In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck.
[1][2][3][4] The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective.
In this case the theorem is true for trivial reasons having nothing to do with the representation of the function as a polynomial: any injection of a finite set to itself is a bijection.
[3] Thus, one can use the arithmetic of finite fields to prove a statement about C even though there is no homomorphism from any finite field to C. The proof thus uses model-theoretic principles such as the compactness theorem to prove an elementary statement about polynomials.
Therefore, a scheme of finite presentation over a base S is a cohopfian object in the category of S-schemes.