Rectifications of A-Algebras

Communications in Algebra®, 35: 2731–2743, 2007 Copyright © Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870701353514 RECTIFICATIONS OF A-ALGEBRAS M. J. Jiménez and P. Real Department of Applied Mathematics, University of Sevilla, Sevilla, Spain In this article, in the setting of connected DG-modules, we prove that, for any A-algebra M mii≥1, there is a chain contraction from a DG-algebra AMonto the DG-module Msuch that the A-algebra structure induced by perturbation theory on Mis precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A-algebra in the sense of Boardman and Vogt (1973). Appropiate dual results are given for A-coalgebras. Key Words: A-Algebra; A-Coalgebra; Contraction; Tilde bar construction, Tilde cobar construction. Mathematics Subject Classification: 18E30; 18G35; 16E45. 1. INTRODUCTION One classical result claims that, given a topological operad , each W-space (strongly homotopy -space) can be “replaced” by a strict -space of the same homotopy type. More concretely, for each W-space, X, there exists a strict -space, MX, together with a strong deformation retraction MX X r  MX →X i  X→MX in which iis a homomorphism of W-spaces (Boardman and Vogt, 1973). Then, MX is called the rectification of X. Taking =Ass, the associative operad, the result means that any A-space is homotopy equivalent to an Ass-space, the latter being the rectification of the former. Basic ideas of the construction of MX on A-spaces are given in Markl et al. (2002). The aim of this article is to construct a similar rectification in the setting of DG-operads and DG-algebras over them, rather than topological operads and algebras over them (which are topological spaces). This way, we assert that, working in the context of connected DG-modules, for any A-algebra M mii≥1, there exists an associative DG-algebra, AM, and a chain contraction from AMto M, that allows AMto be regarded as the rectification of M. In the dual case of A-coalgebras, our framework is the one of simply connected DG-modules. However, we must emphasize some significant aspects of our work in order to make clear the differences with the mentioned topological result: (a) our theorem can be used for A-algebras which do not come from topology, that is, which are not Received February 14, 2006; Revised October 24, 2006. Communicated by K. C. Misra. Address correspondence to M. J. Jiménez, Department of Applied Mathematics, University of Sevilla, Avda. Reina Mercedes, Sevilla, SN 41012, Spain; E-mail: [email protected] 2731 2732 JIMÉNEZ AND REAL chain complexes of topological A-spaces (for example, the A-algebra cohomology of a space H∗X mii≥1); (b) in our situation, the rectification AMof an A- algebra M mii≥1isafree graded algebra, so it provides a free DG-algebra model AM→M mii≥1of an A-algebra (cf. Baues and Lemaire, 1977; Huebschmann and Kadeishvili, 1991, where free models for DG-algebras are considered); (c) it is known that a chain contraction from AMto the chain complex M m1generates, via the tensor trick and the Basic Perturbation Lemma (Gugenheim, 1977; Gugenheim and Stasheff, 1986; Gugenheim et al., 1991), an A-algebra structure on M. In our case, such an A-structure is precisely the initial one. Our method to construct the rectification AMis specific to algebraic framework since the main tool used is the Basic Perturbation Lemma. Namely, we construct first, for a chain complex M m1, an explicit contraction from TcsM, the reduced cobar construction of the tensor coalgebra of the suspension of M,toMand then, using the Basic Perturbation Lemma for the suitable perturbation data defined by the A- operations mii>1, we obtain a contraction from  BM to M, being AM= BM the aimed DG-algebra. We remark that the Munkholm’s (1974) contraction (called trivialized extension) from  BA to a DG-algebra Acan be obtained exactly in the same way, but considering the perturbation data defined by the multiplication of A instead of the A-operations. This article is organized as follows. In the next section we recall the preliminaries needed and state the notation used throughout the article. The third section is devoted to our description of a family of morphisms that provide an A-(co)algebra structure via the tensor trick and the Basic Perturbation Lemma. In the fourth section, we establish the main result of the article by which any A-(co)algebra can be seen as a chain contraction from a (simply) connected DG-(co)algebra onto a DG-module. 