Homological identities and dualizing complexes of commutative differential graded algebras
type: Item
instance of: scholarly article
title: Homological identities and dualizing complexes of commutative differential graded algebras
zbMATH Open document ID: 1490.16020
author: Hiroyuki Minamoto
published in: Israel Journal of Mathematics
publication date: 2021-07-12T00:00:00Z
review text: Let \(R\) be differential graded (DG) algebra. It is said to be \textit{connective} if \(H^i(R)=0\) for \(i>0\). A connective DG algebra is quasi-isomorphic to a DG algebra \(S\) such that \(S^{>0}=0\). In the paper under review, the author considered connective commutative DG algebras \(R\) such that \(R^{>0}\). \(R\) is called a \textit{piece-wise Noetherian} if \(H^0(R)\) is Noetherian and \(H^i(R)\) is finitely generated for all \(i<0\). The author introduced the concept of \textit{dualizing complexes} for piece-wise Noetherian connective commutative DG algebras. In [Isr. J. Math. 245, No. 1, 409--454 (2021; Zbl 1504.16015)], the author introduced the notions of sppj and ifij resolutions for DG modules over a DG algebra. Assume a piece-wise Noetherian connective commutative DG algebra \(R\) has a dualizing complex \(D\) and \(H^0(R)\) is a local ring. The author provided an explicit description of the structure of the minimal ifij resolution of \(D\) in the present paper, which is a generalization of the corresponding result in the classical commutative ring theory. The author also considered the Gorenstein property of a connective commutative DG algebra \(R\). If the cohomology of \(R\) is bounded, that is, \(H^i(R)=0\) for \(i\ll 0\), then the property that \(R\) is Gorenstein DG algebra and that \(H(R)\) is a Cohen-Macaulay graded ring is equivalent to the property that \(H(R)\) is Gorenstein as an ordinary graded commutative ring. Some other homological identities, such as Auslander-Buchsbaum formula and Bass formula, for connective commutative DG algebras were also established in the paper.
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16E65
zbMATH DE Number: 7370890
zbMATH Keywords: connective commutative DG algebra
zbMATH Keywords: dualizing complex
zbMATH Keywords: Gorenstein DG algebra
reviewed by: Ji-Wei He
MaRDI profile type: Publication
full work available at URL: s11856-021-2095-3
OpenAlex ID: W3122770616
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DOI: 10.1007/S11856-021-2095-3
description: scientific article; zbMATH DE number 7370890