Martin's Blog

Algebraic tori

Posted by Martin Orr 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.

1 comment Tags alg-geom, alg-groups, maths, tori Read more...

Complex abelian varieties and Riemann forms

Posted by Martin Orr 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 \mathbb{C} where there are nice applications of complex analysis. In this post I will work over \mathbb{C}, 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.

1 comment Tags abelian-varieties, alg-geom, maths Read more...

Functor of points of non-affine schemes

Posted by Martin Orr 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 X \to Y of k-schemes and natural transformations of the corresponding functors of points.

3 comments Tags alg-geom, maths, points-func Read more...

Functors of points and base ring

Posted by Martin Orr on Thursday, 08 October 2009 at 09:58

So far in my series on functors of points, I have considered functors k\textbf{-Alg} \to \textbf{Set} for some fixed field k. We begin by observing that we may allow k to be any ring. Then I consider whether it is possible to relate functors with base ring k to functors with base ring \mathbb{Z}, with only partial success.

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

Morphisms and functors of points

Posted by Martin Orr on Thursday, 01 October 2009 at 15:45

This post will discuss the fact that A-points of an affine k-scheme X (and more general objects) are the same as morphisms \mathop{\mathrm{Spec}_k} A \to X. James already brought this up in his comment last time. As well as proving this in the affine k-scheme case, I shall attempt to give an intuitive explanation of this fact, although I don't find this entirely satisfying.

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

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