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
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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
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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
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Posted by martin on
Friday, 19 June 2009 at 11:49
Hilbert’s Nullstellensatz is an algebraic result fundamental to algebraic geometry. There are many different proofs of the Nullstellensatz. In this post I will consider the proof given in this year’s Part III Commutative Algebra course, and in particular one section of the proof that seems to contain lots of magic. When I was revising for the exams, I realised that part of the mystery came from the fact that the particular theorem proved in Commutative Algebra does not require an algebraically closed field, unlike the standard statement of Hilbert’s Nullstellensatz.
Thanks are due to Lloyd West for starting me thinking about this, and to Jon Nelson for giving me the courage to believe that it might be true and for supplying the proof of Lemma 4.
Tags alg-geom, maths, nullstellensatz, partiii
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Posted by martin on
Saturday, 16 May 2009 at 19:54
In my last post on the Yoneda lemma for groups, I ignored the naturality part of the lemma. I want to work in detail what this means once - it is a lot of fiddly composing of morphisms and I probably won’t do it again (at least in public). If you’re not in the mood for following such details, then there is little point in reading this, although you could skip to the last paragraph.
Tags categories, groups, maths, yoneda
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Posted by martin on
Sunday, 10 May 2009 at 16:07
When I wrote my first post on Cayley’s theorem, I noticed that Wikipedia claims that the Yoneda lemma is “a vast generalisation of Cayley’s theorem”. In this post I will try to understand why, and end up concluding that this is probably false.
Tags categories, groups, maths, yoneda
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Posted by martin on
Saturday, 02 May 2009 at 17:12
This post explains how we can consider groups as categories, along with treating the G-sets and G-homomorphisms I considered in my last post on group actions as category-theoretic objects. This is preparation for talking about the Yoneda lemma. Before reading this post, you will need to know the definitions of categories, functors and natural transformations.
Tags categories, groups, maths
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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