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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 k-functors satisfying the Galois exactness property, and a morphism of their restrictions to K\textbf{-Alg} which commutes with the action of \mathop{\mathrm{Gal}}(K/k), then it comes from a unique morphism of k-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.

no comments Tags alg-geom, descent, maths, points-func Read more...

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.

no comments Tags alg-geom, descent, maths, points-func Read more...

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