2. NOTATIONS AND PRELIMINARIES Although relevant notions of homological algebra are recalled here, most of common concepts are not explicitly given. They might be consulted, for instance, in Cartan and Eilenberg (1956) and Mac Lane (1995). Take a commutative ground ring, , with unit. A differential graded module or DG-module, M d, is a module M, graded on the non-negative integers, M=n≥0Mn, endowed with a morphism of graded modules dof degree −1 such that, d2=0. A DG-module Mis called connected if M0=and simply connected if it is connected and M1=0. Given a connected DG-module, M, the reduced module Mis such that Mn=Mnfor n>1 and M0=0. In this article, we shall always refer to connected DG-modules in the context of A-algebras and simply connected DG-modules when dealing with A-coalgebras. We will denote the module M⊗n ···⊗Mby M⊗n, with M⊗0=; we will use the notation f⊗nfor the morphism f⊗n ···⊗fM ⊗n→N⊗n We will respect the Koszul convention for signs. RECTIFICATIONS OF A-ALGEBRAS 2733 On the other hand, if fM ⊗i→Nis a DG-module morphism and nis a nonnegative integer, we will denote by fn M ⊗n→N⊗n−i+1the morphism fn = n−i  j=0 1⊗j⊗f⊗1⊗n−i−j The morphism f j≥iM⊗j→k≥1N⊗kwill be the one such that fM⊗n=fn. If M dMis a DG-module, the suspension of Mis defined as the DG-module sM dsM , where sMn=Mn−1and dsM =−dM. The desuspension of Mis given by s−1Mn=Mn+1and differential −dMtoo. We will denote by ↑and ↓the suspension and desuspension morphisms, which shift the degree by +1 and −1, respectively. A given morphism of graded modules of degree k,fM→N, induces sfsM→sN, given by sf =−1k↑f↓. In fact, dsM =sd M=−↑dM↓. ADG-algebra,A dA A, is a DG-module endowed with an associative product, A, compatible with the differential dAand which has a unit A→A, that is, AA⊗1=A1⊗A=1. Sometimes we will use the notation ∗Afor the product on A. If there is no confusion, subscripts will be omitted. A DG-coalgebra C dC C is a DG-module provided with a compatible coproduct and counit CC→(so, C⊗1C=1⊗CC=1). Given a DG-module M d, the tensor module of Mis denoted by TM and is constructed in the following way: TM = n≥0 M⊗n The tensor graduation of TM  t, is given by: a1⊗···⊗ant= n  i=1 ai The differential structure in TM is provided by the tensor differential,dt, which is the morphism d M. A product, , and a coproduct, , can be naturally defined on TM, as follows: •a1⊗···⊗an⊗an+1⊗···⊗an+p =a1⊗···⊗an+p; •a1⊗···⊗an=n i=0a1⊗···⊗ai⊗ai+1⊗···⊗an. Therefore, TM acquires both structures of DG-algebra (denoted by TaM) and DG-coalgebra (TcM), though they are not compatible to each other (that is, TM   is not a Hopf algebra). Every morphism of DG-modules fM→Ninduces another one Tf  TM →TN, such that Tf M⊗n=f⊗n The reduced bar construction of a connected DG-algebra A,BA,isa DG-coalgebra whose module structure is given by TsA = n≥0 sA⊗n A typical element of BA is denoted by ¯a=a1···anand =1∈. The degree of ais given by the sum of the tensor degree a1+···+anand the simplicial degree n, 2734 JIMÉNEZ AND REAL which appeals to the number of components or length of the element (also referred to as simplicial dimension). The total differential dBis given by the sum of the tensor differential, dt(which is the natural one on the tensor product) and the simplicial differential,ds(that depends on the product on A): dt=−↑dA↓ d s=↑A↓⊗2  The coproduct B BA →BA ⊗BA is the natural one on the tensor module. Given a simply connected DG-coalgebra C, the reduced cobar construction, C, is a DG-algebra whose underlying module is Ts−1C = n≥0 s−1C⊗n A typical element of C will be written ¯c=c1···cn, being =1∈. The total degree of ¯cis ¯c=c1+···cn−n. We will refer to the length of the element, n,as the cosimplicial degree (or dimension). The total differential dis given by the sum of the tensor differential and the cosimplicial differential dcos =↓⊗2C↑ The product on C is the natural one on the underlying module. In the context of homological perturbation theory, the main input data are chain contractions (or simply, contractions) (see Eilenberg and Mac Lane, 1953; Huebschmann and Kadeishvili, 1991): a contraction c  N M f g  from a DG- module Nto a DG-module M, consists of a particular homotopy equivalence determined by three morphisms f g, and ; being fN →M(projection) and g M→N(inclusion) two DG-module morphisms and N →N+1a homotopy operator, that is, fg =1Mand dN+dN+gf =1N. Moreover, these data are also required to satisfy f =0g=0=0 Notice that the homology