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Galois ascent for functors of points

Posted by Martin Orr 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.

Tags alg-geom, descent, maths, points-func

Trackbacks

  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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