Injective resolutions of some regular rings
type: Item
instance of: scholarly article
title: Injective resolutions of some regular rings
zbMATH Open document ID: 0945.16009
DOI: 10.1016/S0022-4049(99)00049-3
author: J. J. Zhang
author: Kaushal Ajitabh
author: S. Paul Smith
published in: Journal of Pure and Applied Algebra
publication date: 2000-07-16T00:00:00Z
review text: The authors investigate the extent to which the homogeneity known to hold for the minimal injective resolution of a commutative Noetherian local Gorenstein ring carries over to a noncommutative Noetherian Auslander-Gorenstein ring \(A\). In this context, the desired homogeneity is phrased as a purity condition with respect to the canonical dimension function on \(A\)-modules, given by \(\delta(M)=d-\text{grade}(M)\), where \(d=\text{inj.}\dim(A)\). An \(A\)-module is called \(s\)-pure if all its nonzero Noetherian submodules \(N\) satisfy \(\delta(N)=s\), and essentially \(s\)-pure if it contains an essential submodule which is \(s\)-pure. Finally, \(A\) is said to have a pure (essentially pure) injective resolution if the \(i\)-th term of the minimal injective resolution of \(A_A\) is always \((d-i)\)-pure (essentially \((d-i)\)-pure). The authors showed earlier that a pure minimal injective resolution exists when \(A\) is grade-symmetric and satisfies a polynomial identity [Commun. Algebra 26, No. 7, 2159-2180 (1998; Zbl 0910.16005)]. However, even essentially pure resolutions do not exist in general, as shown by examples of Artin and Stafford [in loc. cit.]. In the present paper, it is proved that pure or essentially pure minimal injective resolutions occur for several classes of examples with \(d\leq 4\). For instance, pure resolutions occur for Auslander-Gorenstein domains \(A\) with injective dimension 2 such that every simple \(A\)-module with grade 2 is annihilated by a completely prime normal element, and that essentially pure resolutions occur for the analogous Auslander-Gorenstein domains with injective dimension 3. Essentially pure graded injective resolutions are obtained for the four-dimensional Sklyanin algebra and for the homogenized enveloping algebra of any three-dimensional Lie algebra (over an algebraically closed field). The authors' route to these results involves some work with noncommutative Cohen-Macaulay conditions that is interesting in it…
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1330822
zbMATH Keywords: injective resolutions
zbMATH Keywords: Gorenstein rings
zbMATH Keywords: Auslander-Gorenstein rings
zbMATH Keywords: purities
zbMATH Keywords: Auslander-regular rings
zbMATH Keywords: Auslander-Gorenstein domains
zbMATH Keywords: injective dimensions
zbMATH Keywords: essentially pure resolutions
Wikidata QID: Q114122615
reviewed by: Kenneth R. Goodearl
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1330822