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

TY - JOUR

T1 - The structure of balanced big Cohen-Macaulay modules over Cohen-Macaulay rings

AU - Holm, Henrik Granau

PY - 2017

Y1 - 2017

N2 - 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 pre-envelope 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 has been proved by Auslander and Buchweitz.

AB - 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 pre-envelope 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 has been proved by Auslander and Buchweitz.

U2 - 10.1017/S0017089516000343

DO - 10.1017/S0017089516000343

M3 - Journal article

AN - SCOPUS:84973557335

SN - 0017-0895

VL - 59

SP - 549

EP - 561

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

IS - 3

ER -