cartier divisors#

Section 82.24 of the Stacks Project, Tag 0ERD. 1 tag here is carried by 3 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ERE, a lemma

Tag 0ERE in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_canonicalSection_flatPullbackMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RelativeEffectiveCartierFlatPullback.lean, line 649. The attribute's comment: “first identity: i^ p^ α = (p|_D)^ α in CH_{k+r}(D), on cycles: i^ is the refined Gysin map of Stacks 0ER1 for (O_X(D), 1_D) (gysinCycleMap at sectionGysinRule), p^ and (p|_D)^ the flat pullbacks of Stacks 0EP8, the classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical); the identity holds already in Z(D)”.

Stacks 0ERE, first identity. In Situation 0EDT let X, Y/B be good, let p : X ⟶ Y be flat of relative dimension r + 1 and let i : D → X be a relative effective Cartier divisor on X/Y (Stacks 0EPP), with ℒ = 𝒪_X(D) and its canonical section 1_D (Stacks 0841), whose vanishing locus Z(1_D) is D (Stacks 0847). Then for every (k+1)-cycle α on Y

i^* p^* α = (p|_D)^* α in CH_{k+r+1}(D),

where p^* is the flat pullback of cycles of Stacks 0EP8 (AlgebraicGeometry.Space.flatPullbackMap), i^* the refined Gysin map of Stacks 0ER1 for the pair (𝒪_X(D), 1_D) on cycles (AlgebraicGeometry.Space.gysinCycleMap at AlgebraicGeometry.Space.sectionGysinRule), and (p|_D)^* the flat pullback along p|_D : D → Y, which is flat of relative dimension r; the classes are taken in CH_{k+r+1}(D) of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical. The identity holds already on cycles, AlgebraicGeometry.Space.gysinCycleMap_sectionGysinRule_flatPullbackMap.

theorem chowClassCanonical_gysinCycleMap_canonicalSection_flatPullbackMap (k : ℤ)
    (α : CycleGroup D sY (k + 1)) :
    chowClassCanonical.{v} D (_ ≫ sX) (isGood_vanishingLocus D sX hX _ _) (k + r + 1)
        (gysinCycleMap D sX (vanishingLocus D sX hX
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)
            (canonicalSection D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1))
          (isGood_vanishingLocus D sX hX _ _) (k + r + 1)
          (sectionGysinRule D sX hX _ _ (k + r + 1)
            (associatedInvertibleSheaf_isInvertible D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1))
          (flatPullbackMap D sX hX sY hY p hbase (r + 1) hflat hr (k + 1) (k + r + 1 + 1)
            (by push_cast; ring) α)) =
      chowClassCanonical.{v} D (_ ≫ sX) (isGood_vanishingLocus D sX hX _ _) (k + r + 1)
        (flatPullbackMap D (_ ≫ sX) (isGood_vanishingLocus D sX hX _ _) sY hY (_ ≫ p)
          (comp_structureMorphism_eq D _ sX sY _ p rfl hbase) r
          (isRelativeEffectiveCartierDivisor_vanishingLocus_canonicalSection D sX hX sY hY p Dv
            hDv).2
          (hasRelativeDimension_ι_comp D sX hX sY hY p hbase r hr _
            (isRelativeEffectiveCartierDivisor_vanishingLocus_canonicalSection D sX hX sY hY p Dv
              hDv)) (k + 1) (k + r + 1) (by ring) α)

AlgebraicGeometry.Space.chowClassCanonical_firstChernCapCycleMap_canonicalSection_flatPullbackMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RelativeEffectiveCartierFlatPullback.lean, line 688. The attribute's comment: “second identity: c₁(O_X(D)) ∩ p^ α = i_((p|_D)^ α) in CH_{k+r}(X), on cycles: c₁ ∩ - is the cap product of Stacks 0EQJ at the canonical choices, p^ and (p|_D)^ the flat pullbacks of Stacks 0EP8, i_ the pushforward along the closed immersion, the classes taken in the Chow group of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical)”.

Stacks 0ERE, second identity. In the situation of AlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_canonicalSection_flatPullbackMap, for every (k+1)-cycle α on Y

c₁(𝒪_X(D)) ∩ p^* α = i_* ((p|_D)^* α) in CH_{k+r+1}(X),

where c₁(𝒪_X(D)) ∩ - is the cycle-level cap product of Stacks 0EQJ at the canonical choices (AlgebraicGeometry.Space.firstChernCapCycleMap at AlgebraicGeometry.Space.canonicalFirstChernRule), i_* the pushforward of cycles along the closed immersion i : D → X, and the classes are taken in CH_{k+r+1}(X) of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical. It follows from the first identity and Stacks 0ER4.

theorem chowClassCanonical_firstChernCapCycleMap_canonicalSection_flatPullbackMap (k : ℤ)
    (α : CycleGroup D sY (k + 1)) :
    chowClassCanonical.{max (u + 1) v} D sX hX (k + r + 1)
        (firstChernCapCycleMap D sX hX (k + r + 1)
          (canonicalFirstChernRule D sX hX
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)
            (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _
              (associatedInvertibleSheaf_isInvertible D.site D.base D.base_mem ⟨X, hX.1⟩ Dv
                hDv.1)) (k + r + 1))
          (flatPullbackMap D sX hX sY hY p hbase (r + 1) hflat hr (k + 1) (k + r + 1 + 1)
            (by push_cast; ring) α)) =
      chowClassCanonical.{max (u + 1) v} D sX hX (k + r + 1)
        (closedSubspaceCyclePushforward D sX hX (vanishingLocus D sX hX
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)
            (canonicalSection D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)) (k + r + 1)
          (flatPullbackMap D (_ ≫ sX) (isGood_vanishingLocus D sX hX _ _) sY hY (_ ≫ p)
            (comp_structureMorphism_eq D _ sX sY _ p rfl hbase) r
            (isRelativeEffectiveCartierDivisor_vanishingLocus_canonicalSection D sX hX sY hY p Dv
              hDv).2
            (hasRelativeDimension_ι_comp D sX hX sY hY p hbase r hr _
              (isRelativeEffectiveCartierDivisor_vanishingLocus_canonicalSection D sX hX sY hY p
                Dv hDv)) (k + 1) (k + r + 1) (by ring) α))

AlgebraicGeometry.Space.chowClassCanonical_flatPullbackMap_canonicalSection_eq_zero_of_hasRelativeDimension_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/RelativeEffectiveCartierFlatPullback.lean, line 805. The attribute's comment: “the case of relative dimension r = 0, where D is empty: every cycle on D is zero, and i^ p^ α = 0 in CH_k(D) and c₁(O_X(D)) ∩ p^* α = 0 in CH_k(X), on cycles, the classes taken in the Chow groups of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (chowGroupCanonical)”.

Stacks 0ERE in relative dimension zero. If p : X ⟶ Y is flat of relative dimension 0 and i : D → X is a relative effective Cartier divisor on X/Y, then D has no points (AlgebraicGeometry.Space.isEmpty_points_of_isRelativeEffectiveCartierDivisor), so every cycle on D is zero, and for every (k+1)-cycle α on Y both sides of the two identities of Stacks 0ERE vanish: i^* p^* α = 0 in CH_k(D) and c₁(𝒪_X(D)) ∩ p^* α = i_*(0) = 0 in CH_k(X), the operations being those on cycles and the classes taken in the Chow groups of Stacks 0EQ6, AlgebraicGeometry.Space.chowGroupCanonical.

theorem chowClassCanonical_flatPullbackMap_canonicalSection_eq_zero_of_hasRelativeDimension_zero
    (Dv : ClosedSubspaces D.site D.base D.base_mem X)
    (hDv : IsRelativeEffectiveCartierDivisor D.site D.base D.base_mem
      (ObjectProperty.homMk p : (⟨X, hX.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
        ⟨Y, hY.1⟩) Dv) (k : ℤ) (α : CycleGroup D sY (k + 1)) :
    (∀ (n : ℤ) (β : CycleGroup D ((vanishingLocus D sX hX
        (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)
        (canonicalSection D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)).1.ι ≫ sX) n), β = 0) ∧
    chowClassCanonical.{v} D (_ ≫ sX) (isGood_vanishingLocus D sX hX _ _) k
        (gysinCycleMap D sX (vanishingLocus D sX hX
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)
            (canonicalSection D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1))
          (isGood_vanishingLocus D sX hX _ _) k
          (sectionGysinRule D sX hX _ _ k
            (associatedInvertibleSheaf_isInvertible D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1))
          (flatPullbackMap D sX hX sY hY p hbase 0 hflat hr₀ (k + 1) (k + 1) (by simp) α)) = 0 ∧
    chowClassCanonical.{max (u + 1) v} D sX hX k
        (firstChernCapCycleMap D sX hX k
          (canonicalFirstChernRule D sX hX
            (associatedInvertibleSheaf D.site D.base D.base_mem ⟨X, hX.1⟩ Dv hDv.1)
            (Stacks.SheafOfModules.IsInvertible.ofAlgebraicGeometry _ _ _
              (associatedInvertibleSheaf_isInvertible D.site D.base D.base_mem ⟨X, hX.1⟩ Dv
                hDv.1)) k)
          (flatPullbackMap D sX hX sY hY p hbase 0 hflat hr₀ (k + 1) (k + 1) (by simp) α)) = 0

Sources

• Section 82.24, Tag 0ERD
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.chowClassCanonical_gysinCycleMap_canonicalSection_flatPullbackMap · AlgebraicGeometry.Space.chowClassCanonical_firstChernCapCycleMap_canonicalSection_flatPullbackMap · AlgebraicGeometry.Space.chowClassCanonical_flatPullbackMap_canonicalSection_eq_zero_of_hasRelativeDimension_zero

Section82.25 Affine bundles#

Section 82.25 of the Stacks Project, Tag 0ERF. 2 tags here are carried by 2 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ERG, a lemma

Tag 0ERG in the Stacks Project.

AlgebraicGeometry.Space.isFlatPullbackSurjective_of_isQuasiCompact_of_hasAffineSpaceFibres, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackSurjectiveAffineFibres.lean, line 132.

Stacks 0ERG. In Situation 0EDT let X, Y/B be good and let f : X → Y be a quasi-compact morphism over the base, flat of relative dimension r, with X_y ≅ 𝐀^r_{κ(y)} for every y ∈ |Y|. Then f^* : CH_k(Y) → CH_{k+r}(X) is surjective: for every k ∈ ℤ and every α ∈ Z_{k+r}(X) there is β ∈ Z_k(Y) with f^*β - α rationally equivalent to zero, as a locally finite sum of principal divisors at the canonical embeddings of Stacks 0ENK.

theorem isFlatPullbackSurjective_of_isQuasiCompact_of_hasAffineSpaceFibres
    (hqc : IsQuasiCompact D.site D.base D.base_mem f) (hfib : HasAffineSpaceFibres D sY hY f r)
    (k : ℤ) : IsFlatPullbackSurjective D sX hX sY hY f hbase r hflat hr k

Tag 0ERH, a lemma

Tag 0ERH in the Stacks Project.

AlgebraicGeometry.Space.bijective_flatPullbackChowMapCanonical_vectorBundle, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/LineBundleFlatPullbackChow.lean, line 340.

Stacks 0ERH. In Situation 0EDT let X/B be good, let ℒ be a finite locally free 𝒪_X-module of rank one and let p : L = \underline{Spec}(Sym^*(ℒ)) → X be its line bundle, flat of relative dimension one. Then the flat pullback p^* : CH_k(X) → CH_{k+1}(L) of Stacks 0EQB, on the Chow groups of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK, is an isomorphism for every k.

theorem bijective_flatPullbackChowMapCanonical_vectorBundle (k : ℤ) :
    Function.Bijective (flatPullbackChowMapCanonical.{u + 1} D _
      (isGood_vectorBundle D W F sW hW) sW hW
      (vectorBundleProjection D.site D.base D.base_mem W F).hom rfl 1
      (isFlat_vectorBundleProjection D.site D.base D.base_mem W F)
      (hasRelativeDimension_vectorBundleProjection D.site D.base D.base_mem W F 1) k (k + 1)
      (by push_cast; rfl))

Sources

• Section 82.25, Tag 0ERF
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.isFlatPullbackSurjective_of_isQuasiCompact_of_hasAffineSpaceFibres · AlgebraicGeometry.Space.bijective_flatPullbackChowMapCanonical_vectorBundle

Section82.26 Bivariant intersection theory#

Section 82.26 of the Stacks Project, Tag 0ERI. 9 tags here are carried by 25 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ERJ, a definition

Tag 0ERJ in the Stacks Project.

AlgebraicGeometry.Space.BivariantClass, a structure in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/Bivariant.lean, line 258. The attribute's comment: “over the parameterized Chow groups of 82.15 and Gysin maps of 82.22”.

Stacks 0ERJ. A bivariant class of degree p for a morphism f : X ⟶ Y of good algebraic spaces over the base: a rule assigning to every good Y' ⟶ Y over the base and every k an additive map CH_k(Y') ⟶ CH_{k-p}(Y' ×_Y X), natural for proper pushforward, for flat pullback of fixed relative dimension, and for the refined Gysin maps of Stacks 0ER1. The group of all of them is denoted A^p(X ⟶ Y).

Stated against an AlgebraicGeometry.Space.ChowGroupSystem, which supplies the Chow groups of Stacks 82.15 and the three operations on them that the axioms quantify over, none of which this repository constructs; see the implementation notes of StacksFrontier.AlgebraicGeometry.Spaces.Chow.Bivariant.

structure BivariantClass (D : BaseSpaceDimensionSituation.{u}) (G : ChowGroupSystem D)
    {X Y : D.Ambient} (sX : X ⟶ D.space) (hX : D.IsGood sX) (sY : Y ⟶ D.space)
    (hY : D.IsGood sY) (f : X ⟶ Y) (hf : f ≫ sY = sX) (p : ℤ)

AlgebraicGeometry.Space.BivariantClassCanonical, a structure in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantCanonical.lean, line 213. The attribute's comment: “the group A^p(X → Y) of bivariant classes of degree p, over the constructed Chow groups of Stacks 0EQ6 and the constructed proper pushforward, flat pullback and refined Gysin maps of Stacks 0EP1, 0EP8 and 0ER1; the three naturality axioms are stated through cycle representatives, equivalently to the source's equations, because the descent of those three operations to Chow groups is Stacks 0EQC, 0EQB and 0ERA and is not proved here”.

