The finite presentation trick: Or, how to get the non-noetherian form of Chevalley’s theorem

In classical algebraic geometry, one defines a subset of a variety over an algebraically closed field to be constructible if it is a union of locally closed subsets (in the Zariski topology). One of the basic results that one proves, which can be called “elimination theory” and is due to Chevalley, states that constructible sets are preserved under taking images: if is a regular map and is constructible, then so is . In general, this is the best one can say: even very nice subsets of (e.g. itself) need not have open or closed (or even locally closed) images.

In the theory of schemes, one can formulate a similar result. A morphism of finite type between noetherian schemes sends constructible sets into constructibles. One proves this result by making a sequence of reductions to considering the case of two integral affine schemes, and then using a general fact from commutative algebra. It turns out, however, that there is a more general form of the Chevalley theorem:

Theorem 1 Let be a finitely presented morphism of schemes. Then if is locally constructible, so is .

I will explain today how one deduces this more general fact from the specific case of noetherian schemes. This will highlight a useful fact: oftentimes, general facts in algebraic geometry can be reduced to the noetherian case since, for instance, every ring is an inductive limit of noetherian rings. This can be developed systematically, as is done in EGA IV-8, but I shall not do so here.

N.B. As a result, this post is written entirely for those whom Ravi Vakil would call “non-noetherian people.” I will simply assume as known the noetherian results (which can be found easily, e.g. in Hartshorne or EGA I) and explain how they can be generalized. Nonetheless, even noetherian readers have a very good reason to care. In fact, it is through such a “finite presentation” argument that Grothendieck proves the general quasi-finite form of Zariski’s main theorem; the finite presentation trick is a very ingenious strategy, about which I hope to say more soon, that can reduce many results not only to the noetherian case, but also to the local case.

 

Categories parametrized by schemes and representation theory in complex rank

[Minor corrections made, 6/21]

I’d now like to discuss my paper “Categories parametrized by schemes and representation theory in complex rank.” My RSI project was rather-open ended: to investigate the categories of representation theory in complex rank. Pavel Etingof told me that it would be expected that they would behave similar in some ways to the integral case (at least if “there was justice in the world”). For instance, we know that Deligne’s has a comibnatorial parametrization of simple objects similar to the classical case. However, as I discovered when I got there, I don’t actually know representation theory. I had looked through some material on finite groups, and knew (in outline, not usually proofs) the basic facts about the symmetric group. I certainly didn’t know anything about Hecke algebras (the literature of which seems rather inaccessible to beginners), and I don’t think I could define a semisimple Lie algebra. Anyway, so what I did was therefore was the easy case: representation theory in transcendental rank. I sort of ended up stumbling into this by accident, so I’ll try to reconstruct the story below, somewhat. I apologize in advance to readers that know algebraic geometry and will probably find this post rather slow-moving (it’s really addressed with a younger version of myself in mind). Readers may wish, however, to review my earlier posts on this topic.