Iterated bar complexes andEn-homology with coefficients

Abstract

The first author proved in a previous paper that the n-fold bar construction for commutative algebras can be generalized to En-algebras, and that one can calculate Enhomology with trivial coefficients via this iterated bar construction. We extend this result to En-homology and En-cohomology of a commutative algebra A with coefficients in a symmetric A-bimodule. IntroductionBoardman-Vogt [BV73] and May [Ma72] showed in the early 70's that n-fold loop spaces are equivalent to algebras over the little n-cubes operad (at least, up to group completion issues, or provided that we restrict ourselves to connected spaces). In this paper, we deal with E noperads in chain complexes, which are defined as operads quasi-isomorphic to the chain operad of little n-cubes, for any 1 n ∞. The category of E n -algebras, algebras over an E n -operad, can be thought of as an algebraic version of the category of n-fold loop spaces. If we choose a Σ * -cofibrant E n -operad, let E n , then we can define homological invariants specifically suited to E n -algebras, namely the E n -homology and E n -cohomology of an E n -algebra A with coefficients in a representation M of A. These invariants are defined via derived Quillen functors. Note that every strictly commutative algebra A is an E n -algebra for any 1 n ∞, and that every symmetric A-bimodule is a representation of the E n -algebra A. The homology of E n -algebras can also be regarded as a particular case of the factorization homology theory (see [Fra13]), and, in the case of commutative algebras, agrees with Pirashvili's higher Hochschild homology theory (see [Pir00]). In the case n = ∞, we retrieve the Γ-homology theory of [RW02].In [Fre11] the first author also proved that the classical iterated bar complex of augmented commutative algebras B n (A) = B • · · · • B(A), such as defined by Eilenberg and MacLane (see [EM53]), computes the E n -homology of A with coefficients in the commutative unital ground ring M = k. The purpose of this paper is to extend this construction to the nontrivial coefficient case in order to get an explicit chain complex computing the E n -homology and the E n -cohomology of commutative algebras with coefficients. In short, we define a chain complex (M ⊗ B n (A), ∂ θ ) by adding a twisting differential ∂ θ : M ⊗ B n (A) → M ⊗ B n (A) to the tensor product of the Eilenberg-MacLane iterated bar complex B n (A) with any symmetric A-bimodule of coefficients M (see Definition 1.2). We then prove that the chain complex B [n] * (A, M ) = (M ⊗ Σ −n B n (A), ∂ θ ), where we use an n-fold desuspension Σ −n to mark an extra degree shift, computes the E n -homology of A with coefficients in M (see Theorem 1.3). Thus, we get 2000 Mathematics Subject Classification. 55P48, 57T30, 16E40. Key words and phrases. En-homology, Iterated bar construction, En-operad, Module over an operad, Hochschild homology.