Some applications of the frobenius in Characteristic 0

Abstract

Several applications are given of the technique of proving theorems in char 0 (as well as char p) by, in some sense, "reducing" to char p and then applying the Frobenius. A "metatheorem" for reduction to char p is discussed and the proof is sketched. This result is used later to give the idea of the proof of the existence of big Cohen-Macaulay modules in the equicharacteristic case. Homological problems related to the existence of big Cohen-Macaulay modules are discussed. A different application of the same circle of ideas is the proof that rings of invariants of reductive linear algebraic groups over fields of char 0 acting on regular rings are Cohen- Macaulay. Despite the fact that this result is false in char p, the proof depends on reduction to char p. A substantial number of examples of rings of invariants is considered, and a good deal of time is spent on the question, what does it really mean for a ring to be Cohen-Macaulay? The paper is intended for nonspecialists. © 1978 American Mathematical Society.