Duality for cochain DG algebras

TL;DR

This paper establishes a duality theory for connected cochain DG algebras, focusing on non-commutative cases, and demonstrates how certain finiteness properties transfer via a dualizing DG module.

Contribution

It introduces a duality framework for non-commutative cochain DG algebras using a dualizing DG module, linking derived categories of modules with finitely generated cohomology.

Findings

Duality between derived categories of DG modules established.

Conditions under which finiteness properties of modules are preserved.

Application to the semi-free resolution of the canonical module.

Abstract

This paper develops a duality theory for connected cochain DG algebras, with particular emphasis on the non-commutative aspects. One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology. As an application, it is proved that if the canonical module $A / A^{\geq 1}$ has a semi-free resolution where the cohomological degree of the generators is bounded above, then the same is true for each DG module with finitely generated cohomology.