Gamma-homology of algebras over an operad

TL;DR

This paper extends Gamma-homology to operads, especially outside characteristic zero, by using cofibrant replacements, and provides explicit complexes for computation, including for the commutative operad.

Contribution

It introduces a generalized Gamma-homology framework for operads using cofibrant replacements, with explicit computational tools for binary Koszul operads.

Findings

Explicit small complex for Gamma-homology when operad is binary and Koszul

Retrieves Robinson's complex for commutative algebras

Framework applicable outside characteristic zero contexts

Abstract

The purpose of this paper is to study generalizations of Gamma-homology in the context of operads. Good homology theories are associated to operads under appropriate cofibrancy hypotheses, but this requirement is not satisfied by usual operads outside the characteristic zero context. In that case, the idea is to pick a cofibrant replacement Q of the given operad P. We can apply to P-algebras the homology theory associated to Q in order to define a suitable homology theory on the category of P-algebras. We make explicit a small complex to compute this homology when the operad P is binary and Koszul. In the case of the commutative operad P=Com, we retrieve the complex introduced by Robinson for the Gamma-homology of commutative algebras.