A bivariant class of degree p for a morphism f : X ⟶ Y of good algebraic spaces over the base: a rule assigning to every good g : Y' ⟶ Y over the base and every k an additive map CH_k(Y') ⟶ CH_{k-p}(Y' ×_Y X), natural for proper pushforward, for flat pullback of fixed relative dimension, and for the refined Gysin maps of Stacks 0ER1. The group of all of them is denoted A^p(X ⟶ Y).

The Chow groups are AlgebraicGeometry.Space.chowGroupCanonical of Stacks 0EQ6, and the three operations are the operations on cycles of Stacks 0EP1, Stacks 0EP8 and Stacks 0ER1; each axiom asserts the source's equation through cycle representatives.

structure BivariantClassCanonical {X Y : D.Ambient} (sX : X ⟶ D.space) (hX : D.IsGood sX)
    (sY : Y ⟶ D.space) (hY : D.IsGood sY) (f : X ⟶ Y) (hf : f ≫ sY = sX) (p : ℤ)

Tag 0ERK, a definition

Tag 0ERK in the Stacks Project.

AlgebraicGeometry.Space.chowCohomologyComponent, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomology.lean, line 89. The attribute's comment: “the degree p part”.

Stacks 0ERK. The degree p part A^p(X) of the Chow cohomology of a good algebraic space X/B: the group A^p(X ⟶ X) of bivariant classes of degree p for the identity morphism of X.

abbrev chowCohomologyComponent (D : BaseSpaceDimensionSituation.{u}) (G : ChowGroupSystem D)
    {X : D.Ambient} (sX : X ⟶ D.space) (hX : D.IsGood sX) (p : ℤ) : Type (u + 1)

AlgebraicGeometry.Space.chowCohomology, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomology.lean, line 97. The attribute's comment: “the graded group, not the ℤ-algebra: composition is absent”.

Stacks 0ERK. The Chow cohomology A^*(X) of a good algebraic space X/B: the graded group whose degree p part is the group A^p(X ⟶ X) of bivariant classes of degree p for the identity morphism of X.

abbrev chowCohomology (D : BaseSpaceDimensionSituation.{u}) (G : ChowGroupSystem D)
    {X : D.Ambient} (sX : X ⟶ D.space) (hX : D.IsGood sX) : Type (u + 1)

AlgebraicGeometry.Space.ChowCohomologyClass, a structure in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyAlgebra.lean, line 166. The attribute's comment: “the degree p component A^p(X) = A^p(X → X): bivariant classes of degree p for id_X over the constructed Chow groups of Stacks 0EQ6, with the fibre product Y' ×_X X taken to be Y'; flat pullback and Gysin maps are the descended ones of Stacks 0EQB and 0ERA, the proper axiom is stated through cycle representatives”.

The group A^p(X) = A^p(X ⟶ X) of bivariant classes of degree p (Stacks 0ERJ) for the identity of a good algebraic space X/B: a rule assigning to every good test space g : Y' ⟶ X over the base and every k an additive map c ∩ - : CH_k(Y') ⟶ CH_{k-p}(Y') (the fibre product Y' ×_X X being Y'), such that

1. for h : Y'' ⟶ Y' proper, c ∩ h_*α'' = h_*(c ∩ α'') for all α'' on Y'';
2. for h : Y'' ⟶ Y' flat of relative dimension r, c ∩ h^*α' = h^*(c ∩ α') for all α';
3. for (ℒ', s', i' : D' → Y') as in Definition 0ER1, c ∩ (i')^*α' = (i')^*(c ∩ α') for all α' on Y'.

The Chow groups are the constructed AlgebraicGeometry.Space.chowGroupCanonical of Stacks 0EQ6; h^* is the flat pullback of Stacks 0EQB, (i')^* the Gysin map of Stacks 0ERA, and h_* the proper pushforward of cycles of Stacks 0EP1, axiom (1) being read through cycle representatives. The cap operation in degree k is indexed by its target degree l, with l + p = k. See the implementation notes of StacksFrontier.AlgebraicGeometry.Spaces.Chow.ChowCohomologyAlgebra.

structure ChowCohomologyClass (hX : D.IsGood sX) (p : ℤ)

AlgebraicGeometry.Space.ChowCohomologyClass.instGRing, an instance in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyAlgebra.lean, line 524. The attribute's comment: “the graded Z-algebra structure on A^*(X): the product of the degree p and q components is the composition A^p(X → X) × A^q(X → X) → A^{p+q}(X → X) of bivariant classes, associative and bilinear, with unit 1 ∈ A^0(X)”.

The graded ring structure on p ↦ A^p(X) = A^p(X ⟶ X): the product of the degree p and q components is the composition A^p(X ⟶ X) × A^q(X ⟶ X) ⟶ A^{p+q}(X ⟶ X) of bivariant classes, associative and bilinear, with unit 1 ∈ A^0(X); the integers act through the multiples of 1.

instance instGRing : DirectSum.GRing (ChowCohomologyClass D sX hX)

AlgebraicGeometry.Space.ChowCohomologyAlgebra, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyAlgebra.lean, line 553. The attribute's comment: “the Chow cohomology A^*(X): the graded Z-algebra ⊕_p A^p(X → X), with multiplication the composition of bivariant classes”.

The Chow cohomology A^*(X) of a good algebraic space X/B: the graded ℤ-algebra ⊕_p A^p(X) whose degree p component is the group A^p(X ⟶ X) of bivariant classes of degree p for id_X over the constructed Chow groups of Stacks 0EQ6, with multiplication the composition of bivariant classes (DirectSum.ring of AlgebraicGeometry.Space.ChowCohomologyClass.instGRing) and its ℤ-algebra structure the one of any ring.

abbrev ChowCohomologyAlgebra : Type _

AlgebraicGeometry.Space.chowCohomologyComponentCanonical, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyCanonical.lean, line 82. The attribute's comment: “the degree p part A^p(X) = A^p(X → X), over the constructed Chow groups of Stacks 0EQ6”.

The degree p part A^p(X) of the Chow cohomology of a good algebraic space X/B: the group A^p(X ⟶ X) of bivariant classes of degree p for the identity morphism of X, over the constructed Chow groups of Stacks 0EQ6.

abbrev chowCohomologyComponentCanonical (D : BaseSpaceDimensionSituation.{u}) {X : D.Ambient}
    (sX : X ⟶ D.space) (hX : D.IsGood sX) (p : ℤ)

AlgebraicGeometry.Space.chowCohomologyCanonical, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyCanonical.lean, line 95. The attribute's comment: “the graded group A^*(X) = ⊕_p A^p(X → X) over the constructed Chow groups of Stacks 0EQ6, not the ℤ-algebra: the product is composition of bivariant classes, which needs proper pushforward along an isomorphism to respect rational equivalence (Stacks 0EQC)”.

The Chow cohomology A^*(X) of a good algebraic space X/B: the graded group whose degree p part is the group A^p(X ⟶ X) of bivariant classes of degree p for the identity morphism of X, over the constructed Chow groups of Stacks 0EQ6.

This is the graded group and not the graded ℤ-algebra, whose product is composition of bivariant classes.

abbrev chowCohomologyCanonical (D : BaseSpaceDimensionSituation.{u}) {X : D.Ambient}
    (sX : X ⟶ D.space) (hX : D.IsGood sX)

Tag 0ERL, a remark

Tag 0ERL in the Stacks Project.

AlgebraicGeometry.Space.chowCohomologyRestrictionCap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomology.lean, line 134. The attribute's comment: “the cap operations of the pullback f^*c”.

Stacks 0ERL. The cap operations of the pullback f^*c of a Chow cohomology class c ∈ A^p(Y) along a morphism f : X ⟶ Y of good algebraic spaces over the base: on a good test space g : X' ⟶ X it is the cap operation of c on X' viewed over Y, carried to X ×_X X' along the comparison isomorphism.

def chowCohomologyRestrictionCap (c : chowCohomologyComponent D G sY hY p)
    (hg : D.IsGood (g ≫ sX)) (hpbY : D.IsGood (pullback.fst (𝟙 Y) (g ≫ f) ≫ sY))
    (hpbX : D.IsGood (pullback.fst (𝟙 X) g ≫ sX))
    (hκ : IsProper D.site D.base D.base_mem (restrictionComparisonIso D f g).hom) (k : ℤ) :
    G.chow (g ≫ sX) hg k →+ G.chow (pullback.fst (𝟙 X) g ≫ sX) hpbX (k - p)

AlgebraicGeometry.Space.chowCohomologyRestrictionCapHom, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomology.lean, line 171. The attribute's comment: “the pullback of cap operations is additive in the class”.

Stacks 0ERL. The pullback along f : X ⟶ Y of the cap operation of a Chow cohomology class on a fixed good test space over X, as an additive map in the class.

def chowCohomologyRestrictionCapHom (hg : D.IsGood (g ≫ sX))
    (hpbY : D.IsGood (pullback.fst (𝟙 Y) (g ≫ f) ≫ sY))
    (hpbX : D.IsGood (pullback.fst (𝟙 X) g ≫ sX))
    (hκ : IsProper D.site D.base D.base_mem (restrictionComparisonIso D f g).hom) (k : ℤ) :
    chowCohomologyComponent D G sY hY p →+
      (G.chow (g ≫ sX) hg k →+ G.chow (pullback.fst (𝟙 X) g ≫ sX) hpbX (k - p))

AlgebraicGeometry.Space.ChowCohomologyAlgebra.comap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyAlgebraPullback.lean, line 205.

Stacks 0ERL. The canonical ℤ-algebra map f^* : A^*(Y) ⟶ A^*(X) attached to a morphism f : X ⟶ Y of good algebraic spaces over the base. On the degree p component it is the pullback AlgebraicGeometry.Space.ChowCohomologyClass.comap: given c ∈ A^p(Y) and a test space X' ⟶ X, the class f^*c caps on CH_k(X') ⟶ CH_{k-p}(X') as c does, X' being thought of as an algebraic space over Y through X' ⟶ X ⟶ Y.

def comap (f : X ⟶ Y) (hf : f ≫ sY = sX) :
    ChowCohomologyAlgebra D sY hY →ₐ[ℤ] ChowCohomologyAlgebra D sX hX

Tag 0ERM, a lemma

Tag 0ERM in the Stacks Project.

AlgebraicGeometry.Space.firstChernBivariantClass, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernBivariantClass.lean, line 685. The attribute's comment: “the rule f : X' → X ↦ c_1(f^*L) ∩ - : CH_k(X') → CH_{k-1}(X'), for X/B good and L an invertible O_X-module, is a bivariant class of degree 1”.

In Situation 0EDT let X/B be good and let ℒ be an invertible 𝒪_X-module. Then the rule that to f : X' ⟶ X assigns c₁(f^*ℒ) ∩ - : CH_k(X') ⟶ CH_{k-1}(X') is a bivariant class of degree 1, in the sense of Stacks 0ERJ (AlgebraicGeometry.Space.BivariantClassCanonical for 𝟙 X, over the constructed Chow groups AlgebraicGeometry.Space.chowGroupCanonical of Stacks 0EQ6). The cap operation is AlgebraicGeometry.Space.firstChernBivariantCap.

def firstChernBivariantClass :
    BivariantClassCanonical.{u + 1} D sX hX sX hX (𝟙 X) (Category.id_comp sX) 1

Tag 0ERN, a lemma

Tag 0ERN in the Stacks Project.

AlgebraicGeometry.Space.flatPullbackBivariantClassCanonical, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FlatPullbackBivariantClass.lean, line 290. The attribute's comment: “the rule g : Y' → Y ↦ (f')^* : CH_k(Y') → CH_{k+r}(X'), X' = X ×_Y Y', for f flat of relative dimension r between good spaces over B, is a bivariant class of degree -r”.

In Situation 0EDT let f : X ⟶ Y be a morphism of good algebraic spaces over B which is flat of relative dimension r. Then the rule that to g : Y' ⟶ Y assigns the flat pullback (f')^* : CH_k(Y') ⟶ CH_{k+r}(X') of Stacks 0EQB along the base change f' : X' = X ×_Y Y' ⟶ Y' is a bivariant class of degree -r, in the sense of Stacks 0ERJ (AlgebraicGeometry.Space.BivariantClassCanonical, over the constructed Chow groups AlgebraicGeometry.Space.chowGroupCanonical of Stacks 0EQ6).

def flatPullbackBivariantClassCanonical :
    BivariantClassCanonical.{u + 1} D sX hX sY hY f hf (-(r : ℤ))

Tag 0ERP, a lemma

Tag 0ERP in the Stacks Project.

AlgebraicGeometry.Space.exists_bivariantClassCanonical_gysin, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/GysinBivariantClass/BivariantClass.lean, line 680. The attribute's comment: “the rule g : X' → X ↦ (i')^ : CH_k(X') → CH_{k-1}(E'), E' = E ×_X X', for X/B good and (L, s, i : E → X) a triple as in 0ER1, is a bivariant class of degree 1 for i; the cap operation is pinned to the Gysin map of the pulled-back triple carried along Z(g^s) ≅ E ×_X X'”.

Stacks 0ERP. In Situation 0EDT let X/B be good and (ℒ, σ, i : E → X) a triple as in Stacks 0ER1: ℒ an invertible 𝒪_X-module, σ a global section and E = Z(σ) its vanishing locus. Then the rule that to g : X' ⟶ X assigns the Gysin map (i')^* : CH_k(X') ⟶ CH_{k-1}(E'), E' = E ×_X X', is a bivariant class of degree 1 for i : E ⟶ X (AlgebraicGeometry.Space.BivariantClassCanonical, Stacks 0ERJ).

