On commutative differential graded algebras

TL;DR

This paper develops foundational results for connective commutative differential graded algebras, introducing new resolutions that extend classical homological algebra tools to the DG setting, including structure theorems and Gorenstein conditions.

Contribution

It introduces sup-projective and inf-injective resolutions for DG-modules, generalizes classical homological formulas, and establishes structure theorems for dualizing complexes in the context of CDGAs.

Findings

Established DG-version of Bass's structure theorem for minimal injective resolutions.

Proved DG-analogues of classical formulas like Auslander-Buchsbaum and Bass.

Generalized Gorenstein conditions for CDGAs in terms of cohomology algebra.

Abstract

In this paper we undertake a basic study on connective commutative differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA, which is a DG-counter part of commutative Noetherian algebra. We establish basic results for example, Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolution for ordinary modules. These are different from DG-projective and DG-injective resolutions which is known DG-version of the projective and injective resolution. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules. Many classical result about the projective and injective resolutions can be generalized to DG-setting by using sppj and ifij resolutions. . Among other things we prove a DG-version of Bass's structure theorem of a minimal injective resolution holds for a minimal ifij resolution and a DG-version of the Bass numbers introduced by the same formula with the classical case. We also prove a structure theorem of a minimal ifij resolution of a dualizing complex $D$, which is completely analogues to the structure theorem of a minimal injective resolution of a dualizing complex over an ordinary commutative algebra. Specializing to results about a dualizing complex, we study a Gorenstein CDGA. We generalize a result by Felix-Halperin-Felix-Thomas and Avramov-Foxby which gives conditions that a CDGA $R$ is Gorenstein in terms of its cohomology algebra $\text{H}(R)$.