Homological perturbation theory and computability of Hochschild and cyclic homologies of CDGAs

International Conferene on Seondary alulus and ohomologial Physis Mosow, August 1997 HOMOLOGICAL PERTURBATION THEORY AND COMPUTABILITY OF HOCHSCHILD AND CYCLIC HOMOLOGIES OF CDGAS  Alvarez V.  , Armario J.A. y , Real P.  and Silva B.  Dpto. Matematia Apliada I, Univ. de Sevilla E-mails: valvarezeuler.e.us.es, armarioia.es, realia.es, silvaia.es Abstrat. We establish an algorithm omputing the homology of ommutative dierential graded algebras (briey, CDGAs). The main to ol in this approah is given by the Homologial Perturbation Theory partiularized for the algebra ategory (see [21℄). Taking into aount these results, we develop and rene some metho ds already known ab out the omputation of the Ho hshild and yli homologies of CDGAs. In the last setion of the pap er, we analyze the p -lo al homology of the iterated bar onstrution of a CDGA ( p prime). 1. Introdution. The desription of eÆient algorithms of homologial omputation might b e onsidered as a very imp ortant question in Homologial Algebra, in order to use those pro esses mainly in the resolution of problems on algebrai top ology; but this sub jet also inuene diretly on the development of non so losedareas as Cohomologial Physis (in this sense, we nd useful referenes in [12℄, [24 ℄, [25℄) and Seondary Calulus ([14 ℄, [27℄, [28℄). Working in the ontext of CDGAs, Homologial Perturbation Theory ([9℄, [11℄), supplies at one a general algorithm omputing the homology of these ob jets (see [16℄), whih often b ears high omputational harges and atually restrits its appliation to the low dimensional homologial alulus. Our rst goal onsists in rening this algorithm, by means of preservation issues over the ategory of CDGAs when applying p erturbation tehniques (see [21℄). Indeed, the pro ess reveal a p olynomial b ehavior when we deal with some onrete families of CDGAs; for instane, we ompute in this pap er the homology of the CDGAs A s; ( r 1 ;  ;r n  1 ) e 1 ;  ;e n (with r i 2 N , 8 i = 1 ;  ; n  1). These and similar p ositive results make us b e sinerely exp etant in the ahievement of an eetive general algorithm omputing the homology of CDGAs. One the previous algorithm is outlined, our interest turns to the alulation of Ho hshild and yli homologies. Key words and phrases: Homology, CDGA, bar onstrution, (minimal) homologial mo del, ontration, p erturbation, twisted tensor pro dut of CDGA. 2  alvarez a., armario j.a., real p. and silva b. The omputation of Ho hshild homology has already b een studied by Guione and Guione in [8℄, where they set reursive formulae in order to determine the dierential of a smal l homologial model for a CDGA given. We onstrut here an alternative approah using the prop er mahinery of Homologial Perturbation Theory; not only from that p oint of view of Lamb e's (see [16 ℄), but also attending to the preservation of algebra strutures (as develop ed in [21℄). The main dierene b etween this metho d and Guione's one is that we get here more expliit homologial information, available in terms of homotopy equivalenes. This fat allows an immediate translation of the attained results as the departure p oint for obtaining the yli homology, via p erturbation. Unlike the Ho hshild homology one, the metho d omputing yli homology is quite more ompliated, sine no algebrai preservation results an b e applied. We study here an interesting partiular instane, in whih the diÆulties derease substantially. It seems to b e a go o d idea to use the  -op erations desrib ed by Lo day ([18℄, [19℄) in order to solve the general problem of the omputation of yli homology: a suggestive way app ears in this sense. The present pap er is organized into the following setions. The rst one is devoted to explain the basi algebrai onepts from whih the pro ess of homologial omputation is onstruted; these ideas help to the ne understanding of the underlying algorithmi onepts. We shall use without further explanation the prop er onepts of our framework, Algebrai Top ology and Homologial Algebra: we refer to some notions like DGA-algebra , DGA-oalgebra , derivation , oderivation , Hopf DGA-algebra and similar ones. Along