The cap operation of the class is pinned down on the class of every k-cycle α on every test space: it is the class of the Gysin cycle of Stacks 0ER1 of the pulled-back triple (g^*ℒ, g^*σ, Z(g^*σ) → X') (Stacks 0ER3) at its own rule, carried to E ×_X X' along the identification e : Z(g^*σ) ≅ E ×_X X' over X' (chowGroupCanonicalPushforwardOfIso). The three axioms of Stacks 0ERJ are Stacks 0ER5, Stacks 0ER6 and Stacks 0ERC for the pulled-back triples, as in the source's proof.

theorem exists_bivariantClassCanonical_gysin :
    ∃ c : BivariantClassCanonical.{u + 1} D (𝔼.1.ι ≫ p) (isGood_vanishingLocus D p hp L σ) p hp
        𝔼.1.ι rfl 1,
      ∀ {X' : D.Ambient} (g : X' ⟶ X) (hX' : D.IsGood (g ≫ p))
        (hE' : D.IsGood (Limits.pullback.fst 𝔼.1.ι g ≫ 𝔼.1.ι ≫ p))
        (e : (vanishingLocus D (g ≫ p) hX'
            (pullbackSheaf D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L)
            (pullbackGlobalSection D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L σ)).1.toFunctor ≅
          Limits.pullback 𝔼.1.ι g)
        (_he : e.hom ≫ Limits.pullback.snd 𝔼.1.ι g =
          (vanishingLocus D (g ≫ p) hX'
            (pullbackSheaf D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L)
            (pullbackGlobalSection D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L σ)).1.ι)
        (hcomp : e.hom ≫ Limits.pullback.fst 𝔼.1.ι g ≫ 𝔼.1.ι ≫ p =
          (vanishingLocus D (g ≫ p) hX'
            (pullbackSheaf D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L)
            (pullbackGlobalSection D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L σ)).1.ι ≫
            g ≫ p)
        (k : ℤ) (α : CycleGroup D (g ≫ p) k),
        c.cap g hX' hE' k (chowClassCanonical.{u + 1} D (g ≫ p) hX' k α) =
          chowGroupCanonicalPushforwardOfIso.{u + 1} D _
            (isGood_vanishingLocus D (g ≫ p) hX' _ _) _ hE' e hcomp (k - 1)
            (chowClassCanonical.{u + 1} D _ (isGood_vanishingLocus D (g ≫ p) hX' _ _) (k - 1)
              (gysinCycleMap D (g ≫ p) (vanishingLocus D (g ≫ p) hX'
                  (pullbackSheaf D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L)
                  (pullbackGlobalSection D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L σ))
                (isGood_vanishingLocus D (g ≫ p) hX' _ _) (k - 1)
                (sectionGysinRule D (g ≫ p) hX' _
                  (pullbackGlobalSection D p hp (g ≫ p) hX' (ObjectProperty.homMk g) L σ)
                  (k - 1)
                  (isInvertibleSheaf_pullbackSheaf D p hp (g ≫ p) hX' (ObjectProperty.homMk g)
                    L hL))
                (cycleGroupDegreeCongr D (g ≫ p) (sub_add_cancel k 1).symm α)))

Tag 0ERQ, a lemma

Tag 0ERQ in the Stacks Project.

AlgebraicGeometry.Space.isProper_bivariantPushforwardBaseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantProperPushforward.lean, line 205. The attribute's comment: “properness of the base change of the proper morphism, so that the rule is defined”.

Stacks 0ERQ. The base change of a proper morphism along a test space is proper, so the rule of Stacks 0ERQ has a proper morphism to push forward along.

theorem isProper_bivariantPushforwardBaseChange (D : BaseSpaceDimensionSituation.{u})
    {X Y Z Z' : D.Ambient} (f : X ⟶ Y) (g : Y ⟶ Z) (q : Z' ⟶ Z)
    (hX : IsAlgebraicSpace D.site D.base D.base_mem X)
    (hY : IsAlgebraicSpace D.site D.base D.base_mem Y)
    (hPX : IsAlgebraicSpace D.site D.base D.base_mem (pullback (f ≫ g) q))
    (hPY : IsAlgebraicSpace D.site D.base D.base_mem (pullback g q))
    (hf : IsProper D.site D.base D.base_mem f) :
    IsProper D.site D.base D.base_mem (bivariantPushforwardBaseChange D f g q)

AlgebraicGeometry.Space.bivariantProperPushforwardCap_proper_naturality, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantProperPushforward.lean, line 337. The attribute's comment: “the proper-pushforward condition of Stacks 0ERJ for the rule, over functoriality of the parameterized proper pushforward”.

Stacks 0ERQ. The proper-pushforward condition of Stacks 0ERJ for the rule of 0ERQ: the cap operation α ↦ f'_*(c ∩ α) commutes with proper pushforward along a morphism of good test spaces over Z. This is the part of the source proof carried by Stacks 0EP2, which enters as functoriality of the parameterized proper pushforward.

theorem bivariantProperPushforwardCap_proper_naturality
    (hfun : G.HasFunctorialProperPushforward) {Z' Z'' : D.Ambient} (q : Z' ⟶ Z)
    (hZ' : D.IsGood (q ≫ sZ)) (hY' : D.IsGood (pullback.fst g q ≫ sY)) (h : Z'' ⟶ Z')
    (hZ'' : D.IsGood ((h ≫ q) ≫ sZ)) (hY'' : D.IsGood (pullback.fst g (h ≫ q) ≫ sY))
    (hh : IsProper D.site D.base D.base_mem h) (h' : pullback g (h ≫ q) ⟶ pullback g q)
    (hsq : IsPullback (pullback.snd g (h ≫ q)) h' h (pullback.snd g q))
    (hfst : h' ≫ pullback.fst g q = pullback.fst g (h ≫ q))
    (hh' : IsProper D.site D.base D.base_mem h') (k : ℤ)
    (α : G.chow ((h ≫ q) ≫ sZ) hZ'' k) :
    bivariantProperPushforwardCap D G sX hX sY hY sZ hZ f g hf hfg hproper hgood p c q hZ'
        hY' k
        (G.properPushforward ((h ≫ q) ≫ sZ) hZ'' (q ≫ sZ) hZ' h (Category.assoc h q sZ).symm
          hh k α) =
      G.properPushforward (pullback.fst g (h ≫ q) ≫ sY) hY'' (pullback.fst g q ≫ sY) hY' h'
        (by rw [← Category.assoc, hfst]) hh' (k - p)
        (bivariantProperPushforwardCap D G sX hX sY hY sZ hZ f g hf hfg hproper hgood p c
          (h ≫ q) hZ'' hY'' k α)

AlgebraicGeometry.Space.eq_of_isBivariantProperPushforward, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantProperPushforward.lean, line 401. The attribute's comment: “uniqueness of the class defined by the rule”.

Stacks 0ERQ. The rule of Stacks 0ERQ determines the pushed-forward class: two bivariant classes of degree p for g that both push c forward along f are equal.

theorem eq_of_isBivariantProperPushforward (hg : g ≫ sZ = sY)
    {d d' : BivariantClass D G sY hY sZ hZ g hg p}
    (hd : IsBivariantProperPushforward D G sX hX sY hY sZ hZ f g hf hfg hproper hgood p c hg d)
    (hd' : IsBivariantProperPushforward D G sX hX sY hY sZ hZ f g hf hfg hproper hgood p c hg
      d') :
    d = d'

AlgebraicGeometry.Space.BivariantClassCanonical.properPushforward, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantProperPushforwardCanonical.lean, line 742. The attribute's comment: “the rule α ↦ f'_(c ∩ α) is a bivariant class of degree p for g: the class f_ ∘ c ∈ A^p(Y → Z) over the constructed Chow groups and operations, with its three axioms proved from those of c, Stacks 0EP2, 0EPD and 0ER5”.

Stacks 0ERQ. In Situation 0EDT let f : X ⟶ Y and g : Y ⟶ Z be morphisms of good algebraic spaces over the base, with f proper, and let c ∈ A^p(X ⟶ Z). The rule which to a test space Z' ⟶ Z assigns α ↦ f'_*(c ∩ α), for f' : Z' ×_Z X ⟶ Z' ×_Z Y the base change of f, is a bivariant class of degree p for g; this is that class f_* ∘ c ∈ A^p(Y ⟶ Z), and AlgebraicGeometry.Space.BivariantClassCanonical.properPushforward_cap_apply says that it caps by the rule.

def BivariantClassCanonical.properPushforward :
    BivariantClassCanonical.{max (u + 1) v} D sY hY sZ hZ g hg p

AlgebraicGeometry.Space.BivariantClassCanonical.properPushforward_cap_apply, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantProperPushforwardCanonical.lean, line 754. The attribute's comment: “the class caps by the rule α ↦ f'_*(c ∩ α)”.

Stacks 0ERQ. The class f_* ∘ c caps by the rule of the source: on a test space z : Z' ⟶ Z, (f_* ∘ c) ∩ α = f'_*(c ∩ α) in CH_{k-p}(Z' ×_Z Y).

theorem BivariantClassCanonical.properPushforward_cap_apply {Z' : D.Ambient} (z : Z' ⟶ Z)
    (hZ' : D.IsGood (z ≫ sZ)) (hY' : D.IsGood (pullback.fst g z ≫ sY)) (k : ℤ)
    (α : chowGroupCanonical.{max (u + 1) v} D (z ≫ sZ) hZ' k) :
    (BivariantClassCanonical.properPushforward D hY hf hg hfP c).cap z hZ' hY' k α =
      chowGroupCanonicalProperPushforward.{max (u + 1) v} D (pullback.fst (f ≫ g) z ≫ sX)
        (isGood_pullback_fst D hX hZ (f ≫ g) z hZ') (pullback.fst g z ≫ sY) hY'
        (bivariantPushforwardBaseChange_comp_structure D hf g z)
        (isProper_bivariantPushforwardBaseChange D f g z hX.1 hY.1
          (isGood_pullback_fst D hX hZ (f ≫ g) z hZ').1 hY'.1 hfP) (k - p)
        (c.cap z hZ' (isGood_pullback_fst D hX hZ (f ≫ g) z hZ') k α)

Tag 0ERR, a lemma

Tag 0ERR in the Stacks Project.

AlgebraicGeometry.Space.firstChernBivariantClass_compCap_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernBivariantCentral.lean, line 70. The attribute's comment: “c_1(L) ∈ A^1(X) commutes with every c ∈ A^p(X): the composites c_1(L) ∘ c and c ∘ c_1(L) have the same cap operation on every test space”.

Stacks 0ERR. In Situation 0EDT let X/B be good and let ℒ be an invertible 𝒪_X-module. Then c₁(ℒ) ∈ A^1(X) (Stacks 0ERM, AlgebraicGeometry.Space.firstChernBivariantClass) commutes with every c ∈ A^p(X): on every good test space g : Y' ⟶ X the composites c₁(ℒ) ∘ c and c ∘ c₁(ℒ) have the same cap operation, c₁(ℒ) ∩ (c ∩ α) = c ∩ (c₁(ℒ) ∩ α) for every α ∈ CH_k(Y'), the two degrees k - p - 1 and k - 1 - p being identified.

theorem firstChernBivariantClass_compCap_comm {p : ℤ}
    (c : BivariantClassCanonical.{u + 1} D sX hX sX hX (𝟙 X) (Category.id_comp sX) p)
    {Y' : D.Ambient} (g : Y' ⟶ X) (hY' : D.IsGood (g ≫ sX)) (k : ℤ)
    (x : chowGroupCanonical.{u + 1} D (g ≫ sX) hY' k) :
    BivariantClassCanonical.compCap D (firstChernBivariantClass D sX hX L hL) c g hY' k x =
      chowGroupCanonicalDegreeCongr D _ _ (by omega)
        (BivariantClassCanonical.compCap D c (firstChernBivariantClass D sX hX L hL) g hY' k
          x)

Tag 0ERS, a lemma

Tag 0ERS in the Stacks Project.

AlgebraicGeometry.Space.zero_capsFundamentalClassesToZero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantVanishing.lean, line 402. The attribute's comment: “the forward implication”.

Stacks 0ERS, the forward implication. The zero class of A^p(X) caps the fundamental class of every integral algebraic space locally of finite type over X to zero.

theorem zero_capsFundamentalClassesToZero :
    BivariantClass.CapsFundamentalClassesToZero D G sX hX p 0

AlgebraicGeometry.Space.cap_integralClosedSubspaceCycle_eq_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantVanishing.lean, line 412. The attribute's comment: “the converse on the cycle of an integral closed subspace”.

Stacks 0ERS, the content of the converse. If c ∈ A^p(X) caps the fundamental class of every integral algebraic space locally of finite type over X to zero, then it caps the cycle of every integral closed subspace of every good test space over X to zero: push forward along the inclusion of the subspace, where the hypothesis applies.

theorem cap_integralClosedSubspaceCycle_eq_zero
    (hpp : G.IsInducedByClosedCyclePushforward) (hbc : D.HasGoodSubspaceBaseChanges sX)
    (c : chowCohomologyComponent D G sX hX p)
    (hc : BivariantClass.CapsFundamentalClassesToZero D G sX hX p c)
    {X' : D.Ambient} (g : X' ⟶ X) (hg : D.IsGood (g ≫ sX))
    (hpb : D.IsGood (pullback.fst (𝟙 X) g ≫ sX))
    (Z : IntegralClosedSubspace D.site D.base D.base_mem X') (k : ℤ)
    (hk : deltaDimension D (g ≫ sX) Z = k) :
    c.cap g hg hpb k
      (G.cycleClass (g ≫ sX) hg k (integralClosedSubspaceCycle D (g ≫ sX) Z k)) = 0

AlgebraicGeometry.Space.eq_zero_iff_capsFundamentalClassesToZero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/BivariantVanishing.lean, line 458. The attribute's comment: “for Chow classes of cycles of finite support, over a cycle-induced pushforward”.

