Results 1 
2 of
2
Transferring Algebra Structures Up to Homology Equivalence
 Math. Scand
, 1998
"... 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] ba ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
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\Omega we mean\Omega R . The usual (Koszul) sign conventions are assumed. The degree of a homogeneous element m of some module is denoted by jmj. 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 Rmodule of maps from M to N (for Rmodules 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 \Gamma (\Gamma1) jf j 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. If A is an algebra and C is a coalgebra, the module hom(C; A) is an algebra with 1 respect to the operation defined by the following diagram C f [ g  A C\Omega C \Delta ? f\Omega g  A\Omega A 6 m (1) This product is called the cup or convolution...
"Coalgebra" Structures on 1Homological Models for Commutative Differential Graded Algebras
"... In [3] "mall" 1homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [79] provides an explicit description of an A1coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination of the map ..."
Abstract
 Add to MetaCart
In [3] "mall" 1homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [79] provides an explicit description of an A1coalgebra structure ( 1 ; 2 ; 3 ; : : :) of H. In this paper, we are mainly interested in the determination of the map 2 : H ! H H as a first step in the study of this structure. Developing the techniques given in [20] (inversion theory), we get an important improvement in the computation of 2 with regard to the first formula given by HPT. In the case of purely quadratic algebras, we sketch a procedure for giving the complete Hopf algebra structure of its 1homology.