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|This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.|
|Y. Yoshino, Hideyuki Matsumura. Chapter 1. Preliminaries In this chapter we will |
review some basic facts without proofs and give some of the basic notation that
will be used throughout the book. For further results in commutative algebra we ...
|The study of Cohen-Macaulay modules over Noetherian local rings originates |
from the theory of integral representations of finite groups (which, on its side,
grew up from a very classical problem of classification of crystallographic groups,
|First, we recall that a module M of finite type over a local ring R is called a Cohen-|
Macaulay module if depth M = dim M. In general, depth M < dim- M < dim R. If
depth M = dim M = dim R we call M a maximal Cohen-Macaulay module. If R
|Definition 3.3.7 (Big Cohen-Macaulay Modules). Let R be a Noetherian local ring, |
and let M be an arbitrary R-module. We call M a big Cohen-Macaulay module, if
there exists a system of parameters in R which is M-regular. If moreover every ...
|Definition of Cohen-Macaulay modules By prop. 7, for every pe Ass(E), we have |
dim(A/p) > depth(E). Since dim E = sup dim(A/p) for p e Ass(E), we have in
particular dim E > depth E. Definition 1. The module E is called a Cohen-
|nonzero finitely generated module M is Cohen-Macaulay if and only if M is |
projective, or equivalently, if and only if M is torsion free; d) if R is a regular ring,
then a finitely generated 72-module M is Cohen-Macaulay if and only if M is
Michiel Hazewinkel - 1994 - Preview
|A regular local ring (and, in general, any Gorenstein ring) is a Cohen - Macaulay |
ring; any Artinian ring, any one- dimensional reduced ring ... A module M over a
local ring A is called a Cohen -Macaulay module if its depth equals its dimension.