Quillen (co)homology of divided power algebras over an operad

TL;DR

This paper explores Quillen cohomology for divided power algebras over an operad, identifying key structures and comparing it with cohomology of P-algebras, with detailed examples.

Contribution

It extends Quillen cohomology to divided power algebras over operads, identifying modules, derivations, and differentials, and compares it with P-algebra cohomology.

Findings

Identified Beck modules, derivations, and Kähler differentials for divided power algebras over an operad

Compared cohomology of divided power algebras with that of P-algebras

Worked out specific examples illustrating the theory

Abstract

Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example Andr\'e--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad $P$: the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad $P$, identifying Beck modules, derivations, and K\"ahler differentials in that setup. We also compare the cohomology of divided power algebras over $P$ with that of $P$-algebras, and work out some examples.