Residue complexes over noncommutative rings.
type: Item
instance of: scholarly article
title: Residue complexes over noncommutative rings.
zbMATH Open document ID: 1035.16004
DOI: 10.1016/S0021-8693(02)00579-3
author: Amnon Yekutieli
author: J. J. Zhang
published in: Journal of Algebra
publication date: 2003-04-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0103075
review text: Grothendieck's concept of a residue complex is generalized to noncommutative rings, where it becomes a dualizing complex of injective bimodules (in the sense of the first author [\textit{A. Yekutieli}, J. Algebra 153, No. 1, 41-84 (1992; Zbl 0790.18005)]) which is rigid in the sense of \textit{M. Van den Bergh} [J. Algebra 195, No. 2, 662-679 (1997; Zbl 0894.16020)] and satisfies the Auslander conditions and related purity conditions. The authors prove uniqueness, existence, and functoriality of residue complexes under appropriate hypotheses. (As they warn, however, algebras such as \(U({\mathfrak{sl}}_2(\mathbb{C}))\) which are ``too noncommutative'' may not have residue complexes.) In particular, if \(A\) is an affine Noetherian PI algebra (over a field) admitting a Noetherian connected filtration, they prove that \(A\) has a residue complex \({\mathcal K}_A\), necessarily unique up to unique isomorphism in \(\text{D}(\text{Mod\,}A\otimes A^{\text{op}})\), and they describe its structure in detail. For instance, \({\mathcal K}_A^{-q}\) is a direct sum of indecomposable bimodules \(\Gamma_Z{\mathcal K}_A^{-q}\) supported on those cliques \(Z\subseteq\text{Spec\,}A\) with \(j_{{\mathcal K}_A}(A/{\mathfrak p})=-q\) for \({\mathfrak p}\in Z\), and as a left module each \(\Gamma_Z{\mathcal K}_A^{-q}\) is the direct sum over \({\mathfrak p}\in Z\) of \(\text{Goldie-rank}(A/{\mathfrak p})\) copies of the unique indecomposable injective with associated prime \(\mathfrak p\).
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 16R99
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 1891330
zbMATH Keywords: residue complexes
zbMATH Keywords: rigid dualizing complexes
zbMATH Keywords: Auslander dualizing complexes
zbMATH Keywords: Cousin functors
zbMATH Keywords: polynomial identity algebras
MaRDI profile type: Publication
arXiv ID: math/0103075
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description: scientific article