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Linearity defects of modules over commutative rings
This article concerns linear parts of minimal resolutions of finitely
generated modules over commutative local, or graded rings. The focus is on the
linearity defect of a module, which marks the point after which the linear part
of its minimal resolution is acyclic. The results established track the change
in this invariant under some standard operations in commutative algebra. As one
of the applications, it is proved that a local ring is Koszul if and only if it
admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective
analogue of the linearity defect is introduced and studied. The main results
express this new invariant in terms of linearity defects of free resolutions,
and relate it to other ring theoretic and homological invariants of the module.Comment: 23 pages, minor modification
Homological dimensions and regular rings
A question of Avramov and Foxby concerning injective dimension of complexes
is settled in the affirmative for the class of noetherian rings. A key step in
the proof is to recast the problem on hand into one about the homotopy category
of complexes of injective modules. Analogous results for flat dimension and
projective dimension are also established.Comment: 8 page
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