Posted by martin on Wednesday, 03 March 2010 at 10:08
I am finally ready to finish my series on algebraic tori, by talking about their representations. I shall show that these representations can be classified by a grading on the vector space of the representation, after extending scalars to the separable closure. I will describe this classification explicitly in a simple case.
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.
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.
Posted by martin on Sunday, 24 January 2010 at 18:10
In this post I will return to the subject of algebraic tori. Just as Pontryagin duality classifies locally compact abelian groups through their characters, so algebraic tori are also classified by their characters.
In order to account for the arithmetic phenomenon of non-split tori, we need to include a Galois action on the character group.
The primary result of this post is that there is an anti-equivalence of categories between {-tori} and {finitely generated free abelian groups with a continuous action of 
}.
Posted by martin on Sunday, 17 January 2010 at 21:12
This post was supposed to be about character groups of algebraic tori. But while writing about that, I found that I wanted to use Hopf algebras, which were something that previously seemed alien to me. So instead I have written about Hopf algebras and why they are useful in the study of algebraic groups.
Posted by martin on Friday, 08 January 2010 at 16:16
Algebraic tori are the simplest examples of algebraic groups. In this post I will define algebraic tori and give some examples. Later I will write about their character groups and representations, and after that I will be able to talk about Hodge structures.
I have been trying to write a post about algebraic tori for several days, mainly because I was trying to sort out the proof that tori over separably closed fields are split. This is complicated and not very important as in practice I only care about perfect fields, so I have left it out.
Note that the algebraic tori considered here have nothing to do with the complex tori in my last post; indeed the complex points of an algebraic torus are not compact in the usual topology! They are called tori because they play the same role in the theory of algebraic groups as real tori play in the theory of Lie groups.
Posted by martin on Wednesday, 30 December 2009 at 21:48
The theory of abelian varieties is very beautiful, both in its arithmetic and geometrical aspects, and also looking just over 
where there are nice applications of complex analysis.
In this post I will work over , and sketch a proof that a complex torus is isomorphic to an abelian variety if and only if it admits a Riemann form.
This will assume some knowledge of the theory of complex manifolds.
Posted by martin on Wednesday, 16 December 2009 at 19:48
Last week I gave a talk on Hodge theory. For the Differential Geometry course, all the students have to give a talk on a topic related to the course. The talk was very long - 1 hour 45 minutes - but this is about the average length of the talks so far. I did my best to shorten it by leaving out unimportant details. Had it not been for the fact that many other talks were longer, I would have removed sections of it entirely, but it did cover about the minimum needed to reach a point of interest to me as an algebraic geometer.
This was the first time I have given a talk of any length in French. This was not too difficult, as I had practised the talk, but probably did slow me down a bit. I am sure the language was far from perfect; for example, I probably should have used the subjunctive all over the place but I didn’t bother with it. But the audience were not too concerned about that.
The first half of the talk contained a lot of analysis, needed to prove the Hodge theorem. This is not my area, but it was fun to learn a little bit; I skipped out all the tedious calculations. The second half contained applications of this to complex manifolds, leading up to the fundamental example of a Hodge structure. I shall need soon to learn about the latter in a more abstract setting; no doubt preparing this talk has given me some of the motivation for them, but I am not sure how useful all the proofs will turn out to be.
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
Posted by martin on Thursday, 08 October 2009 at 09:58
So far in my series on functors of points, I have considered functors 
for some fixed field . We begin by observing that we may allow to be any ring. Then I consider whether it is possible to relate functors with base ring to functors with base ring 
, with only partial success.