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Ax–Grothendieck theorem: Wikis


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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 \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.[1][2]

The full theorem generalizes to any algebraic variety over an algebraically closed field.[1]


Proof via finite fields

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 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.[1]. 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.[2] 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).


Further reading

External links



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