Martin's Blog

Galois ascent for functors of points

Posted by martin on Thursday, 04 February 2010 at 22:10

I was very pleased this weekend when I worked out how to define Galois descent data for functors of points. I was less pleased when I reached the end of this post and discovered that I couldn’t prove that descending morphisms of functors works nicely.

Galois descent relates objects (e.g. vector spaces, varieties) defined over a field $k$ to objects defined over a bigger field $K$ with “descent data”: a “semilinear” action of $\mathop{\mathrm{Gal}}(K/k)$ on the $K$-object.

If we want to do this for functors of points, it is not clear how to define a semilinear morphism. That is what I shall explain in this post, together with how to ascend (go from a functor over the small field to one over the big field). This is all purely formal.

Galois descent for vector spaces

Here is a quick reminder of Galois descent for vector spaces. For more details and proofs, see Keith Conrad’s notes.

Throughout this post, $K/k$ will be a Galois field extension with Galois group $G$.

Given a $k$-vector space $V$, you get an action of $G$ on $V_K = V \otimes_k K$.

This action is not $K$-linear, but it is more than just $k$-linear. Specifically, for each $\sigma \in G$, the corresponding automorphism of $V_K$ is $\sigma$-semilinear: for each $\lambda \in K, v \in V_K$, $\sigma(\lambda v) = \sigma(\lambda) \sigma(v)$.

Furthermore, given a $K$-linear map $f : V_K \to W_K$, it is of the form $f_0 \otimes_k K$ for some $k$-linear $f_0 : V \to W$ iff it commutes with the $G$-actions.

The above (“ascent”) is all formal in nature. Descent (which you have to work to prove) tells you that given a $K$-vector space $W$ equipped with a semilinear $G$-action, there is a $k$-vector space $W_0$ and an isomorphism $W_0 \otimes_k K \to W$ which preseves the $G$-action.

Semilinear morphisms of functors

I shall write ”$K$-functor” to mean a functor $K\textbf{-Alg} \to \textbf{Set}$. You should think of such a functor as being the functor of points of a $K$-scheme (or other geometrical object), though of course $K$-functors are much more general.

Let $\sigma$ be a field automorphism of $K$. We want to define a notion of $\sigma$-semilinear morphism $X \to Y$, where $X$ and $Y$ are $K$-functors.

As is the case with vector spaces, a $\sigma$-semilinear morphism should not be a $K$-morphism (unless $\sigma = 1$). However a $K$-morphism of $K$-functors is the same as a natural transformation of functors, and at the level of generality of functors there is little you can use except natural transformations!

Hence we cannot tweak the definition of a “morphism” $X$ to $Y$ to get something new. The key idea is that we instead tweak the domain and codomain of the morphism: we will define a new functor $\sigma X$, and then the correct notion of “semilinear morphism $X$ to $Y$” is an ordinary natural transformation, but going from $\sigma X$ to $Y$.

Twisted $K$-algebras

Given a $K$-algebra $A$ with structural homomorphism $f : K \to A$, define $F_\sigma(A)$ to be the $K$-algebra with the same underlying ring as $A$, but with structural homomorphism $f \circ \sigma$.

Now $F_\sigma$ is a functor $K\textbf{-Alg} \to K\textbf{-Alg}$, and $F_\tau \circ F_\sigma = F_{\sigma\tau}$.

Note that $F_\sigma(K)$ is isomorphic (as a $K$-algebra) to $K$, by $\sigma : K \to F_\sigma(K)$, but in general $A$ and $F_\sigma(A)$ need not be isomorphic as $K$-algebras.

The importance of the functor $F_\sigma$ is that a $\sigma$-semilinear homomorphism $A \to B$ becomes an ordinary $K$-algebra homomorphism $A \to F_\sigma(B)$.

Twisted $K$-functors and semilinear actions

Given a $K$-functor $X$ and $\sigma \in \mathop{\mathrm{Aut}}(K)$, we set $\sigma X = X \circ F_\sigma$. This defines a functor $\sigma : K\textbf{-Func} \to K\textbf{-Func}$.

This takes affine $K$-schemes to affine $K$-schemes: if $X = \mathop{\mathrm{Spec}_K} B$, then

$ \sigma X(A) = \mathop{\mathrm{Hom}_K}(B, F_\sigma(A)) = \mathop{\mathrm{Hom}_K}(F_{\sigma^{-1}}(B), A), $

so $\sigma X = \mathmop{\mathrm{Spec}_K} F_{\sigma^{-1}}(B)$.

Now we can define a $\sigma$-semilinear morphism $X \to Y$ of $K$-functors to be a natural transformation $\sigma X \to Y$.

(In the affine case, this corresponds to a $\sigma^{-1}$-semilinear homomorphism of $K$-algebras. The $\sigma^{-1}$ makes sense because of the equivalence between algebras and representable functors is contravariant.)

A semilinear action of $\mathop{\mathrm{Gal}}(K/k)$ on $X$ is a collection of morphisms $\phi_\sigma : \sigma X \to X$, for each $\sigma \in G$, such that $\phi_\sigma$ is $\sigma$-semilinear, $\phi_1 = \mathrm{id}_X$ and $\phi_{\sigma\tau} = \phi_\sigma \circ \sigma \phi_\tau$.

($\sigma \phi_\tau$ here means take the morphism $\phi_\tau$, and apply the functor $\sigma : K\textbf{-Func} \to K\textbf{-Func}$. You need to do this to make the domain and codomain agree when you compose.)

Ascending $K$-functors

Let $X$ a $k$-functor. The “extension of scalars” of $X$ to $K$ is the functor $X_K : K\textbf{-Alg} \to \textbf{Set}$ obtained by composing $X$ with the forgetful functor $K\textbf{-Alg} \to k\textbf{-Alg}$.

To get a semilinear action of $\mathop{\mathrm{Gal}}(K/k)$, observe that for any $K$-algebra $B$, $B \cong F_\sigma(B)$ as $k$-algebras, and so $X_K = \sigma X_K$. Hence we can simply let $\phi_\sigma$ be the identity natural transformation $\sigma X_K \to X_K$.

This is a bit weird: we get a non-trivial group action by leaving the points on which $G$ is apparently “acting” fixed, and instead moving the ground underneath those points. I am not sure if that is the right way to think about it.

Descending morphisms of $K$-functors

That was a lot of formal manipulation, for which I am going to give no immediate pay-off. The point is that it gives us the language we need to think about descent, which is much less trivial than ascent (indeed, it is not always possible to descend: you have to impose conditions on the descent data).

However, what I really wanted was to be able to descend morphisms of $K$-functors. In the case of vector spaces, that is much easier than descending vector spaces (assuming that you already know that the relevant vector spaces descend).

I wanted to show that, if $X, Y$ are two $k$-functors and $f : X_K \to Y_K$ a morphism of $K$-functors, then $f$ is the extension of scalars of a morphism $f_0 : X \to Y$ of $k$-functors iff it commutes with the Galois actions.

Unfortunately I can’t prove this, and it looks like it might be false without some condition on $Y$.

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  1. Galois descent for morphisms of functors of points From Martin's Blog
    I was disappointed in my last post that I was unable to prove any results about Galois descent for morphisms of functors. I have now tracked down a fairly mild condition on the functors that you need for this descent to work, which I shall explain...

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