Posted by martin 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.
Tags alg-geom, descent, maths, points-func
Read more...
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 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.
Tags alg-geom, descent, maths, points-func
Read more...
Posted by martin on
Saturday, 07 November 2009 at 16:59
This post was inspired by Monday’s algebraic geometry exercise class, although in fact it fits neatly into my series on functors of points (except that it requires you to know what a scheme is, while previously I have considered only affine schemes). I shall prove the following theorem:
Theorem. There is a canonical bijection between morphisms 
of -schemes and natural transformations of the corresponding functors of points.
Tags alg-geom, maths, points-func
Read more...
Posted by martin on
Wednesday, 02 September 2009 at 22:51
In this article, I will examine in more detail the functor of points of an affine variety,
which I defined in the last article.
I shall show that this functor is the same as a Hom-functor on the category of -algebras,
and that morphisms of varieties correspond to natural transformations of functors.
Tags alg-geom, maths, points-func, yoneda
Read more...
Posted by martin on
Tuesday, 25 August 2009 at 20:56
I think I have made some progress recently in understanding the “functor of points” idea in algebraic geometry. In this article I shall explain how the functor of points of an affine variety arises simply by considering solutions to fixed polynomials over varying rings; this gives the motivating example for considering functors associated to more general algebraic-geometric objects.
Tags alg-geom, maths, points-func
Read more...
for some fixed field 
, with only partial success.
-points of an affine 
(and more general objects) are the same as morphisms 
. James already brought this up in his comment 
for a suitable 
. Only a restricted class of