In mathematics, the Ax–Grothendieck theorem is a result that was proved independently by James Ax and Grothendieck.^{[1]}^{[2]}
The theorem is often given as the special case as follows: If P is a polynomial from to and P is injective then P is bijective. That is, if P is onetoone then P is onto.^{[1]}^{[2]}
The full theorem generalizes to any algebraic variety over an algebraically closed field.^{[1]}
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Grothendieck's proof of the theorem^{[1]} ^{[2]} is based on proving the analogous theorem for finite fields and their algebraic closures. That is, for any field F that is itself finite or that is the closure of a finite field, if a polynomial P from F^{n} to itself is injective then it is bijective.
If F is a finite field, then F^{n} is finite. In this case the theorem is true for trivial reasons, and true more generally for arbitrary functions, not just for polynomials: any injection of a finite set to itself is a bijection. When F is the algebraic closure of a finite field, the result follows from Hilbert's Nullstellensatz. The Ax–Grothendieck theorem for complex numbers can therefore be proven by showing that a counterexample over would translate into a counterexample in some algebraic extension of a finite field.
This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of characteristic 0 translate into algebraic relations over finite fields with large characteristic.^{[1]}. Thus, one can use the arithmetic of finite fields to prove a statement about even though there is no nontrivial homomorphism from any finite field to . The proof thus uses model theoretic principles to prove an elementary statement about polynomials. The proof for the general case uses a similar method.
There are other proofs of the theorem. Armand Borel gave a proof using topology.^{[2]} The case of n = 1 and field follows since is algebraically closed and can also be thought of as a special case of the result that for any analytic function f on , f injective implies f surjective. This is a corollary of Picard's theorem.
Another example of reducing theorems about morphisms of finite type to finite fields can be found in EGA IV: There, it is proved that a radicial Sendomorphism of a scheme X of finite type over S is bijective (10.4.11), and that if X/S is of finite presentation, and the endomorphism is a monomorphism, then it is an automorphism (17.9.6).
