Some homological invariants of local PI algebras
type: Item
instance of: scholarly article
title: Some homological invariants of local PI algebras
zbMATH Open document ID: 0955.16010
author: J. J. Zhang
author: Quan-Shui Wu
published in: Journal of Algebra
publication date: 2001-02-27T00:00:00Z
review text: The following main results for a Noetherian PI-algebra \(A\) are proved: (1) if \(A\) is a local algebra then the Auslander-Buchsbaum formula holds; i.e., if \(X\in D^b_{fg}(\text{Mod }A)\) with \(\text{pd }X<\infty\), then \(\text{pd }X+\text{depth }X=\text{depth }A\); (2) if \(A\) is a semilocal algebra and \(\widehat A\) its completion then (i) for all finite \(A\)-modules \(M\): \(\text{K-dim }M=\text{K-dim}_{\widehat A}\widehat M\); (ii) \(\text{id }X=\text{id}_{\widehat A}\widehat X\) for all \(X\in D^b_{fg}(\text{Mod }A)\); (3) if an \((A,T)\)-bimodule \(_AE_T\) defines a Morita duality between \(A\) and \(T\) then \(T\) is a Noetherian PI-algebra; (4) if \(A\) is a semilocal algebra and \(M\) is a finite \(A\)-module then \(\text{lcd}(M)=\text{K-dim }M\) and \(H^d_{\mathcal M}(M)\) has dual Krull dimension \(d\) where \(d=\text{K-dim }M\).
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16R99
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1429475
zbMATH Keywords: Noetherian PI-algebras
zbMATH Keywords: local algebras
zbMATH Keywords: Auslander-Buchsbaum formula
zbMATH Keywords: semilocal algebras
zbMATH Keywords: completions
zbMATH Keywords: Morita dualities
zbMATH Keywords: dual Krull dimensions
reviewed by: Yu. N. Maltsev
MaRDI profile type: Publication
full work available at URL: jabr.1999.8179
OpenAlex ID: W2033163806
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DOI: 10.1006/JABR.1999.8179
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description: scientific article; zbMATH DE number 1429475