Homological properties of cochain Differential Graded algebras

TL;DR

This paper extends key homological results from local chain Differential Graded algebras to simply connected cochain Differential Graded algebras, broadening the understanding of their algebraic properties.

Contribution

It proves analogous homological theorems for simply connected cochain Differential Graded algebras, inspired by previous results for local chain DGAs.

Findings

Established an Amplitude Inequality for cochain DGAs

Proved an Auslander-Buchsbaum Equality in this context

Demonstrated a Gap Theorem for simply connected cochain DGAs

Abstract

Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander-Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes such analogous results.