On the semifree resolutions of DG algebras over the enveloping DG algebras

TL;DR

This paper develops a method to construct semifree resolutions for certain non-negatively graded DG algebras over their enveloping DG algebras, using bar resolution and tensor algebra techniques.

Contribution

It introduces a new construction of semifree resolutions for DG algebras over their enveloping algebras, expanding the toolkit for homological algebra in DG settings.

Findings

Constructed semifree resolutions using reduced bar resolution.

Applied tensor algebra of the shifted diagonal ideal.

Provided explicit methods for non-negatively graded strongly commutative DG algebras.

Abstract

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra $B$ over the enveloping DG algebra $B\otimes_AB$, where $A\subseteq B$ is a DG subalgebra and $B$ is semifree over $A$. Our construction of such a semifree resolution uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal.