Cohen-Macaulay isolated singularities with a dualizing module.
type: Item
instance of: scholarly article
title: Cohen-Macaulay isolated singularities with a dualizing module.
zbMATH Open document ID: 1109.16012
author: Kenji Nishida
published in: Algebras and Representation Theory
publication date: 2006-09-12T00:00:00Z
review text: Let \(R\) be a complete local Gorenstein domain of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order which is an isolated singularity in the sense of \textit{M. Auslander} [Lect. Notes Math. 1178, 194-242 (1986; Zbl 0633.13007)], i.e. \(\text{gld\,}\Lambda_p=\dim R_p\) for non-maximal primes \(p\). By \(\Lambda\mathbf{CM}\) we denote the category of (maximal) Cohen-Macaulay representations of \(\Lambda\). The author continues his former study (see [Algebr. Represent. Theory 5, No. 2, 137-147 (2002; Zbl 1003.16005)], for the one-dimensional case) of dualizing \((\Lambda,\Lambda)\)-bimodules, generalizing previous results of \textit{H.-B. Foxby} [Math. Scand. 31(1972), 267-284 (1973; Zbl 0272.13009)] to the non-commutative case. By definition, a bimodule \(\omega\in\Lambda\mathbf{CM}\) is called dualizing if it is balanced, (left and right) self-orthogonal with finite injective dimension, such that \(\text{Ext}^d_\Lambda(S,\Hom_R(\omega,R))\not=0\) for simple \(\Lambda\)-modules \(S\). Thus dualizing \(\Lambda\)-modules are fairly similar to cotilting modules. Following Foxby, the author associates two subcategories \(\mathcal A\) and \(\mathcal B\) of modules to any dualizing module \(\omega\) which are equivalent via the Hom- and tensor functor of \(\omega\). Furthermore, he shows that the \(\Lambda\)-modules of finite Gorenstein-dimension belong to \(\mathcal A\). The two categories \(\mathcal A\) and \(\mathcal B\) coincide if and only if \(\Lambda\) is a Gorenstein order. Apart from this criterion, the author provides eight other (essentially asymmetric) characterizations of this property. The results are illustrated by examples related to almost Bass orders (introduced by Hijikata in his study of local orders) and \(k\)-Gorenstein algebras.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 5052960
zbMATH Keywords: complete local Gorenstein domains
zbMATH Keywords: Cohen-Macaulay orders
zbMATH Keywords: maximal Cohen-Macaulay representations
zbMATH Keywords: dualizing bimodules
zbMATH Keywords: cotilting modules
zbMATH Keywords: Gorenstein orders
MaRDI profile type: Publication
full work available at URL: s10468-005-9004-z
OpenAlex ID: W1970495219
cites work: Stable module theory
cites work: Q4012100
cites work: Applications of contravariantly finite subcategories
cites work: Q3240881
cites work: Gorenstein dimensions
cites work: Gorenstein Modules and Related Modules.
cites work: Minimal injective resolution of an order over a local dedekind domain
cites work: Minimal injective resolutions of Cohen-Macaulay isolated singularities
cites work: Finite Modules of Finite Injective Dimension Over a Noetherian Algebra
cites work: Towards a theory of Bass numbers with application to Gorenstein algebras.
cites work: On the derived category of a finite-dimensional algebra
cites work: Bass orders in non semisimple algebras
cites work: A characterization of Gorenstein orders
cites work: Dualizing modules for orders and Artin algebras
cites work: Maximal Orders over Regular Local Rings of Dimension Two
DOI: 10.1007/S10468-005-9004-Z
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description: scientific article