the seond setion, a p ositive answer is given ab out the omputability problem of the homology of CDGAs; the algorithm develop ed here pro dues not only an expliit homotopy equivalene for the redued bar onstrution of a CDGA, but also a omparison map with the bar resolution: i.e., an expliit homotopy equivalene with the \standard resolution". Nevertheless, this is not a ompletely satisfatory solution as this result do es not allow one to implement the algorithm into pratise (without any memory or time limits): the sheme often displays a high omplexity. The subsequent subsetion intro dues a onrete example, treated with sp ei tehniques in order to improve the general pro edure. Arguments whih might improve the algorithm of the homologial omputation are explained at the end of this paragraph. Finally, the obtention of general algorithms omputing Ho hshild and yli homologies and the searh of minimal homologial models of iterated Bar onstrutions of CDGAs will b e the sub jets of the remaining setions. 2. Preliminaries. Although relevant notions of Homologial Algebra are explained through the exp osition of the pap er, most of ommon onepts are not expliitly given (whih might b e onsulted for instane in [4℄ or [15 ℄). Let  b e a ommutative ring with non zero unit and let us assume that  is the ground ring. The notation ( A; d A ;  A ) means that A is a dierential graded algebra (briey, DGA- algebra) endowed with a dierential op erator d A and an asso iative pro dut  A . If there is homology of dgas. 3 no further onfusion, subindexes or even op erators are to b e omitted. Notie that a DGA-algebra is said to b e onneted whenever A 0 is . The degree of an element a 2 A n is denoted by j a j = n . Three partiular algebras are of esp eial interest in the development of the pap er, whih are the exterior, p olynomial and divided p ower algebras. Let n b e a xed integer. The exterior algebra E ( x; 2 n + 1) is the graded algebra with generators 1 and x of degrees 0 and 2 n + 1, resp etively; and trivial pro dut (that is, x 2 = 0 and x  1 = x ). The p olynomial algebra P ( y ; 2 n ) onsists of the graded algebra with generators 1 of degree 0 and eah p ower y i of degree 2 ni , with i 2 N ; the pro dut is given by the usual one of p olynomials, i.e.: y i  y j = y i + j for non negative integers i and j . Finally, the divided p ower algebra ( y ; 2 n ) is the graded algebra with generators 1 and y ( i ) (of ourse, y = y (1) ) of resp etive degrees 0 and 2 ni , i 2 N ; and pro dut dened by the rules y ( i )  y ( j ) = i + j i ! y ( i + j ) , whihever non negative integers i and j b e. Eah of these three typ es of algebras an b e onsidered as a CDGA with trivial dierential. The redued bar onstrution asso iated to a DGA-algebra given is a standard algebrai to ol whih allows one to save the pro dut struture of the initial DGA-algebra along the pro ess of homologial omputation. The redued bar onstrution asso iated to a DGA-algebra A is dened to b e as the DG-mo dule  B ( A ) onsisting of the tensor pro duts of i opies of A at eah dimension i (understanding  at the initial level, i.e., in degree 0), and dierential op erator dep ending on the original dierential (tensor dierential) and pro dut (simpliial dierential) morphisms on A (see [15 ℄). A generi homogeneous element of degree n is usually denoted by the expression [ a r 1 j a r 2 j j a r s ℄, where a r i 2 A r i and s + s X i =1 r i = n ; the tensor degree of suh an element is P s i =1 r i = n , and the simpliial degree is s . Whenever the algebra A is ommutative, it is p ossible to dene a multipliative struture up on  B ( A ) (via an op erator alled shue produt ), so that the redued bar onstrution b eomes also a CDGA. A resolution of  over a DGA-algebra A onsists in a dierential A -mo dule X , whih is pro jetive as an A -mo dule, suh that the homology of X is zero exept for degree 0, where it oinides with . If X is atually a free A -mo dule then X is alled a free resolution . A relevant notion to intro due is the twisted tensor pro dut of DGA-algebras [1℄ (briey, TTP ). Let f A i g i 2 I b e a set of ommutative DGA-algebras. A twisted tensor produt ~   i 2 I A i is a CDGA satisfying the following onditions: i) ~   i 2 I A i oinides with the tensor pro dut  i 2 I A i as a graded algebra. ii) The dierential op erator onsists in the sum of the dierential of the banal tensor pro dut and a derivation  . Atually, a TTP of DGA-algebras is a \p erturb ed" DGA-algebra in whih the \p erturbation" only aets its dierential and not its pro dut. Notie that this onept reveals a little strutural dierene with resp et to that of multipliative prinipal onstrution of Cartan (see [3℄, [20℄). An interesting typ e of free resolutions of  over a ommutative DGA-algebra A onsists in TTPs of the form X = A ~    X , where  X is a free CDGA. In this irumstanes,  X is known to 4  alvarez a., armario j.a., real p. and silva b. b e the redued omplex of the resolution. For instane, if A is a CDGA, the \bar resolution" B ( A ) (see [15℄) b eomes a twisted tensor pro dut of DGA-algebras. More preisely, B ( A ) is the ommutative DGA-algebra A ~    B ( A ), where  ( a  [ a 1 j a 2 j j a n ℄) = ( a  A a 1 )  [ a 2 jj a n ℄ : The algorithmi tehniques that are prop osed in pursuit of solving the problem of the homology omputation are tted into the framework of onstrutive Homologial Algebra. The main input data are the ontrations ([5℄, [13 ℄): a ontration  , N  ) M (sometimes denoted by ( f ; g ;  )) from a DG-mo dule ( N ; d N ) to a DG-mo dule ( M ; d M ) onsists in an homotopy equivalene determined by three morphisms f , g and  ; b eing f : N  ! M  (pro jetion) and g : M  ! N  (inlusion) two DG-mo dule morphisms and  : N  ! N  +1 an homotopy op erator. Moreover, these data are required to satisfy the following rules: f g = 1 M , f  = 0, g = 0, d N + d N  + g f = 1 N and  = 0. Notie that a free resolution K of  over the DGA-algebra A splits o the bar onstrution if there exists a ontration (alled splitting) from B ( A ) to K . This notion is due to Lamb e ([17℄) and is to b e applied later on. There is a relevant partiular typ e of ontrations of DGA-algebras, that we systemat- ially use in the remainder of this pap er. We refer to semiful l algebra ontrations ([21℄), whih onsist of a multipliative inlusion, a quasi-algebra pro jetion and a quasi-algebra homotopy. Given a ontration ( f ; g ;  ) b etween two DGAs A and A 0 , we reall that: 1. The pro jetion f is said to b e a quasi-algebra projetion whenever the following onditions hold: f (   A  ) = 0 ; f (   A g ) = 0 ; f ( g  A  ) = 0 : (1) 2. The homotopy op erator  is said to b e a quasi-algebra homotopy whenever the rules b elow are veried:  (   A  ) = 0 ;  (   A g ) = 0 ;  ( g  A  ) = 0 : (2) For instane, the bar resolution B ( A ) of a DGA-algebra A generates the following semifull algebra ontration: R B ( A ) : f B ( A ) ;  ;  B ( A ) ;  B ( A ) ; s g ; where the homotopy op erator s : B ( A ) ! B ( A ) is given by s ( a  [ a 1 j j a n ℄) = [ a j a 1 jj a n ℄ : The restrition to this partiular typ e of ontrations is due to the fat that they are preserved by most of algebra op erations: for instane, the set of all semifull algebra ontrations is losed by omp osition and tensor pro dut of ontrations. The basi pro edure we state here onsists in the establishment of an expliit ontration from an initial DG-mo dule N to a free nite typ e DG-mo dule M , so that the homology of N is atually omputable from that of M : that is, H  ( N ) = H  ( M ). There exist other imp ortant to ols from whih one reahes extra homologial information: under ertain irumstanes one may \p erturb" a given ontration in order to obtain a new homology of dgas. 5 ontration b etween the initial underlying graded mo dules endowed with distint dierential op erators (to know more ab out Homologial Perturbation Theory, onsult [9℄, [11 ℄, [13℄, [16℄ and [17℄). Notie that semifull algebra ontrations are preserved via p erturbation pro esses. We reall the onept of a p erturbation datum. Let N b e a graded mo dule and let f : N ! N b e a morphism of graded mo dules. The morphism f is dened to b e pointwise nilpotent whenever for all x 2 N , x 6 = 0, a p ositive integer n exists suh that f n ( x ) = 0. Notie that the integer n generally dep ends on the element x . A perturbation of a DGA-module N onsists in a morphism of graded mo dules Æ : N ! N of degree  1, suh that ( d