Stacks 0ERS. A class c ∈ A^p(X) is zero if and only if c ∩ [Y] = 0 in CH_*(Y) for every integral algebraic space Y locally of finite type over X.

theorem eq_zero_iff_capsFundamentalClassesToZero
    (hpp : G.IsInducedByClosedCyclePushforward) (hbc : D.HasGoodSubspaceBaseChanges sX)
    (hfin : ∀ {X' : D.Ambient} (g : X' ⟶ X) (hg : D.IsGood (g ≫ sX)) (k : ℤ)
      (α : G.chow (g ≫ sX) hg k), ∃ β : CycleGroup D (g ≫ sX) k,
        (Function.support β.1).Finite ∧ G.cycleClass (g ≫ sX) hg k β = α)
    (c : chowCohomologyComponent D G sX hX p) :
    c = 0 ↔ BivariantClass.CapsFundamentalClassesToZero D G sX hX p c

AlgebraicGeometry.Space.ChowCohomologyClass.eq_zero_iff_capsFundamentalClassesToZero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowCohomologyVanishing.lean, line 505.

Stacks 0ERS. In Situation 0EDT let X/B be good and let c ∈ A^p(X). Then c is zero if and only if c ∩ [Y] = 0 in CH_*(Y) for every integral algebraic space Y locally of finite type over X.

theorem ChowCohomologyClass.eq_zero_iff_capsFundamentalClassesToZero
    (c : ChowCohomologyClass D sX hX p) : c = 0 ↔ c.CapsFundamentalClassesToZero D sX hX

Sources

• Section 82.26, Tag 0ERI
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.BivariantClass · AlgebraicGeometry.Space.BivariantClassCanonical · AlgebraicGeometry.Space.chowCohomologyComponent · AlgebraicGeometry.Space.chowCohomology · AlgebraicGeometry.Space.ChowCohomologyClass · AlgebraicGeometry.Space.ChowCohomologyClass.instGRing · AlgebraicGeometry.Space.ChowCohomologyAlgebra · AlgebraicGeometry.Space.chowCohomologyComponentCanonical

Section82.27 Projective space bundle formula#

Section 82.27 of the Stacks Project, Tag 0ERT. 3 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ERU, a lemma

Tag 0ERU in the Stacks Project.

AlgebraicGeometry.Space.chowClassCanonical_properCyclePushforward_firstChernCapPowCycleMap_flatPullbackMap_projectiveBundle_eq_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProjectiveBundleTopPower.lean, line 471. The attribute's comment: “the vanishing for s < r - 1, read on a cycle representing α, in the Chow group of Stacks 0EQ6 at the canonical embeddings”.

Stacks 0ERU, the vanishing, in the Chow group at the canonical embeddings. Let ℰ be a finite locally free 𝒪_X-module of rank n + 1 on a good algebraic space X/B and (π : P → X, 𝒪_P(1)) its projective bundle. For s < n and every k-cycle α on X, the class of π_*(c₁(𝒪_P(1))^s ∩ π^*α) in CH_{k+n-s}(X) is zero, the Chow group being that of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK.

theorem chowClassCanonical_properCyclePushforward_firstChernCapPowCycleMap_flatPullbackMap_projectiveBundle_eq_zero
    (s : ℕ) (hs : s < n) (k m : ℤ) (h : k + n = m + s) (α : CycleGroup D sX k) :
    chowClassCanonical.{u + 1} D sX hX m
        (properCyclePushforward D _ (isGood_projectiveBundleProj_comp D sX hX E hfree) sX hX rfl
          (isProper_projectiveBundleProj (X := ⟨X, hX.1⟩) E hfree).2.1.2 m
          (firstChernCapPowCycleMap D _ (isGood_projectiveBundleProj_comp D sX hX E hfree)
            (projectiveBundleTwist (X := ⟨X, hX.1⟩) E hfree).obj
            (isInvertibleModule_projectiveBundleTwist (X := ⟨X, hX.1⟩) E hfree) s m
            (flatPullbackMap D _ (isGood_projectiveBundleProj_comp D sX hX E hfree) sX hX
              (projectiveBundleProj (X := ⟨X, hX.1⟩) E hfree).hom rfl n
              (isFlat_projectiveBundleProj (X := ⟨X, hX.1⟩) E n hr hfree)
              (hasRelativeDimension_projectiveBundleProj (X := ⟨X, hX.1⟩) E n hr hfree) k
              (m + s) h α))) = 0

AlgebraicGeometry.Space.chowClassCanonical_properCyclePushforward_firstChernCapPowCycleMap_flatPullbackMap_projectiveBundle, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProjectiveBundleTopPower.lean, line 494. The attribute's comment: “the top power s = r - 1, read on a cycle representing α, in the Chow group of Stacks 0EQ6 at the canonical embeddings”.

Stacks 0ERU, the top power, in the Chow group at the canonical embeddings. Let ℰ be a finite locally free 𝒪_X-module of rank n + 1 on a good algebraic space X/B and (π : P → X, 𝒪_P(1)) its projective bundle. For every k-cycle α on X, the class of π_*(c₁(𝒪_P(1))^n ∩ π^*α) in CH_k(X) is the class of α, the Chow group being that of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK.

theorem chowClassCanonical_properCyclePushforward_firstChernCapPowCycleMap_flatPullbackMap_projectiveBundle
    (k : ℤ) (α : CycleGroup D sX k) :
    chowClassCanonical.{u + 1} D sX hX k
        (properCyclePushforward D _ (isGood_projectiveBundleProj_comp D sX hX E hfree) sX hX rfl
          (isProper_projectiveBundleProj (X := ⟨X, hX.1⟩) E hfree).2.1.2 k
          (firstChernCapPowCycleMap D _ (isGood_projectiveBundleProj_comp D sX hX E hfree)
            (projectiveBundleTwist (X := ⟨X, hX.1⟩) E hfree).obj
            (isInvertibleModule_projectiveBundleTwist (X := ⟨X, hX.1⟩) E hfree) n k
            (flatPullbackMap D _ (isGood_projectiveBundleProj_comp D sX hX E hfree) sX hX
              (projectiveBundleProj (X := ⟨X, hX.1⟩) E hfree).hom rfl n
              (isFlat_projectiveBundleProj (X := ⟨X, hX.1⟩) E n hr hfree)
              (hasRelativeDimension_projectiveBundleProj (X := ⟨X, hX.1⟩) E n hr hfree) k
              (k + n) rfl α))) =
      chowClassCanonical.{u + 1} D sX hX k α

Tag 0ERV, a lemma

Tag 0ERV in the Stacks Project.

AlgebraicGeometry.Space.bijective_projectiveBundleFormulaMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ProjectiveBundleFormulaBijective.lean, line 426.

Stacks 0ERV: the projective space bundle formula. Let X be a good algebraic space over the base of the theory of cycles, ℰ a finite locally free 𝒪_X-module of rank r and (π : P = 𝐏(ℰ) → X, 𝒪_P(1)) its projective bundle. The map ⊕_{i=0}^{r-1} CH_{k+i}(X) → CH_{k+r-1}(P), (α_0, …, α_{r-1}) ↦ π^*α_0 + c₁(𝒪_P(1)) ∩ π^*α_1 + … + c₁(𝒪_P(1))^{r-1} ∩ π^*α_{r-1}, is an isomorphism.

theorem bijective_projectiveBundleFormulaMap
    (F : Modules.{u, u} D.site D.base D.base_mem X.1 X.2) (r : ℕ)
    (hF : RingedSite.IsFiniteLocallyFreeOfRank
      (smallEtaleRingedSite D.site D.base D.base_mem X.1 X.2) F r) (k : ℤ) :
    Function.Bijective (projectiveBundleFormulaMap D sX hX F r hF k)

Tag 0ERW, a lemma

Tag 0ERW in the Stacks Project.

AlgebraicGeometry.Space.bijective_flatPullbackChowMapCanonical_vectorBundleProjection, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/VectorBundleChowPullbackBijective.lean, line 44.

Stacks 0ERW. In Situation 0EDT let X/B be good, let ℰ be a finite locally free 𝒪_X-module of rank r and let p : E = Spec(Sym^*(ℰ)) → X be the associated vector bundle. Then p^* : CH_k(X) → CH_{k+r}(E) is an isomorphism for all k.

theorem bijective_flatPullbackChowMapCanonical_vectorBundleProjection (k : ℤ) :
    letI := AlgebraicGeometry.SheafOfModules.IsFiniteLocallyFreeOfRank.isFiniteLocallyFree F r
    Function.Bijective (flatPullbackChowMapCanonical.{u + 1} D
      ((vectorBundleProjection D.site D.base D.base_mem W F).hom ≫ sW)
      (isGood_vectorBundle D W F sW hW) sW hW
      (vectorBundleProjection D.site D.base D.base_mem W F).hom
      rfl r (isFlat_vectorBundleProjection D.site D.base D.base_mem W F)
      (hasRelativeDimension_vectorBundleProjection D.site D.base D.base_mem W F r) k
      (k + (r : ℤ)) rfl)

Sources

• Section 82.27, Tag 0ERT
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.chowClassCanonical_properCyclePushforward_firstChernCapPowCycleMap_flatPullbackMap_projectiveBundle_eq_zero · AlgebraicGeometry.Space.chowClassCanonical_properCyclePushforward_firstChernCapPowCycleMap_flatPullbackMap_projectiveBundle · AlgebraicGeometry.Space.bijective_projectiveBundleFormulaMap · AlgebraicGeometry.Space.bijective_flatPullbackChowMapCanonical_vectorBundleProjection

Section82.28 The Chern classes of a vector bundle#

Section 82.28 of the Stacks Project, Tag 0ERX. 5 tags here are carried by 10 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ERY, a lemma

Tag 0ERY in the Stacks Project.

AlgebraicGeometry.Space.existsUnique_chernCap_and_bivariant, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassBivariant.lean, line 1200. The attribute's comment: “existence and uniqueness of c_i(E) ∩ - : CH_k(X') → CH_{k-i}(X'), i = 0..r, with c_0 = id and sum_i (-1)^i c_1(O_{P'}(1))^i ∩ π'^*(c_{r-i}(E) ∩ α) = 0 on P' = P ×_X X', for every morphism X' → X of good spaces over B, E finite locally free of rank r on a good X/B; and the maps c_i(E) ∩ - form a bivariant class of degree i (BivariantClassCanonical, over chowGroupCanonical)”.

Stacks 0ERY. In Situation 0EDT let X/B be good, ℰ a finite locally free 𝒪_X-module of rank r and (π : P → X, 𝒪_P(1)) its projective space bundle. For every morphism g : X' → X of good algebraic spaces over B there are unique maps c_j(ℰ) ∩ - : CH_k(X') → CH_{k-j}(X'), j = 0, …, r, such that c₀(ℰ) ∩ α = α and

∑_{i=0}^{r} (-1)^i c₁(𝒪_{P'}(1))^i ∩ π'^*(c_{r-i}(ℰ) ∩ α) = 0

for every α ∈ CH_k(X'), where π' : P' = P ×_X X' → X' is the base change of π and 𝒪_{P'}(1) the pullback of 𝒪_P(1) (AlgebraicGeometry.Space.chernRelationSum). Moreover these maps define a bivariant class c_j(ℰ) of degree j on X: there is C ∈ A^j(X → X) whose cap operation on every test space X' is c_j(ℰ) ∩ -, read on X ×_X X' ≅ X' along the flat pullback of relative dimension 0 (AlgebraicGeometry.Space.testSpacePullbackChowMap).

The Chow groups are those of Stacks 0EQ6 at the canonical embeddings of Stacks 0ENK (AlgebraicGeometry.Space.chowGroupCanonical), the bivariant classes those of Stacks 0ERJ over them (AlgebraicGeometry.Space.BivariantClassCanonical), c₀ ∩ α = α is read through the identification CH_k = CH_{k-0}, and the class is AlgebraicGeometry.Space.chernClass (Stacks 0ERZ).

theorem existsUnique_chernCap_and_bivariant :
    (∀ {Y : D.Ambient} (g : Y ⟶ X) (hY : D.IsGood (g ≫ sX)),
      ∃! c : ∀ (j : Fin (r + 1)) (k : ℤ), chowGroupCanonical.{u + 1} D (g ≫ sX) hY k →+
          chowGroupCanonical.{u + 1} D (g ≫ sX) hY (k - ((j : ℕ) : ℤ)),
        (∀ k α, c 0 k α = ProjectiveBundleRelation.degreeCast
          (fun m => chowGroupCanonical.{u + 1} D (g ≫ sX) hY m) (by simp) α) ∧
          ∀ k α, chernRelationSum D sX hX E r hE g hY c k α = 0) ∧
      ∀ j : Fin (r + 1), ∃ C : chowCohomologyComponentCanonical.{u + 1} D sX hX ((j : ℕ) : ℤ),
        ∀ {Y : D.Ambient} (g : Y ⟶ X) (hY : D.IsGood (g ≫ sX))
          (hXg : D.IsGood (pullback.fst (𝟙 X) g ≫ sX))
          (c : ∀ (j : Fin (r + 1)) (k : ℤ), chowGroupCanonical.{u + 1} D (g ≫ sX) hY k →+
            chowGroupCanonical.{u + 1} D (g ≫ sX) hY (k - ((j : ℕ) : ℤ))),
          (∀ k α, c 0 k α = ProjectiveBundleRelation.degreeCast
            (fun m => chowGroupCanonical.{u + 1} D (g ≫ sX) hY m) (by simp) α) →
          (∀ k α, chernRelationSum D sX hX E r hE g hY c k α = 0) →
          ∀ k α, C.cap g hY hXg k α = testSpacePullbackChowMap D sX hX g hY hXg _ (c j k α)

Tag 0ERZ, a definition

Tag 0ERZ in the Stacks Project.

