I recently started writing up some material on finite presentation for the CRing project. There seems to be a folk “finitely presented” approach in mathematics: to prove something over a big, scary uncountable field like $\mathbb{C}$, one argues that the problem descends to some much smaller subobject, for instance a finitely generated subring of the complex numbers. It might be possible to prove using elementary methods the analog for such smaller subobjects, from which one can deduce the result for the big object.

One way to make these ideas precise is the characteristic p principle of Abraham Robinson, which I blogged about in the past when describing the model-theoretic approach to the Ax-Grothendieck theorem. Today, I want to describe a slightly different (choice-free!) argument in this vein that I learned from an article of Serre.

Theorem 1 Let ${F: \mathbb{C}^n \rightarrow \mathbb{C}^n}$ be a polynomial map with ${F \circ F = 1_{\mathbb{C}^n}}$. Then ${F}$ has a fixed point.

We can phrase this alternatively as follows. Let ${\sigma: \mathbb{C}[x_1, \dots, x_n] \rightarrow \mathbb{C}[x_1, \dots, x_n]}$ be a ${\mathbb{C}}$-involution. Then the map on the ${\mathrm{Spec}}$‘s has a fixed point (which is a closed point).

In fact, this result can be proved using directly Robinson’s principle (exercise!). The present argument, though, has more of an algebro-geometric feel to it, and it now appears in the CRing project — you can find it in the chapter currently marked “various.(more…)

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 ${f: X \rightarrow Y}$ is a regular map and ${C \subset X}$ is constructible, then so is ${f(C)}$. In general, this is the best one can say: even very nice subsets of ${X}$ (e.g. ${X}$ 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 ${f: X \rightarrow Y}$ be a finitely presented morphism of schemes. Then if ${C \subset X}$ is locally constructible, so is ${f(C)}$.

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.