Student Algebraic Geometry Seminar

Let $k$ be an algebraically closed field and $V$ a variety over $k$. The Ax Grothendieck Theorem states that any algebraic endomorphism of $V$ injective on the $k$-points is bijective. Trivial if $k$ is replaced by a finite field, in the characteristic 0 case even when $k=C$ and $V$ is a complex vector space, the theorem is far from obvious. In this talk we will prove the theorem by showing a method of reducing the char 0 case to char p, and then time permitting discuss how this result can be interpreted as exhibiting the model independence of a certain logical class of algebraic statements, thus allowing a general technique of translating between different field-theoretic contexts. Speaker(s): Matthew Sawoski (UM)