AlgebraicGeometry.Space.chernClass, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassBivariant.lean, line 1151. The attribute's comment: “the i-th Chern class c_i(E) ∈ A^i(X), i = 0, ..., r, the bivariant class of degree i of Lemma 0ERY, for E finite locally free of rank r on a good X/B; A^i(X) = A^i(X → X) over the Chow groups chowGroupCanonical”.

Stacks 0ERZ: the j-th Chern class c_j(ℰ) ∈ A^j(X) of a finite locally free module ℰ of rank r on a good algebraic space X/B, j = 0, …, r: the bivariant class of degree j of Stacks 0ERY (AlgebraicGeometry.Space.existsUnique_chernCap_and_bivariant), in A^j(X) = A^j(X → X) (AlgebraicGeometry.Space.chowCohomologyComponentCanonical). Its cap operation on a test space g : X' → X is c_j(ℰ) ∩ - on X' (AlgebraicGeometry.Space.chernCap), read on X ×_X X' ≅ X' (AlgebraicGeometry.Space.chernClass_cap).

def chernClass (j : Fin (r + 1)) :
    chowCohomologyComponentCanonical.{u + 1} D sX hX ((j : ℕ) : ℤ)

AlgebraicGeometry.Space.totalChernClass, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassBivariant.lean, line 1174. The attribute's comment: “the total Chern class c(E) = c_0(E) + c_1(E) + ... + c_r(E), the formal sum of the Chern classes as a nonhomogeneous class in A^*(X) = ⊕_p A^p(X)”.

Stacks 0ERZ: the total Chern class c(ℰ) = c₀(ℰ) + c₁(ℰ) + … + c_r(ℰ), the formal sum of the Chern classes of ℰ (AlgebraicGeometry.Space.chernClass), a nonhomogeneous class in the Chow cohomology A^*(X) = ⊕_p A^p(X) (AlgebraicGeometry.Space.chowCohomologyCanonical).

def totalChernClass : chowCohomologyCanonical.{u + 1} D sX hX

Tag 0ES0, a lemma

Tag 0ES0 in the Stacks Project.

AlgebraicGeometry.Space.chernCap_one_eq_firstChernBivariantClass_cap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/FirstChernClassBivariantComparison.lean, line 83. The attribute's comment: “c_1(L) ∈ A^1(X) of Definition 0ERZ (chernCap 1 on every good Y' → X) equals the bivariant class c_1(f^*L) ∩ - of Lemma 0ERM, for X/B good and L invertible”.

Stacks 0ES0. In Situation 0EDT let X/B be good and let ℒ be an invertible 𝒪_X-module. Then the first Chern class c₁(ℒ) of Definition 0ERZ equals the bivariant class of Stacks 0ERM (AlgebraicGeometry.Space.firstChernBivariantClass).

Both are classes in A^1(X), and the equality is that of their operations on the Chow groups of every good test space g : Y' ⟶ X: for every k and α ∈ CH_k(Y'), the cap operation CH_k(Y') ⟶ CH_{k-1}(X ×_X Y') of the class of Stacks 0ERM, followed by the identification of X ×_X Y' with Y' (pushforward along the isomorphism X ×_X Y' ⟶ Y'), is c₁(g^*ℒ) ∩ α for the first Chern class of Definition 0ERZ of the rank-one module g^*ℒ on Y' (AlgebraicGeometry.Space.PrivateChern.chernCap, the operation on CH_*(Y') = ⨁_k CH_k(Y'); chowGroupCanonicalEquivChowGroup identifies the Chow groups of Stacks 0EQ6 on which the bivariant classes act with the summands of CH_*(Y')).

theorem chernCap_one_eq_firstChernBivariantClass_cap
    (L : Modules.{u, u} D.site D.base D.base_mem X hX.1)
    (hL : IsInvertibleModule (⟨X, hX.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) L)
    {Y' : D.Ambient} (g : Y' ⟶ X) (hY' : D.IsGood (g ≫ sX))
    (hX' : D.IsGood (pullback.fst (𝟙 X) g ≫ sX)) (k : ℤ)
    (α : chowGroupCanonical.{u + 1} D (g ≫ sX) hY' k) :
    properPushforwardChow.{u, u + 1} D (pullback.fst (𝟙 X) g ≫ sX) hX' (g ≫ sX) hY'
        (pullback.snd (𝟙 X) g) (by rw [← Category.assoc, ← pullback.condition, Category.comp_id])
        (isProper_of_isIso D.site D.base D.base_mem (pullback.snd (𝟙 X) g))
        (DirectSum.of _ (k - 1) (chowGroupCanonicalEquivChowGroup.{u + 1} D _ hX' (k - 1)
          ((firstChernBivariantClass D sX hX L hL).cap g hY' hX' k α))) =
      chernCap.{u, u + 1} (hX := hY') (pullbackLine D (g ≫ sX) hY' sX hX g L) 1
        (isFiniteLocallyFreeOfRank_one_pullbackLine D sX hX g hY' L hL) 1
        (DirectSum.of _ k (chowGroupCanonicalEquivChowGroup.{u + 1} D _ hY' k α))

Tag 0ES1, a lemma

Tag 0ES1 in the Stacks Project.

AlgebraicGeometry.Space.ChowCohomologyClass.chernCap_commute_capChowHomology, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassCommutation.lean, line 462. The attribute's comment: “the main assertion: for every good test space g : X' → X, every finite locally free E' of rank r on X' (in particular E' = g^E) and every j, the cap operation of c ∈ A^p(X) on CH_(X') commutes with c_j(E') ∩ -; this is c ∘ c_j(E) = c_j(E) ∘ c in A^*(X). The source's c_j(L) is a typo for c_j(E)”.

Stacks 0ES1. In Situation 0EDT let X/B be good and let c ∈ A^p(X) (Stacks 0ERK, AlgebraicGeometry.Space.ChowCohomologyClass). Then c commutes with the Chern classes of Stacks 0ERZ (AlgebraicGeometry.Space.PrivateChern.chernCap): for every good test space g : X' ⟶ X, every finite locally free 𝒪_{X'}-module ℰ' of rank r and every j,

c ∩ (c_j(ℰ') ∩ α) = c_j(ℰ') ∩ (c ∩ α) for all α ∈ CH_*(X'),

where c ∩ - is the cap operation of c on X' (capChowHomology). For a finite locally free ℰ of rank r on X the bivariant class c_j(ℰ) ∈ A^j(X) caps on X' as c_j(ℰ|_{X'}) ∩ -, so taking ℰ' = ℰ|_{X'} this is the identity c ∘ c_j(ℰ) = c_j(ℰ) ∘ c in A^*(X) (two bivariant classes being equal when their cap operations on all test spaces are); the statement here allows any ℰ' on X'.

theorem chernCap_commute_capChowHomology {Y : D.Ambient} (g : Y ⟶ X) (hY : D.IsGood (g ≫ sX))
    (E : Modules.{u, u} D.site D.base D.base_mem Y hY.1) (r : ℕ)
    (hE : PrivateChern.IsFiniteLocallyFreeOfRank D (g ≫ sX) hY E r) (j : ℕ) :
    Commute (chernCap.{u, u + 1} E r hE j) (c.capChowHomology g hY)

AlgebraicGeometry.Space.chernCap_chernCap_comm, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassCommutation.lean, line 523. The attribute's comment: “in particular: c_i(E) ∩ c_j(F) ∩ α = c_j(F) ∩ c_i(E) ∩ α for every α ∈ CH_k(X), as an identity in CH_*(X) = ⊕_k CH_k(X) (hence of the components in CH_{k-i-j}(X))”.

Stacks 0ES1, in particular. In Situation 0EDT let X/B be good and let ℰ, ℱ be finite locally free 𝒪_X-modules of ranks r and s. Then

c_i(ℰ) ∩ c_j(ℱ) ∩ α = c_j(ℱ) ∩ c_i(ℰ) ∩ α

for all α ∈ CH_k(X), the Chern classes being those of Stacks 0ERZ (AlgebraicGeometry.Space.PrivateChern.chernCap), operations on CH_*(X) = ⨁_k CH_k(X) carrying CH_k(X) to CH_{k-n}(X). The identity is stated in CH_*(X), where α is placed in degree k; both sides lie in the summand CH_{k-i-j}(X), and an identity in CH_*(X) is in particular one of its components in CH_{k-i-j}(X).

theorem chernCap_chernCap_comm (hX : D.IsGood sX)
    (E F : Modules.{u, u} D.site D.base D.base_mem X hX.1) (r s : ℕ)
    (hE : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX E r)
    (hF : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX F s)
    (i j : ℕ) (k : ℤ) (α : chowGroup.{u + 1} D sX hX k) :
    chernCap.{u, u + 1} E r hE i
        (chernCap.{u, u + 1} F s hF j (DirectSum.of (fun k => chowGroup.{u + 1} D sX hX k) k α)) =
      chernCap.{u, u + 1} F s hF j
        (chernCap.{u, u + 1} E r hE i (DirectSum.of (fun k => chowGroup.{u + 1} D sX hX k) k α))

Tag 0ES2, a remark

Tag 0ES2 in the Stacks Project.

AlgebraicGeometry.Space.chernClassOfFiniteLocallyFree, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassNonConstantRank.lean, line 280. The attribute's comment: “the definition: c_i(E) ∈ A^i(X) for E finite locally free of non-constant rank, the bivariant class preserving CH_(X') = ∏_r CH_(X'_r) and acting by c_i(E|_{X_r}) ∩ - on the factors”.

Stacks 0ES2: the i-th Chern class c_i(ℰ) ∈ A^i(X) of a finite locally free module ℰ of possibly non-constant rank on a good algebraic space X/B: the bivariant class of degree i whose cap operation on a good test space g : X' ⟶ X preserves the product decomposition CH_*(X') = ∏_r CH_*(X'_r) along the pullback X'_r = g^{-1}(X_r) of the rank decomposition X = ∐_r X_r of ℰ and acts on the factor of X'_r by c_i(ℰ|_{X_r}) ∩ -, the Chern class of Stacks 0ERZ of the module ℰ|_{X_r} of constant rank r (zero for i > r) (AlgebraicGeometry.Space.chernCapOfFiniteLocallyFree); the cap operation is read on X ×_X X' ≅ X' (AlgebraicGeometry.Space.chernClassOfFiniteLocallyFree_cap).

def chernClassOfFiniteLocallyFree (i : ℕ) :
    chowCohomologyComponentCanonical.{u + 1} D sX hX (i : ℤ)

AlgebraicGeometry.Space.rankDecomposition_chowGroup_chernClassOfFiniteLocallyFree, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassNonConstantRank.lean, line 322. The attribute's comment: “the whole remark: X = ∐_r X_r by the rank of E (open and closed), its pullback to every morphism X' → X of good spaces over B, CH_k(X') = ∏_r CH_k(X'_r) on Chow groups, and c_i(E) ∈ A^i(X) as the unique bivariant class preserving these decompositions and acting by c_i(E|_{X_r}) ∩ - on the factors”.

Stacks 0ES2. In Situation 0EDT let X/B be good and let ℰ be a finite locally free 𝒪_X-module, of possibly non-constant rank.

1. X = X_0 ⨿ X_1 ⨿ X_2 ⨿ …, where X_r ⊆ X is the open and closed subspace where ℰ has rank r.
2. For every morphism g : X' ⟶ X of good algebraic spaces over B, the inverse images X'_r = g^{-1}(X_r) form the corresponding decomposition of X', and
3. the restrictions to the pieces identify CH_k(X') with ∏_r CH_k(X'_r) for every k.
4. For every i, c_i(ℰ) ∈ A^i(X) (AlgebraicGeometry.Space.chernClassOfFiniteLocallyFree) is a bivariant class which preserves these direct product decompositions and acts by c_i(ℰ|_{X_r}) ∩ - on the factors (AlgebraicGeometry.Space.rankPieceChernCap: the Chern class of Stacks 0ERZ of ℰ|_{X_r}, of constant rank r, and 0 for i > r), and it is the only bivariant class of degree i with this property.

The cap operation of a class C ∈ A^i(X) on X' lands in CH_{k-i}(X ×_X X'), which is read on X' along the flat pullback by the isomorphism X ×_X X' ≅ X' (AlgebraicGeometry.Space.testSpacePullbackChowMap).

theorem rankDecomposition_chowGroup_chernClassOfFiniteLocallyFree :
    IsClopenSubspaceDecomposition D sX
        (fun r : ULift.{u} ℕ => (rankSubspace (Y := ⟨X, hX.1⟩) E r.down).ι) ∧
      (∀ {Y : D.Ambient} (g : Y ⟶ X) (hY : D.IsGood (g ≫ sX)),
        ∃ hdec : IsClopenSubspaceDecomposition D (g ≫ sX)
            (fun r : ULift.{u} ℕ => ((rankSubspace (Y := ⟨X, hX.1⟩) E r.down).preimage g).ι),
          ∀ k : ℤ, Function.Bijective (chowGroupCanonicalDecompositionRestrict hdec k)) ∧
      ∀ i : ℕ,
        let P : chowCohomologyComponentCanonical.{u + 1} D sX hX (i : ℤ) → Prop := fun C =>
          ∀ {Y : D.Ambient} (g : Y ⟶ X) (hY : D.IsGood (g ≫ sX))
            (hXg : D.IsGood (pullback.fst (𝟙 X) g ≫ sX)) (k : ℤ)
            (α : chowGroupCanonical.{u + 1} D (g ≫ sX) hY k),
            ∃ β : chowGroupCanonical.{u + 1} D (g ≫ sX) hY (k - (i : ℤ)),
              C.cap g hY hXg k α = testSpacePullbackChowMap D sX hX g hY hXg _ β ∧
                ∀ r : ULift.{u} ℕ,
                  chowGroupCanonicalPieceRestrict
                      ((isClopenSubspaceDecomposition_rankSubspace D sX hX E hE).preimage g hY) r
                      (k - (i : ℤ)) β =
                    rankPieceChernCap D sX hX E hE i r
                      ((rankSubspace (Y := ⟨X, hX.1⟩) E r.down).fromPreimage g)
                      (((isClopenSubspaceDecomposition_rankSubspace D sX hX E hE).preimage g
                        hY).isGoodPiece r) k
                      (chowGroupCanonicalPieceRestrict
                        ((isClopenSubspaceDecomposition_rankSubspace D sX hX E hE).preimage g hY)
                        r k α)
        P (chernClassOfFiniteLocallyFree D sX hX E hE i) ∧
          ∀ C, P C → C = chernClassOfFiniteLocallyFree D sX hX E hE i

AlgebraicGeometry.Space.isClopenSubspaceDecomposition_preimage, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ClopenDecomposition.lean, line 279. The attribute's comment: “the decomposition of X' obtained by pullback”.

