Martin's Blog

Galois descent for morphisms of functors of points

Posted by Martin Orr on Saturday, 20 February 2010 at 21:58

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 below. Importantly, this condition is satisfied automatically by the functors of points of a scheme (though I won't prove this).

This tells us that if you have two -functors satisfying the Galois exactness property, and a morphism of their restrictions to which commutes with the action of , then it comes from a unique morphism of -functors.

I shall not discuss descending functors, only morphisms. But a small modification to the Galois exactness condition should allow you to descend functors themselves.

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 to objects defined over a bigger field with "descent data": a "semilinear" action of on the -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.