N + Æ ) 2 = 0 and  N Æ = 0. A perturbation datum of the ontration  : f N ; M ; f ; g ;  g is a p erturbation Æ of the DGA-mo dule N verifying that the omp osition Æ is p ointwise nilp otent. In the partiular ase that b oth of N and M are DGA-algebras, a new notion app ears: an algebra perturbation datum Æ of a ontration  from N to M onsists in a p erturbation datum Æ of  whih is also a derivation. We deal with p erturbation results partiularized for the DGA-algebra ategory (see [21℄). In this sense, we give b elow an analogous theorem to the lassial \Algebra Perturbation Lemma" ([9℄, [11℄, [13 ℄) appropriated for semifull algebra ontrations: Theorem 2.1 ( SF-APL ) ([21℄) Let  : f N ; M ; f ; g ;  g be a semiful l algebra ontration and Æ : N ! N be an algebra perturbation datum of  . Then, a new semiful l algebra ontration  Æ : f ( N ; d N + Æ;  N ;  N ) ; ( M ; d M + d Æ ;  M ;  M ) ; f Æ ; g Æ ;  Æ g is dened by the formulas: d Æ = f Æ  Æ  g ; f Æ = f (1  Æ  Æ   ) ; g Æ =  Æ  g ;  Æ =  Æ   ; where  Æ  = X i  0 (  1) i ( Æ ) i = 1  Æ + Æ Æ     + (  1) i ( Æ ) i +  : Let us note that  Æ  ( x ) is a nite sum for eah x 2 N , b eause of the p ointwise nilp oteny of the omp osition Æ . Moreover, it is obvious that the morphism d Æ b eomes a p erturbation of the DGA-mo dule ( M ; d M ;  M ;  M ). In order to alulate the homology of a DGA-algebra, the notion of \homologial mo del" arises as a omputational alternative to the redued bar onstrution. An homologial model for a ommutative DGA-algebra A is a free DGA-algebra of nite typ e H B A suh that there exists a semifull algebra ontration from  B ( A ) to H B A . In partiular, it is veried that H  (  B ( A )) = H  ( H B A ). At this stage, it is neessary to b orrow Cartan's denition of the susp ension and p - transp oteny additive funtions. Let A b e a ommutative DGA-algebra. The following morphisms of graded mo dules an b e dened:  The suspension ,  : A !  B ( A ); dened as  ( a ) = [ a ℄, for a 2 A .  The p -transpoteny (b eing p a prime integer), ' p : A !  B ( A ); dened as ' p ( a ) = [ a j a p  1 ℄, with a 2 A . 6  alvarez a., armario j.a., real p. and silva b. 3. Computability of the homology of CDGAs. General algorithm. It is ommonly known that every ommutative DGA-algebra A \fatorizes" into a tensor pro dut of exterior and p olynomial algebras endowed with a dierential-derivation; in the sense that there exists an homomorphism onneting b oth strutures, whih indues an isomorphism in homology (onsult [2℄ if neessary). In fat, the ob jet we start from is a generi ommutative DGA-algebra A , fatorized into the mo del ab ove as a TTP of exterior and p olynomial algebras. The studies explained b elow are redued to the nitely generated ase; that is, a twisted tensor pro dut of algebras ~   i 2 I A i , where I denotes a nite set of indies,  is a dierential-derivation and A i is an exterior or a p olynomial algebra, for every i 2 I . Taking into aount the ideas expressed in the setion b efore, the prinipal goal that arises is to obtain a \hain" of ontrations starting at the redued bar onstrution  B ( A ) and ending up at free DGA-algebra of nite typ e. Three almost-ful l algebra ontrations (i.e., semifull algebra ontrations endowed with multipliative pro jetions) are used in order to nd the struture of graded mo dule of an homologial mo del for the DGA-algebra A :  The ontration dened in [6℄ from  B ( A  A 0 ) to  B ( A )   B ( A 0 ), b eing A and A 0 two ommutative DGA-algebras; whih is briey denoted by C  B  . Though Eilenb erg and Ma Lane only set expliit formulae for the pro jetion and inlusion morphisms, an expliit formula of the reursive denition of the homotopy op erator given by Eilenb erg and Ma Lane in [5℄ is established in [21℄. A relevant prop erty onerning the homotopy op erator  B  of the ontration ab ove is that  B  ([ a 1  a 0 1 j a 2 jj a m ℄) =  [ a 0 1 j a 1 j a 2 jj a m ℄ ; (3) with a i 2 A , 8 1  i  m and a 0 1 2 A 0 . Given a tensor pro dut  i 2 I A i of ommutative DGA-algebras, a ontration from  B (  i 2 I A i ) to  i 2 I  B ( A i ) is easily determined, by applying C  B  several times in a suitable way. This new ontration is also denoted by C  B  .  