Differential graded algebras with divided powers and homotopy Lie algebras

TL;DR

This paper constructs a resolution of a quotient algebra as a differential graded algebra with divided powers, enabling the study of homotopy Lie algebras without Noetherian constraints.

Contribution

It introduces a new construction of PD dg algebra resolutions using symmetric tensors, applicable beyond Noetherian rings, and explores their homotopy Lie algebra properties.

Findings

Resolution has lifting properties useful for homotopy Lie algebra analysis

Connects complete intersection cases to Yoneda algebra finite generation

Provides a framework for studying homotopy Lie algebras in non-Noetherian settings

Abstract

Given a commutative algebra $A$ and a quotient $A$-algebra $A/I$, we construct a resolution of $A/I$ as an $A$-module such that it is also a differential graded (dg) algebra with divided powers (PD). This construction makes use of symmetric tensors in the symmetric tensor category of dg $A$-modules and does not require a Noetherian assumption on $A$. Moreover, the resolution has many lifting properties which we leverage to study the homotopy Lie algebra associated to the pair $(A,A/I)$, which is defined as the cohomology of the PD derivations of this PD dg algebra. Finally we investigate the complete intersection case in more details as well as connect it to the finite generation of the Yoneda algebra.