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.
Tags alggeom, descent, maths, points-func
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Posted by Martin Orr 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 
}.
Tags alg-groups, alggeom, maths, tori
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Posted by Martin Orr 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.
Tags alg-groups, alggeom, maths
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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.
Tags alg-groups, alggeom, maths, tori
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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 
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.
Tags abelian-varieties, alggeom, maths
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