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# Ax–Grothendieck theorem: Wikis

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# Encyclopedia

In mathematics, the Ax–Grothendieck theorem is a result that was proved independently by James Ax and Grothendieck.

The theorem is often given as the special case as follows: If P is a polynomial from $\mathbf{C}^n$ to $\mathbf{C}^n$ and P is injective then P is bijective. That is, if P is one-to-one then P is onto.

The full theorem generalizes to any algebraic variety over an algebraically closed field.

## Proof via finite fields

Grothendieck's proof of the theorem  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 Fn to itself is injective then it is bijective.

If F is a finite field, then Fn 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 $\mathbf{C}$ 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.. Thus, one can use the arithmetic of finite fields to prove a statement about $\mathbf{C}$ even though there is no non-trivial homomorphism from any finite field to $\mathbf{C}$. The proof thus uses model theoretic principles to prove an elementary statement about polynomials. The proof for the general case uses a similar method.

## Other proofs

There are other proofs of the theorem. Armand Borel gave a proof using topology. The case of n = 1 and field $\mathbf{C}$ follows since $\mathbf{C}$ is algebraically closed and can also be thought of as a special case of the result that for any analytic function f on $\mathbf{C}$, f injective implies f surjective. This is a corollary of Picard's theorem.

## Related results

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 S-endomorphism 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).