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In this note, I want to discuss the work of Weil as presented in Milne’s papers (in particuar this one). Given a finite Galois extension $L/k$ of fields, many readers will be familiar with the usual “Galois descent” procedure giving an equivalence between (for example) affine varieties over $k$ and affine varieties over $L$ with a suitable $Gal(L/k)$ action. Indeed, such extensions are finite etale, and the result is immediate from the general theory of etale descent and the “Galois isomorphism” $L \otimes_k L \cong \prod_{g \in Gal(L/k)} L$, which one can unravel and check implies that to give a $Gal(L/k)$-action is to give a descent datum.

If $L/k$ is no longer finite, this method breaks down. It will still be an fpqc cover, so descent theory works fine, but the Galois isomorphism breaks down. This can be seen using infinite Galois theory: for example by considering $\bar{Q}/Q$ and noting that only the open index 2 subgroups give rise to degree 2 etale algebras. In the case where $L/k$ is algebraic and ind-Galois, $Aut(L/k)$ is profinite and imposing continuity conditions like the one we saw above, one is able to recover Galois descent theory from the finite case. However, if the extension is transcendental, it is less clear how to proceed. For example, a key obviously interesting example is when one has an object over $\mathbb{C}$ together with an action of $Aut(\mathbb{C}/\mathbb{Q})$. Here $\mathbb{C}$ has uncountable transcendence degree.

Before we state our theorem, let us first (following Milne) make the question at hand a little more precise. Let $V$ be a variety (geometrically reduced separated scheme of finite type) over a field $\Omega$ which we assume is algebraically closed (to avoid having to define “Galois”). Let $k \subset \Omega$ be a subfield. A descent system for $V$ is a collection of isomorphisms $\theta_\sigma: \sigma V \stackrel{\cong}{\rightarrow} V$ one for each $\sigma \in Aut(\Omega/k)$ satisfying the cocycle condition $\theta_{\sigma_1 \sigma_2} = \theta_{\sigma_1} \sigma_1(\theta_{\sigma_2})$. We emphasise that such a thing is generally not the same as a descent datum. If there is a model $V_0/k$ giving rise to $(V,\theta)$ by base change, we say $(V,\theta)$ is effective.

Theorem (Weil): Let $\Omega/k$ be an algebraically closed extension of infinite transcendence degree, and suppose we are given $(V,\theta)$ a quasiprojective variety over $\Omega$ with descent system to $k$ as above. Then for $(V,\theta)$ to be effective it suffices for there to be a finitely generated subextension $L/k$ such that the restriction of $\theta$ to $\Omega/L$ is effective.

Milne notes the following consequence. Suppose your variety over $\Omega$ has the following properties.

1. There is a finite set of points $S$ such that any automorphism of $V$ which fixes each point of $S$ must be the identity.
2. There is a finitely generated extension $L/k$ with the property that $\theta_\sigma (\sigma P) = P$ for all $P \in S, \sigma \in Aut(\Omega/L)$.

Then the given pair is effective for descent $\Omega/k$.

Indeed since $V$ is quasiprojective we may always fix any model for it and enlarge $L$ to contain the coefficients of the polynomials describing the variety and the points of $S$. Thus we obtain a model $V_1/L$ and the condition (2) implies that the isomorphisms coming from the model and the given $\theta$ agree on where they send the points $P$. They therefore agree by condition (1), so we have established effectivity of the restriction of $\theta$ to $Aut(\Omega/L)$. The desired result now follows from the theorem.

As far as I can see, this seems to give a very practical tool for performing descent along transcendental extensions, and is necessary for Milne to show that Langlands’ conjecture on conjugation of Shimura varieties actually really implies the existence of canonical models.

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