The structure of balanced big Cohen-Macaulay modules over Cohen-Macaulay rings

TL;DR

This paper characterizes balanced big Cohen-Macaulay modules over Cohen-Macaulay rings as direct limits of finitely generated maximal CM modules, providing a structure theorem and insights into module approximations.

Contribution

It establishes a new structure theorem for balanced big CM modules as direct limits, extending classical module theory results.

Findings

Balanced big CM modules are direct limits of small CM modules.

Every finitely generated module admits a preenvelope with respect to maximal CM modules.

Provides a dual perspective to maximal CM approximations.

Abstract

Over a Cohen-Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a "structure theorem" for balanced big CM modules. (2) Every finitely generated module has a preenvelope with respect to the class of finitely generated maximal CM modules. This result is, in some sense, dual to the existence of maximal CM approximations, which is proved by Auslander and Buchweitz.