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Archive for the ‘algebraic geometry’ Category

In this post we’ll compute the (topological) cohomology of smooth projective (complex) hypersurfaces in \mathbb{CP}^n. When n = 3 the resulting complex surfaces give nice examples of 4-manifolds, and we’ll make use of various facts about 4-manifold topology to try to say more in this case; in particular we’ll be able to compute, in a fairly indirect way, the ring structure on cohomology. This answers a question raised by Akhil Mathew in this blog post.

Our route towards this result will turn out to pass through all of the most common types of characteristic classes: we’ll invoke, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes.

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Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by \text{Spec } R I mean a commutative ring R regarded as an object in the opposite category \text{CRing}^{op}.

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For two categories C, D let D^C denote the functor category, whose objects are functors C \to D and whose morphisms are natural transformations. For C a locally small category, the Yoneda embedding is the functor C \to \text{Set}^{C^{op}} sending an object x \in C to the contravariant functor \text{Hom}(-, x) and sending a morphism x \to y to the natural transformation \text{Hom}(-, x) \to \text{Hom}(-, y) given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.

Below, whenever we talk about the Yoneda lemma we implicitly restrict our attention to locally small categories.

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Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of \text{SO}(3) might object as follows: “that’s nice and all, but there has to be a general version of this construction for more general Lie groups, right? You can’t always depend on the nice properties of division algebras.” And that someone would be right. Today we’ll begin to describe the appropriate generalization, the exponential map from a Lie algebra to its Lie group. To simplify the exposition, we’ll restrict to the case of matrix groups; that is, nice subgroups of \text{GL}_n(\mathbb{F}) for \mathbb{F} = \mathbb{R} or \mathbb{C}, which will allow us to mostly avoid differential geometry.

The theory of Lie groups and Lie algebras is regarded to be one of the most beautiful in mathematics, and it is also fundamental to many areas, so today’s post is an extended discussion motivating the definition of a Lie algebra. In the next post we will actually do something with them.

For studying the hydrogen atom, our interest in Lie algebras comes from the following. If a Lie group G acts smoothly on a smooth manifold M, its Lie algebra acts by differential operators on the space C^{\infty}(M) of smooth functions, and these differential operators are the “infinitesimal generators” which give us conserved quantities for the evolution of a quantum system on M (in the case that G consists of symmetries of the Hamiltonian). Despite the fact that Lie algebras are commonly sold as a tool for understanding Lie groups, arguably in quantum mechanics the Lie algebra of symmetries of a Hamiltonian is more fundamental. This is important in sitations where the Lie algebra can sometimes exist without an associated Lie group.

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Suppose I hand you a commutative ring R. I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of R, and you also can’t refer directly to the multiplication or addition maps R \times R \to R, since these aren’t morphisms.) Geometrically, I might equivalently say that you are only allowed to work in the language of the category of affine schemes, since the two are dual. Can you recover R as a set, and can you recover the ring operations on R?

The answer turns out to be yes. Today we’ll discuss how this works, and along the way we’ll run into some interesting ideas.

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In this post I’d like to give a better (by which I mean category-theoretic) definition of the lattice of ideals than the standard one. We know that the lattice of ideals has meets and joins defined by intersection and sum, respectively, and that if a lattice is viewed as a category whose arrows are the order relation, then meet and join are the product and coproduct, respectively. But we also know that the lattice of radical ideals of a finitely-generated reduced integral \mathbb{C}-algebra R is dual to the lattice of algebraic subsets of \text{MaxSpec } R (and that the lattice of prime ideals is dual to the lattice of algebraic subvarieties), and there is a very general category-theoretic formalism for understanding subobjects in a category. It turns out that this formalism reproduces the lattice of ideals of an arbitrary commutative ring – as long as we run it in the opposite category \text{CRing}^{op}.

Edit, 2/9/10: The above claim is wrong. But let me tell you the construction I had in mind and you can judge whether it is more natural than the usual definition.

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A basic idea in topology and analysis is to study a space by restricting attention to arbitrarily small neighborhoods of a point. It is desirable, therefore, to have a notion of looking at small neighborhoods of a point which can be stated in entirely ring-theoretic terms. More generally, we’d like to have a way to ignore some points and only think about others. The tool that allows us to do this is called localization, and it offers a conceptual proof of the strong Nullstellensatz from the weak Nullstellensatz, which, as you’ll recall, is the tool that allows us to describe the category of affine varieties as the opposite of a category of algebras.

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