Homological identities and dualizing complexes of commutative differential graded algebras
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.
scientific article; zbMATH DE number 7370890
scientific article