160 research outputs found
Acyclicity versus total acyclicity for complexes over noetherian rings
It is proved that for a commutative noetherian ring with dualizing complex
the homotopy category of projective modules is equivalent, as a triangulated
category, to the homotopy category of injective modules. Restricted to compact
objects, this statement is a reinterpretation of Grothendieck's duality
theorem. Using this equivalence it is proved that the (Verdier) quotient of the
category of acyclic complexes of projectives by its subcategory of totally
acyclic complexes and the corresponding category consisting of injective
modules are equivalent. A new characterization is provided for complexes in
Auslander categories and in Bass categories of such rings.Comment: 29 pages. The main changes are the addition of Lemma 2.6, needed in
the proof of Theorem 2.7, and Remark 5.11, replacing an incorrect statement
in an earlier version of the paper. The paper will now appear in Documenta
Mathematic
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
Gorenstein dimension of modules over homomorphisms
Given a homomorphism of commutative noetherian rings R --> S and an S-module
N, it is proved that the Gorenstein flat dimension of N over R, when finite,
may be computed locally over S. When, in addition, the homomorphism is local
and N is finitely generated over S, the Gorenstein flat dimension equals sup{m
| Tor^R_m(E,N) \noteq 0} where E is the injective hull of the residue field of
R. This result is analogous to a theorem of Andr\'e on flat dimension.Comment: 14 pp. To appear in J. Pure Appl. Algebra. Also available from
http://www.math.unl.edu/~lchristensen3/index.htm
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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