Base change

The “theorem” we want is that if \(L/K\) is a finite extension of number fields, then \(\mathbb{A}_L=L\otimes_K\mathbb{A}_K\). This isn't a theorem though, this is actually a definition (the map between the two objects) and a theorem about the definition (that it's an isomorphism). In fact the full claim is that it is both a homeomorphism and an \(L\)-algebra isomorphism. Before we can prove the theorem, we need to make the definition.

Recall that the adeles \(\mathbb{A}_K\) of a number field is a product \(\mathbb{A}_K^\infty\times K_\infty\) of the finite adeles and the infinite adeles. So our “theorem” follows immediately from the “theorem”s that \(\mathbb{A}_L^\infty=L\otimes_K\mathbb{A}_K^\infty\) and \(L_\infty=L\otimes_KK_\infty\) (both of these equalities mean an algebraic and topological isomorphism). We may thus treat the finite and infinite results separately.

Base change for nonarchimedean completions.

As pointed out above, the theory of finite adeles works fine for Dedekind domains. So for the time being let \(A\) be a Dedekind domain. Recall that the height one spectrum of \(A\) is the nonzero prime ideals of \(A\). Note that because we stick to the literature, rather than to common sense, fields are Dedekind domains in mathlib, and the height one spectrum of a field is empty. The reason I don't like allowing fields to be Dedekind domains is that geometrically the definition of Dedekind domain in the literature is “smooth affine curve, or a point”. But many theorems in algebraic geometry begin “let \(C\) be a smooth curve”, rather than “let \(C\) be a smooth curve or a point”.

Let \(K\) be the field of fractions of \(A\). If \(v\) is in the height one spectrum of \(A\), then we can put the \(v\)-adic topology on \(A\) and on \(K\), and consider the completions \(A_v\) and \(K_v\). The finite adele ring \(\mathbb{A}_{A,K}^\infty\) is defined to be the restricted product of the \(K_v\) with respect to the \(A_v\), as \(v\) runs over the height one spectrum of \(A\). It is topologised by making \(\prod_v A_v\) open with the product topology (here \(A_v\) has the \(v\)-adic topology).

Now let \(L/K\) be a finite separable extension, and let \(B\) be the integral closure of \(A\) in \(L\). We want to relate the finite adeles of \(K\) and of \(L\). We work place by place, starting by fixing one place \(w\) of \(B\) and analysing the relation of \(L_w\) and \(B_w\) to the completions \(K_v\) and \(A_v\) where \(v\) is the place of \(A\) dividing \(w\).

So let \(w\) be a nonzero prime ideal of \(B\). Say \(w\) lies over \(v\), a prime ideal of \(A\). Then we can put the \(w\)-adic topology on \(L\) and the \(v\)-adic topology on \(K\). Furthermore we can equip \(K\) with an additive \(v\)-adic valuation, that is, a function also called \(v\) from \(K\) to \(\mathbb{Z}\cup\{\infty\}\) normalised so that if \(\pi\) is a uniformiser for \(v\) then \(v(\pi)=1\). Similarly we consider \(w\) as a function from \(L\) to \(\mathbb{Z}\cup\{\infty\}\). The next lemma explains how these valuations are related.

IsDedekindDomain.HeightOneSpectr… to NumberField.InfinitePlace.Comple…

NumberField.InfinitePlace.Comple… to NumberField.AdeleRing.baseChange…

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