Given a function f in Z[x_1, ..., x_n], for each prime p, we can consider the image f_p in F_p[x_1, ..., x_n]. This family of prime characteristic models of f give us data about the original function f: for example, if any of the polynomials f_p are irreducible over F_p, then f is irreducible over Q.

In this talk, we'll explain how we can construct a family of prime characteristic models to any algebraic variety X over C. These models encode tremendous amounts of information about the original variety. For instance, X is smooth if and only if (X mod p) is smooth for infinitely many primes p. We will discuss various applications of this perspective, including the following theorem:

Let S be a regular ring containing Q or F_p and R a subring which is also a direct summand of S. Then R is Cohen-Macaulay.