Operadic comodules and (co)homology theories

TL;DR

This paper develops a framework using operadic comodules to compute (co)homology of algebras, introduces spectral sequences from filtrations, and applies these methods to classical and new algebraic structures.

Contribution

It introduces a novel operadic comodule approach to (co)homology, generalizes spectral sequence computations, and provides new proofs and calculations for various operads.

Findings

Operadic comodules can represent (co)homology of algebraic structures.

Filtrations of comodules lead to spectral sequences for (co)homology.

New proof of Hodge decomposition for Hochschild cohomology of commutative algebras.

Abstract

An operad describes a category of algebras and a (co)homology theory for these algebras may be formulated using the homological algebra of operads. A morphism of operads $f:\mathcal{O}\rightarrow\mathcal{P}$ describes a functor allowing a $\mathcal{P}$-algebra to be viewed as an $\mathcal{O}$-algebra. We show that the $\mathcal{O}$-algebra (co)homology of a $\mathcal{P}$-algebra may be represented by a certain operadic comodule. Thus filtrations of this comodule result in spectral sequences computing the (co)homology. As a demonstration we study operads with a filtered distributive law; for the associative operad we obtain a new proof of the Hodge decomposition of the Hochschild cohomology of a commutative algebra. This generalises to many other operads and as an illustration we compute the post-Lie cohomology of a Lie algebra.