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THE STRUCTURE OF BALANCED BIG COHEN–MACAULAY MODULES OVER COHEN–MACAULAY RINGS

Part of: Commutative algebra: Homological methods Theory of modules and ideals

Published online by Cambridge University Press:  10 June 2016

HENRIK HOLM
HENRIK HOLM*
Affiliation:
Department of Mathematical Sciences, Universitetsparken 5, University of Copenhagen, 2100 Copenhagen Ø, Denmark e-mail: holm@math.ku.dk
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Abstract

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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.

MSC classification

Primary: 13C14: Cohen-Macaulay modules
Secondary: 13D05: Homological dimension 13D07: Homological functors on modules (Tor, Ext, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 549 - 561
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

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