The isomorphism of DGA-algebras (hene, a ontration) C  B E from  B ( E ( u; 2 n + 1)) to (  ( u ) ; 2 n + 2), also desrib ed in [6℄. Note that the generator v of the previous divided p ower algebra is denoted by  ( u ), sine g  B E ( v ) =  ( u ), where  is the Cartan's susp ension op erator. Last isomorphism might b e onsidered as a ful l algebra ontration (that is, an almostfull algebra ontration endowed with an algebra homotopy op erator), in an obvious way.  The ontration C  B P from  B ( P ( y ; 2 n )) to E (  ( y ) ; 2 n ) ([6℄). The generator of the exterior algebra is denoted by  ( y ) sine b oth elements orresp ond one to eah other by the pro jetion and inlusion of the ontration. homology of dgas. 7 All the previous ontrations have already b een deep er desrib ed for deades, so we will not explain them further. Thanks to these three ontrations, it is p ossible to establish by omp osition the following semifull algebra ontration C 0 = ( f ; g ;  ):  B (  i 2 I A i ) )  i 2 I  B ( A i ) )  i 2 I H B A i ; where H B A i represents an exterior or a divided p olynomial algebra, dep ending on whether A i is a p olynomial or an exterior algebra. Obviously, the pro dut on  i 2 I H B A i is the natural one. The next step is p erturbing C 0 , in order to obtain an homologial mo del of the initial TTP ~   i 2 I A i . The morphism  pro dues a p erturbation-derivation Æ onto the tensor dierential of  B (  i 2 I A i ). It seems neessary to emphasize that the p ointwise nilp oteny of Æ is guaranteed by the following fats:  The homotopy op erator  inreases the simpliial degree of  B (  i 2 I A i ) by one.  The p erturbation Æ do es not hange the simpliial degree. Indeed, Æ lowers the tensor degree by one. Therefore, by applying SF-APL it is onstruted a new semifull algebra ontration ( f Æ ; g Æ ;  Æ ):  B ( ~   i 2 I A i ) ( C 0 ) Æ ) (  i 2 I H B A i ; d Æ ) ; where the dierential d Æ is determined by the p erturbation pro ess. That means that H B A =  i 2 I ( H B A i ; d Æ ) is an homologial mo del of A = ~  i 2 I A i . Notie that the homotopy operator  Æ inreases the simpliial degree at least by one. This fat is to b e applied in the sequel. Theorem 3.1 Given a nite type TTP A of exterior and polynomial algebras, there exists a semiful l algebra ontration from the redued bar onstrution  B ( A ) to a free DGA-algebra H of nite type. That is, we have a homologial model for A . Moreover, H onsists in a TTP of exterior and divided power algebras suh that, at graded algebra level, it is veried that:  eah E ( u; 2 n  1) fator in A ontributes with a (  ( u ) ; 2 n ) fator in H .  eah P ( u; 2 n ) fator in A ontributes with an E (  ( u ) ; 2 n + 1) fator in H . In this way, it is desrib ed a general algorithm omputing the homology of CDGAs. Obviously, the homology of the homologial mo del obtained an b e omputed using an algorithm based up on the establishment of the Smith's normal form of the matries representing the dierentials at eah degree ([26 ℄). The omputational ost to onstrut the ontration ( C 0 ) Æ is high: notie that b oth the inlusion and homotopy op erators of the ontration C  B  give an answer in exp onential time, when evaluated on eah element. In fat, the formula of the dierential op erator d Æ pro dued by the homologial p erturbation mahinery is given by: d Æ = f Æ Æ Æ (1   Æ Æ +  Æ Æ Æ  Æ Æ     ) Æ g : (4) 8  alvarez a., armario j.a., real p. and silva b. Attending to the previous remarks ab out the eÆieny on the evaluation of morphisms, a rst impression is that the evaluation of d Æ up on any generator b eomes in general a pro ess of exp onential nature. In spite of this, it is p ossible to take advantage of d Æ b eing a derivation (that is, a morphism ompatible with the pro dut of the homologial mo del), in order to improve the omplexity problem: indeed, the fat that d Æ is a derivation implies that it is only neessary to know the value of this morphism applied up on the generators of the mo del (notie that there are so many generators as the ardinal of the set of indies I indiates). This is an enormous improvement in the omputation of the dierential on the small mo del. Finally, Theorem 3.1 an b e used to derive small free resolutions of  over A , as it is shown