Transferring Algebra Structures Up to Homology Equivalence

Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in [16]. Applications to de Rham theory and Massey products are given. 1 Preliminaries and Notation Throughout this paper, R will denote a commutative ring with unit. The term (co)module is used to mean a differential graded (co)module over R and maps between modules are graded maps. When we write ⊗ we mean ⊗R. The usual (Koszul) sign conventions are assumed. The degree of a homogeneous element m of some module is denoted by |m|. Algebras are assumed to be connected and coalgebras simply connected. (Co)algebras are assumed to have (co)units.(Co)algebras are, unless otherwise stated, assumed to be (co)augmented. The differential in an (co)algebra is a graded (co)derivation. The R-module of maps from M to N (for R-modules M and N) is denoted by hom(M,N) (if the context requires it, we will use a subscript to denote the ground ring). The differential in this module is given by D(f) = df − (−1)|f |fd. Note that D is a derivation with respect to the composition operation whenever it is defined. In particular, End(M) = hom(M,M) is an algebra.