A decomposition of a good algebraic space over the base into good open and closed subspaces pulls back along a morphism: if X = V_0 ⊔ V_1 ⊔ V_2 ⊔ … is such a decomposition and f : X' ⟶ X is a morphism whose source is good and whose preimages of the pieces are good, then the preimages form such a decomposition of X'.

theorem isClopenSubspaceDecomposition_preimage {sX : X ⟶ D.space} (f : X' ⟶ X)
    (h : IsClopenSubspaceDecomposition D sX fun i => (V i).ι) (hX' : D.IsGood sX')
    (hpiece : ∀ i, D.IsGood (((V i).preimage f).ι ≫ sX')) :
    IsClopenSubspaceDecomposition D sX' fun i => ((V i).preimage f).ι

AlgebraicGeometry.Space.cycleGroupDecompositionEquiv, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ClopenDecomposition.lean, line 398. The attribute's comment: “the clause CH_(X') = ∏ CH_(X'_r), at the level of cycles”.

Along a decomposition X = X_0 ⊔ X_1 ⊔ X_2 ⊔ … of a good algebraic space over the base into good open and closed subspaces, the groups of k-cycles form a direct product Z_k(X) = ∏_i Z_k(X_i), the projections being restriction along the inclusions of the pieces.

noncomputable def cycleGroupDecompositionEquiv (h : IsClopenSubspaceDecomposition D sX j)
    (k : ℤ) : CycleGroup D sX k ≃+ ∀ i, CycleGroup D (j i ≫ sX) k

Sources

• Section 82.28, Tag 0ERX
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.existsUnique_chernCap_and_bivariant · AlgebraicGeometry.Space.chernClass · AlgebraicGeometry.Space.totalChernClass · AlgebraicGeometry.Space.chernCap_one_eq_firstChernBivariantClass_cap · AlgebraicGeometry.Space.ChowCohomologyClass.chernCap_commute_capChowHomology · AlgebraicGeometry.Space.chernCap_chernCap_comm · AlgebraicGeometry.Space.chernClassOfFiniteLocallyFree · AlgebraicGeometry.Space.rankDecomposition_chowGroup_chernClassOfFiniteLocallyFree

Section82.29 Polynomial relations among Chern classes#

Section 82.29 of the Stacks Project, Tag 0ES3. 1 tag here is carried by 1 declaration of the repository (in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ES4, a lemma

Tag 0ES4 in the Stacks Project.

AlgebraicGeometry.Space.chernCap_tensorObj_of_isInvertibleModule, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassInvertibleTwist.lean, line 320.

Stacks 0ES4, the Chern classes of a twist by an invertible module. In Situation 0EDT let X/B be good, let ℰ be a finite locally free 𝒪_X-module of rank r and let ℒ be an invertible 𝒪_X-module; ℰ ⊗ ℒ is finite locally free of rank r. Then for every i

c_i(ℰ ⊗ ℒ) = ∑_{j=0}^{i} C(r + j - i, j) c_{i-j}(ℰ) c₁(ℒ)^j.

For i ≤ r the binomial coefficient is the source's C(r - i + j, j); for i > r both sides vanish, which is the convention c_i = 0 above the rank, the left side by vanishing above the rank and the right side termwise (c_{i-j}(ℰ) = 0 for i - j > r and C(r + j - i, j) = 0 otherwise).

The Chern classes are those of Stacks 0ERZ, c_n(ℰ) ∩ - on CH_*(X) (chernCap). An identity in A^*(X) is an identity of the operations on CH_*(Y) for every good Y ⟶ X; the operation of c_n(ℰ) on CH_*(Y) is c_n(f^*ℰ) ∩ -, and f^*(ℰ ⊗ ℒ) ≅ f^*ℰ ⊗ f^*ℒ with f^*ℰ of rank r and f^*ℒ invertible, so the statement for all good X is the statement in A^*(X).

theorem chernCap_tensorObj_of_isInvertibleModule {X : D.Ambient} {sX : X ⟶ D.space}
    (hX : D.IsGood sX) {E L : Modules.{u, u} D.site D.base D.base_mem X hX.1} {r : ℕ}
    (hE : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX E r)
    (hL : IsInvertibleModule (goodSpace hX) L) (i : ℕ) :
    chernCap.{u, u + 1} (E ⊗ L) r (isFiniteLocallyFreeOfRank_tensorObj_of_isInvertibleModule hE hL)
        i =
      ∑ j ∈ Finset.range (i + 1), ((r + j - i).choose j : ℤ) •
        (chernCap.{u, u + 1} E r hE (i - j) *
          chernCap.{u, u + 1} L 1
            (isFiniteLocallyFreeOfRank_one_of_isInvertibleModule (goodSpace hX) hL) 1 ^ j)

Sources

• Section 82.29, Tag 0ES3
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.chernCap_tensorObj_of_isInvertibleModule

Section82.30 Additivity of Chern classes#

Section 82.30 of the Stacks Project, Tag 0ES6. 4 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ES7, a lemma

Tag 0ES7 in the Stacks Project.

AlgebraicGeometry.Space.chernCap_eq_of_shortExact_tensorUnit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassTrivialSubmodule.lean, line 80.

Stacks 0ES7. In Situation 0EDT let X/B be good and let ℰ, ℱ be finite locally free 𝒪_X-modules of ranks r + 1 and r fitting into a short exact sequence 0 ⟶ 𝒪_X ⟶ ℰ ⟶ ℱ ⟶ 0. Then

c_{r+1}(ℰ) = 0 and c_j(ℰ) = c_j(ℱ) for j = 0, …, r.

The Chern classes are those of Stacks 0ERZ, c_n(ℰ) ∩ - on CH_*(X) (chernCap), and 𝒪_X is the unit of the tensor product of 𝒪_X-modules (AlgebraicGeometry.Space.modulesTensorUnit_eq). A relation in A^*(X) is equality of the operations on CH_*(Y) for every good Y ⟶ X (Section 0ES3); the pullback of the sequence to Y is again such a sequence (with 𝒪_Y as first term and the same ranks), so the statement over every good X is the relation in A^*(X).

theorem chernCap_eq_of_shortExact_tensorUnit (hX : D.IsGood sX)
    {E F : Modules.{u, u} D.site D.base D.base_mem X hX.1}
    (i : 𝟙_ (Modules.{u, u} D.site D.base D.base_mem X hX.1) ⟶ E) (q : E ⟶ F) (hiq : i ≫ q = 0)
    (hC : (ShortComplex.mk i q hiq).ShortExact) (r : ℕ)
    (hE : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX E (r + 1))
    (hF : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX F r) :
    chernCap.{u, u + 1} E (r + 1) hE (r + 1) = 0 ∧
      ∀ j ≤ r, chernCap.{u, u + 1} E (r + 1) hE j = chernCap.{u, u + 1} F r hF j

Tag 0ES8, a lemma

Tag 0ES8 in the Stacks Project.

AlgebraicGeometry.Space.PrivateChern.chernCap_succ_eq_of_shortExact, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/PrivateChern/ChernCapInvertibleSubmodule.lean, line 306.

Stacks 0ES8. Let X/B be a good algebraic space (Situation 0EDT) and 0 ⟶ ℒ ⟶ ℰ ⟶ ℱ ⟶ 0 a short exact sequence of 𝒪_X-modules with ℒ invertible and ℰ, ℱ finite locally free of ranks r + 1 and r. Then c(ℰ) = c(ℒ) c(ℱ), degree by degree:

c_{n+1}(ℰ) ∩ - = c_{n+1}(ℱ) ∩ - + c₁(ℒ) ∩ (c_n(ℱ) ∩ -) on CH_*(X).

A relation between bivariant classes in A^*(X) is equality of the operations on CH_*(Y) for every good Y ⟶ X (Section 0ES3); since the pullback of the sequence to Y is again such a sequence, the statement over every good X is the relation in A^*(X).

theorem chernCap_succ_eq_of_shortExact {L E F : Modules.{u, u} D.site D.base D.base_mem X hX.1}
    (i : L ⟶ E) (q : E ⟶ F) (hiq : i ≫ q = 0) (hC : (ShortComplex.mk i q hiq).ShortExact)
    (hL : IsInvertibleModule (goodSpace hX) L) (r : ℕ)
    (hE : IsFiniteLocallyFreeOfRank D sX hX E (r + 1)) (hF : IsFiniteLocallyFreeOfRank D sX hX F r)
    (n : ℕ) :
    chernCap.{u, u + 1} E (r + 1) hE (n + 1) =
      chernCap.{u, u + 1} F r hF (n + 1) +
        chernCap.{u, u + 1} L 1 (isFiniteLocallyFreeOfRank_one_of_isInvertibleModule
          (goodSpace hX) hL) 1 * chernCap.{u, u + 1} F r hF n

Tag 0ES9, a lemma

Tag 0ES9 in the Stacks Project.

AlgebraicGeometry.Space.totalChernCap_shortExact, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassWhitney.lean, line 45.

Stacks 0ES9, the Whitney formula. In Situation 0EDT let X/B be good and let 0 ⟶ ℰ₁ ⟶ ℰ ⟶ ℰ₂ ⟶ 0 be a short exact sequence T of finite locally free 𝒪_X-modules, ℰ₁ of rank r₁ and ℰ₂ of rank r₂; then ℰ is finite locally free of rank r₁ + r₂ (AlgebraicGeometry.RingedSite.IsFiniteLocallyFreeOfRank.of_shortExact). The total Chern classes satisfy c(ℰ) = c(ℰ₁) c(ℰ₂).

The Chern classes are those of Stacks 0ERZ, c_n(ℰ) ∩ - on CH_*(X) (chernCap), and c(ℰ) = c_0(ℰ) + ⋯ + c_r(ℰ) (totalChernCap). An identity in A^*(X) is an identity of the operations on CH_*(Y) for every good Y ⟶ X; the operation of c_n(ℰ) on CH_*(Y) is c_n(f^*ℰ) ∩ -, and the inverse image along f of a short exact sequence of finite locally free modules is again one, of modules of the same ranks, so the statement for all good X is the statement in A^*(X).

theorem totalChernCap_shortExact {X : D.Ambient} {sX : X ⟶ D.space} (hX : D.IsGood sX)
    {T : ShortComplex (Modules.{u, u} D.site D.base D.base_mem X hX.1)} (hT : T.ShortExact)
    {r₁ r₂ : ℕ} (h₁ : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX T.X₁ r₁)
    (h₂ : PrivateChern.IsFiniteLocallyFreeOfRank D sX hX T.X₃ r₂) :
    totalChernCap.{u, u + 1} T.X₂ (r₁ + r₂)
        (RingedSite.IsFiniteLocallyFreeOfRank.of_shortExact _ hT h₁ h₂) =
      totalChernCap.{u, u + 1} T.X₁ r₁ h₁ * totalChernCap.{u, u + 1} T.X₃ r₂ h₂

Tag 0ESA, a lemma

Tag 0ESA in the Stacks Project.

AlgebraicGeometry.Space.PrivateChern.totalChernCap_eq_prod_one_add_chernCap_one, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/PrivateChern/TotalChernClassFiltration.lean, line 51.

Stacks 0ESA, the total Chern class of a module with a filtration with invertible quotients. In Situation 0EDT let X/B be good, let ℒ_1, …, ℒ_r be invertible 𝒪_X-modules and let ℰ be a finite locally free 𝒪_X-module of rank r with a filtration 0 = ℰ_0 ⊂ ℰ_1 ⊂ ⋯ ⊂ ℰ_r = ℰ such that ℰ_i / ℰ_{i-1} ≅ ℒ_i (a monotone family F of subobjects of ℰ with F 0 = ⊥, F r = ⊤ and identifications e i of the successive quotients). With x_i = c₁(ℒ_i), c(ℰ) = ∏_{i=1}^r (1 + x_i).

The Chern classes are those of Stacks 0ERZ, c_n(ℰ) ∩ - on CH_*(X) (chernCap), and c(ℰ) = c_0(ℰ) + ⋯ + c_r(ℰ) (totalChernCap). An identity in A^*(X) is an identity of the operations on CH_*(Y) for every good Y ⟶ X; the operation of c_n(ℰ) on CH_*(Y) is c_n(f^*ℰ) ∩ -, and the inverse image along f of the filtration is again a filtration of f^*ℰ with invertible quotients f^*ℒ_i, so the statement for all good X is the statement in A^*(X).

theorem totalChernCap_eq_prod_one_add_chernCap_one
    {r : ℕ}
    (L : Fin r → Modules.{u, u} D.site D.base D.base_mem X hX.1)
    (hL : ∀ i, IsInvertibleModule (goodSpace hX) (L i))
    (E : Modules.{u, u} D.site D.base D.base_mem X hX.1)
    (hE : IsFiniteLocallyFreeOfRank D sX hX E r)
    (F : Fin (r + 1) → Subobject E) (hF : Monotone F) (h0 : F 0 = ⊥) (hr : F (Fin.last r) = ⊤)
    (e : ∀ i : Fin r, cokernel (ofLE (F i.castSucc) (F i.succ) (hF i.castSucc_le_succ)) ≅ L i) :
    totalChernCap.{u, u + 1} E r hE = (List.ofFn fun i ↦ 1 +
      chernCap.{u, u + 1} (L i) 1
        (isFiniteLocallyFreeOfRank_one_of_isInvertibleModule (goodSpace hX) (hL i)) 1).prod

Sources

• Section 82.30, Tag 0ES6
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.chernCap_eq_of_shortExact_tensorUnit · AlgebraicGeometry.Space.PrivateChern.chernCap_succ_eq_of_shortExact · AlgebraicGeometry.Space.totalChernCap_shortExact · AlgebraicGeometry.Space.PrivateChern.totalChernCap_eq_prod_one_add_chernCap_one

Section82.31 The splitting principle#

Section 82.31 of the Stacks Project, Tag 0ESB. 3 tags here are carried by 3 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ESC, a lemma

Tag 0ESC in the Stacks Project.