in the following theorem (whih is a graded-ommutative version of Th. 8.1.3 of [16℄): Theorem 3.2 [1℄ Let A be a onneted ommutative DGA-algebra. Given a semiful l algebra ontration from the redued bar onstrution  B ( A ) to a free ommutative DGA-algebra H (in whih the homotopy operator inreases the simpliial degree at least by one); there exists a free resolution K whih splits o of the bar onstrution. Moreover, this resolution K is a twisted tensor produt of the DGA-algebras A and H and the splitting is a semiful l algebra ontration. In the light of the theorem ab ove, it is p ossible to state the following one: Theorem 3.3 Given a nite type TTP A of exterior and polynomial algebras, there exists a free resolution A ~   0 H that split o of the bar onstrution. More preisely, the splitting is a semiful l algebra ontration and the redued omplex H is determined by Theorem 3.1. Now, a partiular ase is examined in detail, in order to larify the general metho d. 3.1. An example: Homology of the algebras A s; ( r 1 ;r 2 ;:::;r n  1 ) e 1 ;e 2 ;:::;e n . This setion is devoted to the desription of the homologial mo dels of a partiular family of CDGAs. We make use of prop erties of algebra struture preservation. The ground ring is supp osed to b e Z in this subsetion. Let A s; ( r 1 ;r 2 ;:::;r n  1 ) e 1 ;e 2 ;:::;e n denote the CDGA-algebra whih onsists of the DG-mo dule E ( x; 2 s + 1) ~   1 P ( y 1 ; 2 s + 2) ~   2 P ( y 2 ; (2 s + 2)( r 1 + 1)) ~   3 ~   3  ~   n P ( y n ; (2 s + 2)( r 1 + 1)  ( r n  1 + 1)) ; and the dierential op erator  i ( y i ) = e i x  y r 1 1  y r 2 2      y r i  1 i  1 ; 8 1  i  n; where e i 2 N with e 1 > e 2 >  > e n and r i 2 N [ f 0 g . homology of dgas. 9 Let us onsider the following ontration C 0 = ( f ; g ;  ) as starting p oint:  B ( E ( x; 2 s + 1)  P ( y 1 ; 2 s + 1)      P ( y n ; (2 s + 2)( r 1 + 1)  ( r n  1 + 1))) R  B  )  B ( E ( x; 2 s + 1))   B ( P ( y 1 ; 2 s + 2))       B ( P ( y n ; (2 s + 2)( r 1 + 1)  ( r n  1 + 1))) R  B E  R  B P  ) (  ( x ) ; 2 s + 2)  E (  ( y 1 ) ; 2 s + 3)      E (  ( y n ) ; (2 s + 2)( r 1 + 1)  ( r n  1 + 1) + 1)) : The question is to apply the general metho d desrib ed b efore to this initial ontration. From now on, the degrees of the algebra generators are omitted in orden to simplify the notation. The dierentials  1 ,  2 , ::: ,  n pro due a p erturbation-derivation Æ in  B ( E ( x )  P ( y 1 )       P ( y n ) : Applying the p erturbation mahinery, the following semifull algebra ontration arises:  B ( A s; ( 2 r 1 ;:::; n r n  1 ) e 1 ;:::;e n ) ( C 0 ) Æ ) ((  ( x ))  E (  ( y 1 ))      E (  ( y n )) ; d Æ ) ; where d Æ is the dierential determined in the p erturbation pro ess. Hene, H  B A s; ( 2 r 1 ;:::; n r n  1 ) e 1 ;:::;e n = ((  ( x ))  E (  ( y 1 ))      E (  ( y n )) ; d Æ ) onstitutes an homologial mo del for this CDGA. Therefore, it is only neessary to evaluate the dierential on eah generator  ( y 1 ),  ( y 2 ), ::: ,  ( y n ). Conretely, omputing the homology of A s; () e 1 , the formula (4) redues to f Æ Æ Æ g , sine  Æ Æ Æ g = 0 . If we go on alulating the image of the dierential, we get: First at all, d Æ (  ( y 1 )) =  e 1   ( x ). Considering  ( y 2 ), d Æ (  ( y 2 )) = f Æ Æ Æ Æ g (  ( y 2 )) =  e 2  f Æ ([ x  y 1 r 1 ℄) : Note that it is veried that f Æ ([ x  y 1 r 1 ℄) is 0 if r 1  1 and  ( x ) otherwise ( r 1 = 0). Attending to (3), we dedue that  ([ x  y 1 r 1 ℄) =  [ y 1 r 1 j x ℄. And applying Æ up on this element generates r 1  e 1 [ x  y 1 r 1  1 j x ℄. By applying alternatively the morphisms  and Æ on the last element, it is obtained r 1 !  e 1 r 1 [ x j r 1 + 1 times    j x ℄ after r  1 iterations. There is only one non zero p ossible ation of a morphism over this element, whih is f . This fat leads diretly to the formula f Æ ([ x  y 1 r 1 ℄) = r 1 !  e 1 r 1   ( x ) ( r 1 +1) : To sum up, the dierential d Æ applied over  ( y 2 ) is d Æ (  ( y 2 )) =  r 1 !  