A Theorem About Maximal Cohen–Macaulay Modules

Thomas Polstra

doi:10.1093/imrn/rnaa154

Abstract

Abstract It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

Highlights

Let R be a Noetherian commutative ring of prime characteristic p > 0 and for each e ∈ N let F e : R → R be the eth iterate of the Frobenius endomorphism of R
The main contribution of this article is the following uniform property concerning the class of finitely generated Cohen-Macaulay modules in strongly F -regular rings
A new application of Theorem A is that the torsion subgroup of the divisor class group of any local strongly F -regular ring is finite, see Corollary 3.3

Summary

Introduction

Let R be a Noetherian commutative ring of prime characteristic p > 0 and for each e ∈ N let F e : R → R be the eth iterate of the Frobenius endomorphism of R. Studying the class of Cohen-Macaulay modules in a strongly F -regular ring is a warranted venture. The main contribution of this article is the following uniform property concerning the class of finitely generated Cohen-Macaulay modules in strongly F -regular rings. Let (R, m, k) be a local F -finite and strongly F -regular ring of prime characteristic p > 0. We show any minimal generator of a maximal Cohen-Macaulay module M can be sent to 1 in R under an R-linear map F∗e0M → R in Theorem 3.1. A new application of Theorem A is that the torsion subgroup of the divisor class group of any local strongly F -regular ring is finite, see Corollary 3.3. We utilize Theorem A to rederive two other important properties of local strongly F regular rings.

Preliminary results and notation

Main results