Iterated bar complexes and E_n-homology with coefficients

TL;DR

This paper extends the iterated bar complex method to compute E_n-homology and cohomology with coefficients in symmetric bimodules for commutative algebras, generalizing previous results with trivial coefficients.

Contribution

It introduces a new approach to calculate E_n-homology and cohomology with coefficients, broadening the applicability of the iterated bar construction.

Findings

Extended the iterated bar complex to E_n-homology with coefficients

Provided a method to compute E_n-cohomology with coefficients

Generalized previous trivial coefficient results to symmetric bimodules

Abstract

The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized to E_n-algebras, and that one can calculate E_n-homology with trivial coefficients via this iterated bar construction. We extend this result to E_n-homology and E_n-cohomology of a commutative algebra A with coefficients in a symmetric A-bimodule.