e 1 r 1  e 2   ( x ) ( r 1 +1) : 16  alvarez a., armario j.a., real p. and silva b. semifull algebra ontration from  B ( A ) to a TTP of n exterior and divided p ower algebras is obtained. It seems interesting to study the omputability of the homology of  B n ( A ), with n 2 N . It is neessary to reall a theorem proved by Gugenheim, Lamb e and Stashe in whih a ontration from the redued bar onstrution of A to the bar tilde onstrution of M (see [10℄, [23℄) arises, taking as input datum a ontration from a DG-algebra A to a DG-mo dule M . Theorem 6.2 [11℄ Let A and M be a DGA-algebra and a DGA-module respetively, and  : f A; M ; f ; g ;  g be a ontration. Hene, the fol lowing ful l oalgebra ontration an be established:  B (  ) : f  B ( A ) ; ~ B ( M ) ;  B ( f ) ;  B ( g ) ;  B (  ) g ; (10) where d A s is the simpliial dierential operator of  B ( A ) and the DG-module ~ B ( M ) is the Stashe 's bar tilde onstrution. Whenever M = A 0 is a DGA-algebra and  is an algebra ontration, then the ontration  B (  ) \onnets" two bar onstrutions (see [11 ℄). For instane, if g is a morphism of DGA- algebras then  B (  ) is a ontration from  B ( A ) to  B ( A 0 ), and its inlusion  B ( g ) oinides with T ( S (  g )). Sine this morphism preserves shue pro duts,  B (  ) b eomes an algebra ontration with multipliative inlusion. With no further assumptions, it an b e proved that  B ( g ) always preserves shue pro duts (see [21℄). In other words, the previous theorem might b e enrihed, by establishing the b ehavior of the ontration  B (  ) regarding to the algebra strutures determined by the shue pro dut. Theorem 6.3 [21℄ Let  : f A; M ; f ; g ;  g be a ontration, where A is a ommutative DGA-algebra and M is a DGA-module. Then  B (  ) : f  B ( A ) ; ~ B ( M ) ;  B ( f ) ;  B ( g ) ;  B (  ) g onstitutes a semiful l algebra ontration. Applying the redued bar onstrution to an exterior algebra generates a divided p ower algebra; due to this fat and in order to iterate the pro ess ab ove, it is neessary to establish a ontration for the bar onstrution of a divided p ower algebra with one generator of even degree. Working in Z ( p ) , b eing p a prime numb er, a \ p -minimal homologial mo del" of this algebra is obtained, onsisting in a tensor pro dut of an exterior algebra and an innite numb er of \ p -minimal" 2-indeomp osable TTPs of exterior and divided p ower algebras. Let us supp ose, therefore, that the ground ring is given by Z ( p ) , that is, Z lo alized on a prime integer p : Z ( p ) = f r s ; suh that m::d: ( p; s ) = 1 g : homology of dgas. 17 Definition 6.4 [13℄ Let M b e a DGA-mo dule over Z ( p ) . We say that a morphism of DGA- mo dules h : M ! M is p -minimal whenever h ( M )  p  M . We say that a DGA-mo dule M is p -minimal whenever it is free, of nite typ e as graded mo dule over Z ( p ) and its dierential d M is p -minimal. Notie that a ontration b etween two p -minimal DGA-mo dules onstitutes an isomorphism of DGA-mo dules. Taking into aount the previous denitions, a ommutative DGA-algebra H is said to b e a p -minimal homologial model of a given CDGA A whenever H is minimal and there exists a ontration from  B ( A ) to H . Hene, the following result states: Theorem 6.5 [1℄ Let Z ( p ) be the ground ring. There exists a semiful l algebra ontration from the redued bar onstrution of a p -minimal ` -indeomposable twisted tensor produt A of exterior and divided power algebras (with `  2 ) to a tensor produt H of p -minimal k -indeomposable twisted tensor produts (with k  ` ) of exterior and divided power algebras (equipped with the natural produt). That is, H is a p -minimal homologial model of A . Moreover, attending only to the graded algebra level, it is veried that  eah E ( u; 2 n  1) fator in A ontributes with a (  ( u ) ; 2 n ) fator in H .  eah ( u; 2 n ) fator in A ontributes with E (   p i ( u ) ; 2 np i + 1) (taking i values upon the non negative integers) and ( ' p  p i  1 ( u ) ; 2 np i + 2) ( i  1 ), fators in H . What the previous theorem really means is that there exists an homologial preservation (or non-degeneration) result of the ` -indeomp osability of TTPs of that kind. Finally, the following theorem arises: Theorem 6.6 Let Z ( p ) be the ground ring. Let A be a TTP of n exterior and polynomial algebras and let m be a natural number. There is a semiful l algebra ontration from the iterated bar onstrution  B m ( A ) to a tensor produt H of i -indeomposable TTPs of exterior and divided power algebras, with i  n . In