of both DG-modules are isomorphic. We will also use the notation f g   N ⇒Mor simply Nc ⇒Mwhen confusion cannot arise. Given a DG-module contraction, c  N M f g  we can establish the following ones (Gugenheim and Lambe, 1989; Gugenheim et al., 1991): (i) The suspension contraction of c,sc, which consists of the suspended DG-modules and the induced morphisms. sc  sN sM sf sg s being sf =↑f↓,sg=↑g↓and s =−↑↓, which are briefly expressed by f, gand −. RECTIFICATIONS OF A-ALGEBRAS 2735 (ii) The tensor module contraction,Tc, between the tensor modules of Nand M. Tc  TN TM Tf Tg T where TN⊗n=⊗n = n−1  i=0 1⊗i⊗⊗gf⊗n−i−1 Now, we recall the concept of perturbation datum. Let Nbe a graded module and let fN→Nbe a morphism of graded modules. The morphism fis defined to be pointwise nilpotent whenever for all x∈N,x= 0, there exists a positive integer nsuch that fnx =0. A perturbation of a DG-module Nconsists in a morphism of graded modules N→Nof degree −1, such that dN+2=0. A perturbation datum of the contraction c  N M f g  is a perturbation of the DG-module Nsatisfying that the composition  is pointwise nilpotent. The main tool when dealing with contractions is the Basic Perturbation Lemma (Brown, 1967; Gugenheim, 1972; Gugenheim and Stasheff, 1986; Lambe and Stasheff, 1987; Gugenheim and Lambe, 1989; Huebschmann and Kadeishvili, 1991; Real, 2000; Shih, 1962), which is an algorithm whose input is a contraction of DG-modules c N M f g  and a perturbation datum of cand whose output is a new contraction c N dN+ M dM+d fg  defined by the formulas d=f cg f=f1− c g= cg = c where  c=i≥0−1ii The pointwise nilpotency of the composition  guarantees that the sums are finite for each particular element. 3. FROM CONTRACTIONS TO A-STRUCTURES We find the origin of A-(co)algebras in Stasheff (1963), where Stasheff set the concept of strongly homotopy associativity in the search of a homotopy invariant that plays the role of associativity. We recall here the definition of A-algebra (respectively, A-coalgebra) (Kadeishvili, 1980; Prouté, 1984). An A-algebra (respectively, A-coalgebra), is a DG-module M m1(respectively, M 1) endowed with a family of morphisms of graded modules miM ⊗i→M(respectively, iM→M⊗i of degree i−2 such that, for i≥1, i  n=1 i−n  k=0 −1n+k+nkmi−n+11⊗k⊗mn⊗1⊗i−n−k=0(1) respectively, i  n=1 i−n  k=0 −1n+k+nk1⊗i−n−k⊗n⊗1⊗ki−n+1=0(2) 2736 JIMÉNEZ AND REAL Starting from a contraction between a connected DG-algebra Aand a DG-module M, the application of the tensor trick (Gugenheim, 1977; Gugenheim and Stasheff, 1986; Gugenheim et al., 1991) and the Basic Perturbation Lemma provide a way of constructing a family of morphisms that makes the module inherit an A-algebra structure. In fact, the first transference of an A-algebra structure, in this sense, was made by Kadeishvili (1980) for the case M=HA. Using this technique, in the following theorem we will express these morphisms with regard to the component morphisms of the initial contraction. We draft a proof of the theorem with the only purpose of showing the tools used in the context of homological perturbation theory. Theorem 3.1 (Gugenheim et al., 1991; Kadeishvili, 1980).Let A dAand M dM be a connected DG-algebra and a DG-module, respectively, and c  A M f g  a contraction between them. Then the DG-module Mis endowed with an A-algebra structure by the morphisms m1=−dM mn=−1n+1f1⊗22···⊗n−1n−1g⊗nn≥2 where k = k−1  i=0 −1i+11⊗i⊗A⊗1⊗k−i−1 Proof. Starting from c  A M f g , we can construct the contraction Tsc  TcsA TcsM Tf Tg T− Now, in order to get the differential of the bar construction on the initial DG-module, we consider the simplicial differential as a perturbation datum of this contraction. We can easily check the pointwise nilpotency of T−dssince T− does not affect the simplicial dimension of the element, while dsdecreases this amount by one and so will be zero after a finite number of steps. Then, by applying the Basic Perturbation Lemma, a new contraction is obtained, ˜ f˜g ˜   BA ⇒TcsM ˜ d where TcsM ˜ d is called the tilde bar construction of M(Stasheff, 1963), denoted by  BM and the formula obtained for the perturbed differential is ˜ d=dt+ i≥0 −1iTf dsT−dsiTg We call ˜ d1=dt˜ di=−1iTf dsT−dsi−2Tg for i≥2 RECTIFICATIONS OF A-ALGEBRAS 2737 Let us consider the induced morphisms mnM ⊗n→Mof degree n−2, with the formula mn=−1n/2↓˜ dnsM⊗n↑⊗n where the brackets refer to the integer part. Then, it is easy to check that mn=−1n+1f1⊗22···⊗n−1n−1g⊗n Finally, the fact that ˜ d2=0 can be translated into relations between the morphisms mnwhich are, actually, the ones recalled in Eq. (1).  