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A new homological dimension, called GCM-dimension, will be defined
for any finitely generated module $M$ over a local Noetherian ring
$R$. GCM-dimension (short for Generalized Cohen-Macaulay
dimension) characterizes Generalized Cohen-Macaulay rings in the
sense that: a ring $R$ is Generalized Cohen-Macaulay if and only
if every finitely generated $R$-module has finite GCM-dimension.
This dimension is finer than CM-dimension and we have equality if
CM-dimension is finite. Our results will show that this dimension
has expected basic properties parallel to those of the homological
dimensions. In particular, it satisfies an analog of the
Auslander-Buchsbaum formula. Similar methods will be used for
introducing quasi-Buchsbaum and Almost Cohen-Macaulay dimensions,
which reflect corresponding properties of their underlying rings.
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