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 1Let 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.