AlgebraicGeometry.Space.exists_projectiveBundleTower_hasInjectiveChowPullbacks_filtration, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/SplittingPrinciple.lean, line 68.

Stacks 0ESC, the splitting principle. Let X/B be good and ℰ_i, i ∈ ι finite, finite locally free 𝒪_X-modules of ranks r_i. There are a good algebraic space P/B and a morphism π : P ⟶ X over B, flat of some relative dimension d and a finite composite of projective bundle projections, such that

• for every morphism Y ⟶ X of good algebraic spaces over B the flat pullback CH_k(Y) ⟶ CH_{k+d}(Y ×_X P) is injective (HasInjectiveChowPullbacks), and
• every π^*ℰ_i has a filtration 0 = F_0 ⊆ F_1 ⊆ ⋯ ⊆ F_{r_i} = π^*ℰ_i by subobjects whose successive quotients F_{j+1}/F_j are invertible 𝒪_P-modules.

The Chow groups are those whose rational equivalences are indexed by families in the universe u + 1 of the integral closed subspaces of the spaces involved, which is the universe in which rational equivalence on algebraic spaces over the universe-u site is defined.

theorem exists_projectiveBundleTower_hasInjectiveChowPullbacks_filtration
    {ι : Type*} [Finite ι] (X : (BigFppfOver D.site D.base D.base_mem)ᵒᵖ ⥤ Type u)
    (sX : X ⟶ D.space) (hX : D.IsGood sX)
    (E : ι → Modules.{u, u} D.site D.base D.base_mem X hX.1) (r : ι → ℕ)
    (hE : ∀ i, PrivateChern.IsFiniteLocallyFreeOfRank D sX hX (E i) (r i)) :
    ∃ (P : (BigFppfOver D.site D.base D.base_mem)ᵒᵖ ⥤ Type u) (sP : P ⟶ D.space)
      (hP : D.IsGood sP) (π : P ⟶ X) (hπ : π ≫ sX = sP)
      (d : ℕ) (hflat : IsFlat D.site D.base D.base_mem π)
      (hd : HasRelativeDimension d D.site D.base D.base_mem π),
      IsProjectiveBundleTower π ∧ HasInjectiveChowPullbacks.{u + 1} D sX hX sP hP π hπ d hflat hd ∧
        ∀ i, ∃ (F : Fin (r i + 1) → Subobject ((spaceModulePullback D hX.1 hP.1 π).obj (E i)))
          (hF : Monotone F), F 0 = ⊥ ∧ F (Fin.last (r i)) = ⊤ ∧
            ∀ j : Fin (r i), IsInvertibleModule ⟨P, hP.1⟩
              (cokernel (Subobject.ofLE (F j.castSucc) (F j.succ) (hF j.castSucc_le_succ)))

Tag 0ESD, a lemma

Tag 0ESD in the Stacks Project.

AlgebraicGeometry.Space.PrivateChern.chernCap_moduleDual, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/PrivateChern/ChernClassDual.lean, line 49.

Stacks 0ESD: the Chern classes of the dual. Let X/B be good and ℰ a finite locally free 𝒪_X-module with dual ℰ^∨ = ℋom(ℰ, 𝒪_X). Then c_i(ℰ^∨) = (-1)^i c_i(ℰ) in A^i(X): for every morphism f : Y ⟶ X of good algebraic spaces over B along which f^*ℰ has rank r, c_i(f^*ℰ^∨) ∩ - = (-1)^i c_i(f^*ℰ) ∩ - on CH_*(Y).

theorem chernCap_moduleDual (E : Modules.{u, u} D.site D.base D.base_mem X hX.1)
    (hE : SheafOfModules.IsFiniteLocallyFreeOnSite E) {Y : D.Ambient} (f : Y ⟶ X)
    (hY : D.IsGood (f ≫ sX)) (r : ℕ)
    (hr : IsFiniteLocallyFreeOfRank D (f ≫ sX) hY ((spaceModulePullback D hX.1 hY.1 f).obj E) r)
    (i : ℕ) :
    chernCap.{u, u + 1} (hX := hY)
        ((spaceModulePullback D hX.1 hY.1 f).obj ((ihom E).obj (𝟙_ _))) r
        (isFiniteLocallyFreeOfRank_modulePullback_dual
          (ObjectProperty.homMk f : (⟨Y, hY.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
            ⟨X, hX.1⟩) E hE hr) i =
      (-1 : ℤ) ^ i •
        chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj E) r hr i

Tag 0ESE, a lemma

Tag 0ESE in the Stacks Project.

AlgebraicGeometry.Space.PrivateChern.chernCap_tensorObj, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChernClassTensorUnconditional.lean, line 64. The attribute's comment: “c₂ corrected to the degree-two part of ∏ (1 + x_i + y_j); the printed c₂ is false”.

Stacks 0ESE. For finite locally free ℰ, ℱ of ranks r, s on a good algebraic space X/B:

• on every good f : Y ⟶ X, as operations on CH_*(Y), c₁(f^*(ℰ ⊗ ℱ)) = r c₁(f^*ℱ) + s c₁(f^*ℰ) and c₂(f^*(ℰ ⊗ ℱ)) = r c₂(f^*ℱ) + s c₂(f^*ℰ) + C(r,2) c₁(f^*ℱ)² + (r s - 1) c₁(f^*ℱ) c₁(f^*ℰ) + C(s,2) c₁(f^*ℰ)²;
• "and so on (see proof)": on a splitting π : P ⟶ X (flat, with injective flat pullback after every base change) with invertible ℒ_i, 𝒩_j and x_i = c₁(ℒ_i), y_j = c₁(𝒩_j), c(π^*ℰ) = ∏ (1 + x_i), c(π^*ℱ) = ∏ (1 + y_j) and c(π^*(ℰ ⊗ ℱ)) = ∏_{i,j} (1 + x_i + y_j).

The source prints c₂(ℰ ⊗ ℱ) = r² c₂(ℱ) + r s c₁(ℱ) c₁(ℰ) + s² c₂(ℰ), which is false: for r = s = 1, ℰ = 𝒪(1, 0) and ℱ = 𝒪(0, 1) on 𝐏¹ × 𝐏¹, ℰ ⊗ ℱ is invertible so c₂(ℰ ⊗ ℱ) = 0, while c₁(ℱ) c₁(ℰ) ∩ [𝐏¹ × 𝐏¹] is the class of a point, which is not zero. The identity stated here is the degree two part of the source's own product formula ∏_{i,j} (1 + x_i + y_j) (Stacks 0FA6; Fulton, Intersection theory, Example 3.2.2).

theorem chernCap_tensorObj
    (E F : Modules.{u, u} D.site D.base D.base_mem X hX.1) (r s : ℕ)
    (hr : IsFiniteLocallyFreeOfRank D sX hX E r) (hs : IsFiniteLocallyFreeOfRank D sX hX F s) :
    (∀ {Y : D.Ambient} (f : Y ⟶ X) (hY : D.IsGood (f ≫ sX)),
      let hrY := isFiniteLocallyFreeOfRank_modulePullback D.site D.base D.base_mem
        (ObjectProperty.homMk f : (⟨Y, hY.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
          ⟨X, hX.1⟩) hr
      let hsY := isFiniteLocallyFreeOfRank_modulePullback D.site D.base D.base_mem
        (ObjectProperty.homMk f : (⟨Y, hY.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
          ⟨X, hX.1⟩) hs
      let hrsY := isFiniteLocallyFreeOfRank_modulePullback D.site D.base D.base_mem
        (ObjectProperty.homMk f : (⟨Y, hY.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
          ⟨X, hX.1⟩) (isFiniteLocallyFreeOfRank_tensorObj D.site D.base D.base_mem X hX.1 hr hs)
      chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj (E ⊗ F)) (r * s)
          hrsY 1 =
          (r : ℤ) • chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj F) s
              hsY 1 +
            (s : ℤ) • chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj E) r
              hrY 1 ∧
        chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj (E ⊗ F)) (r * s)
            hrsY 2 =
          (r : ℤ) • chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj F) s
              hsY 2 +
            (s : ℤ) • chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj E) r
              hrY 2 +
            (r.choose 2 : ℤ) • chernCap.{u, u + 1} (hX := hY)
              ((spaceModulePullback D hX.1 hY.1 f).obj F) s hsY 1 ^ 2 +
            ((r : ℤ) * s - 1) •
              (chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj F) s hsY 1 *
                chernCap.{u, u + 1} (hX := hY) ((spaceModulePullback D hX.1 hY.1 f).obj E) r
                  hrY 1) +
            (s.choose 2 : ℤ) • chernCap.{u, u + 1} (hX := hY)
              ((spaceModulePullback D hX.1 hY.1 f).obj E) r hrY 1 ^ 2) ∧
    ∃ (P : D.Ambient) (π : P ⟶ X) (hP : D.IsGood (π ≫ sX))
      (d : ℕ) (_ : IsFlat D.site D.base D.base_mem π)
      (_ : HasRelativeDimension d D.site D.base D.base_mem π),
      HasInjectiveChowBaseChange.{u + 1} D sX π d ∧
      ∃ (L : Fin r → Modules.{u, u} D.site D.base D.base_mem P hP.1)
        (N : Fin s → Modules.{u, u} D.site D.base D.base_mem P hP.1)
        (hL : ∀ i, IsFiniteLocallyFreeOfRank D (π ≫ sX) hP (L i) 1)
        (hN : ∀ j, IsFiniteLocallyFreeOfRank D (π ≫ sX) hP (N j) 1),
        (∀ i, IsInvertibleModule ⟨P, hP.1⟩ (L i)) ∧ (∀ j, IsInvertibleModule ⟨P, hP.1⟩ (N j)) ∧
        totalChernCap.{u, u + 1} ((spaceModulePullback D hX.1 hP.1 π).obj E) r
            (isFiniteLocallyFreeOfRank_modulePullback D.site D.base D.base_mem
              (ObjectProperty.homMk π : (⟨P, hP.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
                ⟨X, hX.1⟩) hr) =
          ((List.finRange r).map fun i => 1 + chernCap.{u, u + 1} (L i) 1 (hL i) 1).prod ∧
        totalChernCap.{u, u + 1} ((spaceModulePullback D hX.1 hP.1 π).obj F) s
            (isFiniteLocallyFreeOfRank_modulePullback D.site D.base D.base_mem
              (ObjectProperty.homMk π : (⟨P, hP.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
                ⟨X, hX.1⟩) hs) =
          ((List.finRange s).map fun j => 1 + chernCap.{u, u + 1} (N j) 1 (hN j) 1).prod ∧
        totalChernCap.{u, u + 1} ((spaceModulePullback D hX.1 hP.1 π).obj (E ⊗ F)) (r * s)
            (isFiniteLocallyFreeOfRank_modulePullback D.site D.base D.base_mem
              (ObjectProperty.homMk π : (⟨P, hP.1⟩ : AlgebraicSpaces D.site D.base D.base_mem) ⟶
                ⟨X, hX.1⟩) (isFiniteLocallyFreeOfRank_tensorObj D.site D.base D.base_mem X hX.1
                  hr hs)) =
          ((List.finRange r).flatMap fun i => (List.finRange s).map fun j =>
            1 + (chernCap.{u, u + 1} (L i) 1 (hL i) 1 +
              chernCap.{u, u + 1} (N j) 1 (hN j) 1)).prod

Sources

• Section 82.31, Tag 0ESB
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.exists_projectiveBundleTower_hasInjectiveChowPullbacks_filtration · AlgebraicGeometry.Space.PrivateChern.chernCap_moduleDual · AlgebraicGeometry.Space.PrivateChern.chernCap_tensorObj

Section82.32 Degrees of zero cycles#

Section 82.32 of the Stacks Project, Tag 0ESF. 3 tags here are carried by 6 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0ESG, a definition

Tag 0ESG in the Stacks Project.

AlgebraicGeometry.Space.chowGroupZeroDegree, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ChowGroupZeroDegree.lean, line 102.

Let p : X ⟶ Y be a proper morphism of good algebraic spaces over the base, where Y plays the role of Spec(k): its space of points is a single point, the dimension function vanishes there, and W is its integral closed subspace. The degree of a zero cycle on X is the composite

deg : CH_0(X) ⟶ CH_0(Y) ⟶ ℤ

of proper pushforward p_*, AlgebraicGeometry.Space.chowGroupProperPushforward, with the natural isomorphism CH_0(Y) ≅ ℤ sending the class of [W] to 1, AlgebraicGeometry.Space.chowGroupZeroEquivInt.

def chowGroupZeroDegree (sX : X ⟶ D.space) (hX : D.IsGood sX) (sY : Y ⟶ D.space)
    (hY : D.IsGood sY) {p : X ⟶ Y} (hbase : p ≫ sY = sX)
    (hp : IsProper D.site D.base D.base_mem p)
    (hpt : Subsingleton (points D.site D.base D.base_mem Y))
    (hδ : ∀ y : points D.site D.base D.base_mem Y, D.deltaValue sY y = 0)
    (W : IntegralClosedSubspace D.site D.base D.base_mem Y) :
    chowGroup.{v} D sX hX 0 →+ ℤ

Tag 0ESH, a lemma

Tag 0ESH in the Stacks Project.

