Ax-Grothendieck Theorem

The Ax-Grothendieck Theorem says that if $$V$$ is a variety over an algebraically closed field $$K$$, and $$f : V \to V$$ is a morphism of varieties such that $$V(K) \to V(K)$$ is injective, then $$V(K) \to V(K)$$ is bijective. Here, "variety" can be interpreted as finite-type scheme over $$K$$.

The Ax-Grothendieck theorem has a relatively straightforward proof using model theory, and is often listed as an example of a theorem that is easy to prove using mathematical logic, and harder to prove directly using algebraic geometry.

Proof sketch
Varieties can be seen as (special) definable sets, and morphisms of varieties yield definable maps. Therefore, it suffices to show that for every model $$K$$ of ACF, the following condition holds:


 * (*) If $$D \subset K^n$$ is definable (with parameters) and $$f : D \to D$$ is definable (with parameters), and injective, then $$f$$ is a bijection.

Conditition (*) is equivalent to a small conjunction of first-order statements (an easy exercise). In other words, the set of models of ACF satisfying (*) is an elementary class.

Suppose $$K \models ACF$$ has the property that the definable closure of any finite subset is finite. Then (*) holds. Indeed, suppose $$f : D \hookrightarrow D$$ is definable and injective, and $$p \in D$$. Let $$S$$ be a finite set over which $$f, D, p$$ are defined. Then $$f$$ induces an injective map from $$D(S) := D \cap \operatorname{dcl}(S)$$ to itself. Since $$D(S)$$ is finite, $$f$$ is a bijection, by the pigeonhole principle. So $$p \in f(D)$$, and $$f$$ is surjective.

The algebraic closure $$\overline{\mathbb{F}_p}$$ of $$\mathbb{F}_p$$ is a model of ACF in which every finite set has finite definable closure. (The perfect field generated by any finite set is finite.) So (*) holds in $$\overline{\mathbb{F}_p}$$. Every characteristic $$p$$ model of ACF (every model of $$ACF_p$$) is elementarily equivalent to $$\overline{\mathbb{F}_p}$$. Since (*) is a conjunction of first-order statements, (*) holds in all models of $$ACF_p$$. Then by compactness, it also holds in at least one model of $$ACF_0$$, hence in all models of $$ACF_0$$. So (*) holds in all models of ACF.