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

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.