AlgebraicGeometry.Space.cycleDegree_eq_finsum, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ZeroCycleDegree.lean, line 195. The attribute's comment: “deg(Z) read as the degree [R(Z) : k] of the extension of function fields, which is the weight of Stacks 0EP0, rather than as dim_k Γ(Z, 𝒪_Z): the two are identified on a scheme by Stacks.Chow.isIso_Γevaluation_of_isIntegral_of_subsingleton, and for an algebraic space there is no evaluation of a global function in a residue field. Spec(k) is modelled as a good algebraic space over the base with one point and vanishing dimension function. The degree is the cycle-level degree, not the one on CH_0 of Stacks 0ESG, which needs Stacks 0EQC”.

Stacks 0ESH, in the shape in which this library presents a cycle. Let p : X ⟶ Y be a proper morphism of good algebraic spaces over the base with |Y| a single point and with the dimension function induced on Y vanishing — the situation of Y = Spec(k). A zero cycle α on X is the coefficient family Z ↦ n_Z on the integral closed subspaces of X, so the formal sum α = ∑ n_Z [Z] of the source is α itself, and the degree of α is

deg(α) = ∑ n_Z deg(Z),

the sum running over every integral closed subspace of X and deg(Z) being the degree [R(Z) : k] of Z → Spec(k).

This is the definition of proper pushforward, Stacks 0EP1, after two identifications. The sum defining the pushforward coefficient at W runs over the integral closed subspaces of X whose integral closed image is W, and that is all of them, by AlgebraicGeometry.Space.cycleImageSubspace_eq_of_subsingleton_points. And at a subspace Z with a nonzero coefficient the weight of Stacks 0EP0 is the degree rather than zero: the δ-dimension of Z is zero because α is a zero cycle, and the δ-dimension of its image is zero because the dimension function of Y vanishes.

theorem cycleDegree_eq_finsum (sX : X ⟶ D.space) (hX : D.IsGood sX) (sY : Y ⟶ D.space)
    (hY : D.IsGood sY) {p : X ⟶ Y} (hbase : p ≫ sY = sX)
    (hp : IsProper D.site D.base D.base_mem p)
    (hpt : Subsingleton (points D.site D.base D.base_mem Y))
    (hδ : ∀ y : points D.site D.base D.base_mem Y, D.deltaValue sY y = 0)
    (W : IntegralClosedSubspace D.site D.base D.base_mem Y) (α : CycleGroup D sX 0) :
    cycleDegree D sX hX sY hY hbase hp W α =
      ∑ᶠ Z : IntegralClosedSubspace D.site D.base D.base_mem X, α.1 Z *
        (cycleResidueDegree D sX hX sY hY p
          (Z.ambientGenericPoint D.site D.base D.base_mem) : ℤ)

AlgebraicGeometry.Space.cycleDegree_eq_sum, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ZeroCycleDegree.lean, line 234. The attribute's comment: “the same restriction on deg(Z_i) as on AlgebraicGeometry.Space.cycleDegree_eq_finsum; this is its restatement for a cycle given as a finite sum over pairwise distinct integral closed subspaces”.

Stacks 0ESH, in the source's shape. In the situation of AlgebraicGeometry.Space.cycleDegree_eq_finsum, let the zero cycle α be presented as a finite sum α = ∑ n_i [Z_i] over pairwise distinct integral closed subspaces Z_i ⊆ X — that is, the coefficient of α at Z_i is n_i and its coefficient at a subspace which is no Z_i vanishes. Then

deg(α) = ∑ n_i deg(Z_i),

with deg(Z_i) the degree [R(Z_i) : k] of Z_i → Spec(k).

The sum over all integral closed subspaces of AlgebraicGeometry.Space.cycleDegree_eq_finsum is supported on the Z_i, and the Z_i are distinct, so it is the finite sum over the index type.

theorem cycleDegree_eq_sum (sX : X ⟶ D.space) (hX : D.IsGood sX) (sY : Y ⟶ D.space)
    (hY : D.IsGood sY) {p : X ⟶ Y} (hbase : p ≫ sY = sX)
    (hp : IsProper D.site D.base D.base_mem p)
    (hpt : Subsingleton (points D.site D.base D.base_mem Y))
    (hδ : ∀ y : points D.site D.base D.base_mem Y, D.deltaValue sY y = 0)
    (W : IntegralClosedSubspace D.site D.base D.base_mem Y) (α : CycleGroup D sX 0)
    {J : Type v} [Fintype J] (Z : J → IntegralClosedSubspace D.site D.base D.base_mem X)
    (hZ : Function.Injective Z) (n : J → ℤ) (hn : ∀ i, α.1 (Z i) = n i)
    (hsupp : ∀ T : IntegralClosedSubspace D.site D.base D.base_mem X,
      (∀ i, Z i ≠ T) → α.1 T = 0) :
    cycleDegree D sX hX sY hY hbase hp W α =
      ∑ i : J, n i * (cycleResidueDegree D sX hX sY hY p
        ((Z i).ambientGenericPoint D.site D.base D.base_mem) : ℤ)

AlgebraicGeometry.Space.cycleDegree_eq_finsum_finrank, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ZeroCycleDegreeGlobalFunctions.lean, line 192. The attribute's comment: “deg(Z) is dim_k Γ(Z, 𝒪_Z), as in the source. Spec(k) is modelled as a reduced good algebraic space over the base with one point and vanishing dimension function (which by Stacks 047Z is the spectrum of the field k = Γ(Y, 𝒪_Y)); the cycle is the coefficient family on the integral closed subspaces, so the sum runs over all of them”.

Stacks 0ESH, in the shape in which this library presents a cycle. Let p : X ⟶ Y be a proper morphism of good algebraic spaces over the base, with Y reduced, with a single point and with vanishing induced dimension function — the situation of Y = Spec(k). Writing a zero cycle α on X as the coefficient family Z ↦ n_Z on the integral closed subspaces of X,

deg(α) = ∑ n_Z dim_k Γ(Z, 𝒪_Z),

where k = Γ(Y, 𝒪_Y).

theorem cycleDegree_eq_finsum_finrank (sX : X ⟶ D.space) (hX : D.IsGood sX) (sY : Y ⟶ D.space)
    (hY : D.IsGood sY) {p : X ⟶ Y} (hbase : p ≫ sY = sX)
    (hp : IsProper D.site D.base D.base_mem p)
    (hpt : Subsingleton (points D.site D.base D.base_mem Y))
    (hYred : HasEtaleLocalProperty D.site D.base D.base_mem Y
      (fun T : Scheme.{u} => IsReduced T))
    (hδ : ∀ y : points D.site D.base D.base_mem Y, D.deltaValue sY y = 0)
    (W : IntegralClosedSubspace D.site D.base D.base_mem Y) (α : CycleGroup D sX 0) :
    cycleDegree D sX hX sY hY hbase hp W α =
      ∑ᶠ Z : IntegralClosedSubspace D.site D.base D.base_mem X, α.1 Z *
        (letI : Algebra (globalFunctions' D.site D.base D.base_mem Y)
              (globalFunctions' D.site D.base D.base_mem Z.subspace.toFunctor) :=
            (globalFunctionsMap D.site D.base D.base_mem
              (ObjectProperty.homMk (Z.subspace.ι ≫ p) :
                (⟨Z.subspace.toFunctor, Z.isClosedSubspace.1⟩ :
                    AlgebraicSpaces D.site D.base D.base_mem) ⟶ ⟨Y, hY.1⟩)).toAlgebra
        (Module.finrank (globalFunctions' D.site D.base D.base_mem Y)
          (globalFunctions' D.site D.base D.base_mem Z.subspace.toFunctor) : ℤ))

AlgebraicGeometry.Space.cycleDegree_eq_sum_finrank, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/ZeroCycleDegreeGlobalFunctions.lean, line 225. The attribute's comment: “the same modelling of Spec(k) as in AlgebraicGeometry.Space.cycleDegree_eq_finsum_finrank; this is its restatement for a cycle given as a finite sum over pairwise distinct integral closed subspaces”.

Stacks 0ESH, in the source's shape. In the situation of AlgebraicGeometry.Space.cycleDegree_eq_finsum_finrank, let the zero cycle α be presented as a finite sum α = ∑ n_i [Z_i] over pairwise distinct integral closed subspaces Z_i ⊆ X. Then

deg(α) = ∑ n_i dim_k Γ(Z_i, 𝒪_{Z_i}),

where k = Γ(Y, 𝒪_Y).

theorem cycleDegree_eq_sum_finrank (sX : X ⟶ D.space) (hX : D.IsGood sX) (sY : Y ⟶ D.space)
    (hY : D.IsGood sY) {p : X ⟶ Y} (hbase : p ≫ sY = sX)
    (hp : IsProper D.site D.base D.base_mem p)
    (hpt : Subsingleton (points D.site D.base D.base_mem Y))
    (hYred : HasEtaleLocalProperty D.site D.base D.base_mem Y
      (fun T : Scheme.{u} => IsReduced T))
    (hδ : ∀ y : points D.site D.base D.base_mem Y, D.deltaValue sY y = 0)
    (W : IntegralClosedSubspace D.site D.base D.base_mem Y) (α : CycleGroup D sX 0)
    {J : Type v} [Fintype J] (Z : J → IntegralClosedSubspace D.site D.base D.base_mem X)
    (hZ : Function.Injective Z) (n : J → ℤ) (hn : ∀ i, α.1 (Z i) = n i)
    (hsupp : ∀ T : IntegralClosedSubspace D.site D.base D.base_mem X,
      (∀ i, Z i ≠ T) → α.1 T = 0) :
    cycleDegree D sX hX sY hY hbase hp W α =
      ∑ i : J, n i *
        (letI : Algebra (globalFunctions' D.site D.base D.base_mem Y)
              (globalFunctions' D.site D.base D.base_mem (Z i).subspace.toFunctor) :=
            (globalFunctionsMap D.site D.base D.base_mem
              (ObjectProperty.homMk ((Z i).subspace.ι ≫ p) :
                (⟨(Z i).subspace.toFunctor, (Z i).isClosedSubspace.1⟩ :
                    AlgebraicSpaces D.site D.base D.base_mem) ⟶ ⟨Y, hY.1⟩)).toAlgebra
        (Module.finrank (globalFunctions' D.site D.base D.base_mem Y)
          (globalFunctions' D.site D.base D.base_mem (Z i).subspace.toFunctor) : ℤ))

Tag 0ESI, a lemma

Tag 0ESI in the Stacks Project.

AlgebraicGeometry.Space.intersectionNumber_eq_chowGroupZeroDegree_iteratedFirstChernCapCycle, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Chow/IntersectionNumberChowDegree.lean, line 57.

Stacks 0ESI. Let k be a field, X a proper algebraic space over k, Z ⊆ X a closed subspace of dimension d and ℒ₁, …, ℒ_d invertible 𝒪_X-modules. Then

(ℒ₁ ⋯ ℒ_d · Z) = deg(c₁(ℒ₁) ∩ ⋯ ∩ c₁(ℒ_d) ∩ [Z]_d),

the left hand side the intersection number of Stacks 0EDF, [Z]_d the d-cycle of Z (Stacks 0EE9), the cap products those of Stacks 0EQJ and deg : CH_0(X) → ℤ the degree of Stacks 0ESG, in Situation 0EDT over Spec k with the zero dimension function.

theorem intersectionNumber_eq_chowGroupZeroDegree_iteratedFirstChernCapCycle
    (X : AlgebraicSpaces B (Spec (.of k)) hk) (f : X.1 ⟶ yoneda.obj (baseObject k B hk))
    (hf : IsProper B (Spec (.of k)) hk f) (Z : ClosedSubspaces B (Spec (.of k)) hk X.1) (d : ℕ)
    (hd : dimension B (Spec (.of k)) hk Z.1.toFunctor Z.2.1 = d)
    (L : Fin d → Modules.{u, u} B (Spec (.of k)) hk X.1 X.2)
    [hL : ∀ j, Stacks.SheafOfModules.IsInvertible (structurePresheaf B (Spec (.of k)) hk X.1)
      (Stacks.SheafOfModules.Tensor.ringSheaf
        (smallEtaleRingedSite B (Spec (.of k)) hk X.1 X.2).O).property (L j)] :
    intersectionNumber.{u, w} k B hk X (closedSubspaceToSpace B (Spec (.of k)) hk X Z) d f hf
        (closedSubspaceInclusion B (Spec (.of k)) hk X Z) Z.2.2 hd L =
      chowGroupZeroDegree.{v} (fieldSpaceSituation k B hk) f
        (isGood_fieldSpaceSituation_of_isProper k B hk X hf) (𝟙 _)
        (isGood_fieldSpaceSituation_id k B hk) (Category.comp_id f) hf inferInstance
        (fieldSpaceSituation_deltaValue_id k B hk) (fieldSpacePoint k B hk)
        (chowClass.{v} (fieldSpaceSituation k B hk) f
          (isGood_fieldSpaceSituation_of_isProper k B hk X hf) 0
          (iteratedFirstChernCapCycle (fieldSpaceSituation k B hk) f
            (isGood_fieldSpaceSituation_of_isProper k B hk X hf) d L hL 0
            (cycleGroupCast (fieldSpaceSituation k B hk) f (zero_add _).symm
              (closedSubspaceCycle (fieldSpaceSituation k B hk) f
                (isGood_fieldSpaceSituation_of_isProper k B hk X hf) Z.1 Z.2 d
                (deltaDimensionSubspace_le_of_dimension_eq k B hk X hf Z hd)))))

Sources

• Section 82.32, Tag 0ESF
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Space.chowGroupZeroDegree · AlgebraicGeometry.Space.cycleDegree_eq_finsum · AlgebraicGeometry.Space.cycleDegree_eq_sum · AlgebraicGeometry.Space.cycleDegree_eq_finsum_finrank · AlgebraicGeometry.Space.cycleDegree_eq_sum_finrank · AlgebraicGeometry.Space.intersectionNumber_eq_chowGroupZeroDegree_iteratedFirstChernCapCycle

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