536 research outputs found
Vanishing of cohomology over Cohen--Macaulay rings
A 2003 counterexample to a conjecture of Auslander brought attention to a
family of rings - colloquially called AC rings - that satisfy a natural
condition on vanishing of cohomology. Several results attest to the remarkable
homological properties of AC rings, but their definition is barely operational,
and it remains unknown if they form a class that is closed under typical
constructions in ring theory. In this paper, we study transfer of the AC
property along local homomorphisms of Cohen--Macaulay rings. In particular, we
show that the AC property is preserved by standard procedures in local algebra.
Our results also yield new examples of Cohen-Macaulay AC rings.Comment: Updated references. Final version to appear in Manuscripta Math.; 9
p
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
Ascent Properties of Auslander Categories
Let R be a homomorphic image of a Gorenstein local ring. Recent work has
shown that there is a bridge between Auslander categories and modules of finite
Gorenstein homological dimensions over R.
We use Gorenstein dimensions to prove new results about Auslander categories
and vice versa. For example, we establish base change relations between the
Auslander categories of the source and target rings in a homomorphism R -> S of
finite flat dimension.Comment: Minor corrections; example added; 30 pp. To appear in Canad. J. Math.
Also available from authors' homepages
http://www.math.unl.edu/~lchristensen3/publications.html and
http://home.imf.au.dk/holm/publications.htm
Descent via Koszul extensions
Let R be a commutative noetherian local ring with completion R^. We apply
differential graded (DG) algebra techniques to study descent of modules and
complexes from R^ to R' where R' is either the henselization of R or a pointed
\'etale neighborhood of R: We extend a given R^-complex to a DG module over a
Koszul complex; we describe this DG module equationally and apply Artin
approximation to descend it to R.
This descent result for Koszul extensions has several applications. When R is
excellent, we use it to descend the dualizing complex from R^ to a pointed
\'etale neighborhood of R; this yields a new version of P. Roberts' theorem on
uniform annihilation of homology modules of perfect complexes. As another
application we prove that the Auslander Condition on uniform vanishing of
cohomology ascends to R^ when R is excellent, henselian, and Cohen--Macaulay.Comment: Updated references and made minor changes. Final version, to appear
in J. Algebra; 19 p
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