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 Edit

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.