The Yoneda lemma I

For two categories let denote the functor category, whose objects are functors and whose morphisms are natural transformations. For a locally small category, the Yoneda embedding is the functor sending an object to the contravariant functor and sending a morphism to the natural transformation given by composition. The goal of the next few posts is [...]

Read Full Post »

The quaternions and Lie algebras I

Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of 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 [...]

Read Full Post »

Structures on hom-sets

Suppose I hand you a commutative ring . 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 , and you also can’t refer directly to the multiplication or addition [...]

Read Full Post »

Fractional linear transformations and elliptic curves

The following two lemmas might be encountered in a basic course in complex analysis (the first in a basic course in group theory, even). Lemma 1: Fix a field . The group of fractional linear transformations acts triple transitively on and the stabilizer of any triplet of distinct points is trivial. Lemma 2: The group [...]

Read Full Post »

Ideals and the category of commutative rings

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 [...]

Read Full Post »

Localization and the strong Nullstellensatz

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 [...]

Read Full Post »

MaxSpec is not a functor

For commutative unital C*-algebras and for finitely-generated reduced integral -algebras, we have seen that is a functor which sends homomorphisms to continuous functions. However, this is not true for general commutative rings. What we want is for a ring homomorphism to be sent to a continuous function via contraction. Unfortunately, the contraction of a maximal [...]

Read Full Post »

Affine varieties and regular maps

I have to admit I’ve been using somewhat unconventional definitions. The usual definition of an affine variety is as an irreducible Zariski-closed subset of , affine -space over an algebraically closed field . A generic Zariski-closed subset is usually referred to instead as an algebraic set (although some authors also call these varieties), and the [...]

Read Full Post »

The ideal-variety correspondence

I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging [...]

Read Full Post »

The Noetherian condition as compactness

Let’s think more about what an abstract theory of unique factorization of primes has to look like. One fundamental property it has to satisfy is that factorizations should be finite. Another way of saying this is that the process of writing elements as products of other elements (up to units) should end in a finite [...]

Read Full Post »