An analogous theorem can be established for the case of an A-coalgebra structure, whose proof is completely dual to the one given above. Theorem 3.2 (Gugenheim et al., 1991; Kadeishvili, 1980).Let C dC and M dMbe a simply connected DG-coalgebra and a DG-module, respectively, and c  C M f g  a contraction between them. Then the DG-module Mis endowed with an A-coalgebra structure by the morphisms 1=−dM n=−1n/2+n+1f⊗nn⊗n−1n−1···⊗22g n ≥2 where k = k−2  i=0 −1i1⊗i⊗C⊗1⊗k−i−2 In this case, the simple connection of the DG-coalgebra guarantees that the formulas implied in the contraction C ⇒ M, obtained by the Basic Perturbation Lemma, are finite. Now we are concerned about the inverse process: given an A-(co)algebra, finding a suitable contraction that generates, in the sense given in this section, such a structure. 4. FROM A-STRUCTURES TO CONTRACTIONS In this section, we explicitly construct a rectification of an A-algebra. As a result, we establish that such a structure can be structurely represented as a contraction from a DG-algebra onto a DG-module. Recall that by “DG-module” we shall mean a connected DG-module. Munkholm (1974) obtained a contraction between the reduced cobar construction of the reduced bar construction of a DGA-algebra A,BA, and the DGA-algebra itself. cABAA A Ah A 2738 JIMÉNEZ AND REAL The same contraction can be obtained by means of perturbation theory, for the connected case: starting from a “basic” contraction TcsA A   h and taking the perturbation datum =↓−ds↑ , the contraction cAis obtained by applying the Basic Perturbation Lemma. This technique will allow us to extend this result to the case of Abeing an A-algebra, obtaining a contraction between the reduced cobar construction of the tilde bar construction of A, BA, and A. Now we expose the main theorem of the article. Afterwards, we will dualize the result for A-coalgebras. Theorem 4.1. Let Mbe a DG-module endowed with an A-algebra structure. Then there exists a contraction, cMA MMfg, between a connected DG-algebra AM and the DG-module M, such that the application of the tensor trick (followed by the Basic Perturbation Lemma) to cMprovides the original A-algebra structure on M. Proof. Let M mii≥1be an A-algebra and consider the contraction TcsMMh  whose component morphisms are described below. In order to make clearer the formulas, we denote by cithe ith component an element of the cobar, c1···cn, with length ki ≥1, that is, ci=ai1···aiki and we will only specify the whole expression ai1···aikiin the case we want to emphasize its composition. Particularly, when ki =1, we write ai. •TcsM →M, c1···cn=0ifn≥1ork1≥1 a=a •M→TcsM, a =a •hTcsM →TcsM hc1···cn=0ifn=0orn=1ork1≥1 ha1c2···cn=−1a1+1a1a21···a2k2c3···cn Now the perturbation will consist in including m2m 3 in the tensor differential of the cobar, so that TcsMd+, becomes  BM: =↓− k≥2↑mk↓⊗k ↑  RECTIFICATIONS OF A-ALGEBRAS 2739 The composition h is pointwise nilpotent, since decreases the simplicial degree of the components on the bar construction and hdecreases the number of components on the cobar. Then, we apply the Basic Perturbation Lemma, obtaining the contraction cM BM M f g  where  BM will be the algebra AMsatisfying the theorem. Notice that for an element =a1c2···cn  h =−1a1+k2+1mk2+1a1⊗a21⊗···⊗a2k2c3···cn +other summands with k1≥1 where k2+1=k2a1+k2−1a21+···+a2k2−1+k2+1/2 Then, we can describe recursively hi, up to sign: hi =mki+1+111hi−1 ⊗ai+11⊗···⊗ai+1ki+1ci+2···cn where 11a11···a1k1c2···cn=a11. Taking into account these notes, we can describe the morphisms of the contraction above: •f =±mkn+111hn−2 ⊗an1⊗···⊗ankn. In the particular case of the element a1c2, fa1c2=−11+a1+k2+1mk2+1a1⊗a21⊗···⊗a2k2(3) •g=, since a=0. • =n−1 i=1−1i−1hhi−1a1c2···cn =−1a1+1a1a21···a2k2c3···cn + n−1  i=2 ±mki+111hi−2 ⊗ai1⊗···aikiai+11···ai+1ki+1 ci+2···cn Particularly, for an element a1c2, a1c2=−1a1+1a1a21···a2k2(4) Besides, the perturbed differential on Mis zero since  =0, what means that M still remains with the same differential structure. We can check now that the structure of A-algebra generated on Mvia the tensor trick coincides with the original infinite structure on M. Take the contraction TscM  TsAM TsM Tf Tg T− and the simplicial differential, ds, that depends on the juxtaposing product ,asa perturbation datum. The pointwise nilpotency of T−dsis due to the fact that ds