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.
Theorem 1 Let ${f: X \rightarrow Y}$ be a finitely presented morphism of schemes. Then if ${C \subset X}$ is locally constructible, so is ${f(C)}$.