fat, the departure p oint ab ove is onstituted by a semifull algebra ontration from  B ( A ) to a TTP of n algebras (exterior and divided p ower algebras), whih has already b een established b efore. Combining appropriately Theorem 6.3 and Theorem 6.5, the result follows. Theorems ab ove admits an immediate translation to the language of free resolutions, via Theorem 6.3: Theorem 6.7 Let A be a TTP of n exterior and polynomial algebras and let m be a natural number. There exists a free resolution  B m  1 ( A ) ~   H that splits o of the bar resolution B (  B m  1 ( A )) . More preisely, the splitting is a semiful l algebra ontration and the redued omplex H is determined by Theorem 6.6. 18  alvarez a., armario j.a., real p. and silva b. Referenes [1℄ Armario J.A., Real P. and Silva B.: A generalization of a result of J.C. Moore about Cartan's little onstrutions , ommuniation presented at Conferene about Higher Homotopy Strutures in Topology and Mathematial Physis , Vassar ollege, New York (June, 1996). [2℄ Burghelea D. and Vigue Poirrier M.: Cyli homology of ommutative algebras I , Leture Notes in Mathematis, Algebrai Top ology Rational Homotopy, 1318 , pp. 51-72 (1986). [3℄ Cartan H.: Algebres d'Eilenberg-Ma Lane , Seminaire H. Cartan 1954/55, (exp ose 2 a 11 ), E. Normale Sup eriere, Paris (1.956). [4℄ Cartan H. and Eilenb erg S.: Homologial Algebra , Prineton University Press, Prineton (1956). [5℄ Eilenb erg S. and Ma Lane S.: On the groups H (  ; n ) , I , Annals of Math., 58 , pp. 55-139 (1953). [6℄ Eilenb erg S. and MaLane S.: On the groups H (  ; n ) , II , Annals of Math., vol. 60 , pp. 49-139 (1954). [7℄ Getzler E., Jones J.D.S. and Petrak S.: Dierential forms on loop spaes and the yli bar omplex , Top ology, vol. 30 , n. 3, pp. 339-371 (1991). [8℄ Guione J.A. and Guione J.J.: Hohshild homology of ommutative algebras in positive harateristi , Communiations in algebra, 22 (15), pp. 6037-6046 (1994). [9℄ Gugenheim V.K.A.M. and Lamb e L.A.: Perturbation theory in Dierential Homologial Algebra, I , Illinois J. Math., vol. 33 , pp. 556-582 (1989). [10℄ Gugenheim V.K.A.M., Lamb e L.A. and Stashe J.D.: Algebrai aspets of Chen's twisting ohain , Illinois J. Math., 34 (2), pp. 485-502 (1990). [11℄ Gugenheim V.K.A.M., Lamb e L.A. and Stashe J.D.: Perturbation theory in dierential homologial algebra, II , Illinois J. Math., 35 (3), pp. 357-373 (1991). [12℄ Henneaux M. and Teitelb oim C.: Generalitation of gauges systems , Prinenton Univ. Press (1992). [13℄ Huebshmann J. and Kadeishvili T.: Smal l models for hain algebras , Math. Zeit., 207 , pp. 245-280 (1991). [14℄ Krasil'shhik I.S.: Some new ohomologial invariants for nonlinear dierential equations , Di. Geom. Appl. 2 , pp. 307-350 (1992). [15℄ MaLane S.: Homology , Classis in Mathematis, Springer (1995). [16℄ Lamb e L.A.: Homologial perturbation theory, Hohshild homology and formal groups , Contemp. Math., 134 , pp. 183-218 (1992). [17℄ Lamb e L.A.: Resolutions whih split o of the bar onstrution , J. Pure Appl. Algebra 84 , pp. 311-329 (1993). [18℄ Lo day J-L. y Quillen D.: Cyli homology and the Lie algebra of matries , Comment. Math. Helv., 59 , pp. 565-591 (1984). [19℄ Lo day J-L.: Operations sur l'homologie ylique des algebres ommutatives , Invent. Math., 96 , pp. 205- 230 (1989). [20℄ Proute A.: Algebres dierentiel les fortement homotopiquement assoiatives ( A 1 -algebres) , Ph. D. Thesis, Universite Paris VI I. [21℄ Real P.: Homologial Perturbation Theory and assoiativity , Prepubliaion: MA1-05-96I. Enviado a J. Pure App. Alg . [22℄ Seibt P.: Cyli homology of algebras , World Sienti Publishing, Singap ore (1987). [23℄ Stashe J.D.: Homotopy assoiativity of H-spaes, II , Transations of A.M.S., vol 108 , pp. 293-312 (1963). [24℄ Stashe J.D: Constrained Hamiltonians, BRS and homologial algebra , Pro . Conf. on Ellipti Funtions in Algebrai Top ology (1986). [25℄ Stashe J.D: Cohomologial Physis , Leture Notes in Mathematis, Algebrai Top ology Rational Homotopy, 1318 , Springer Verlag, pp. 228-237 (1988). [26℄ Veblen O.: Analysis Situs , A.M.S. Publiations, vol. 5 (1931). [27℄ Vinogradov A.M., Krasil'shhik I.S. and Lyhagin V.V.: Introdution to the Geometry of Nonlinear Dierential Equations , Nauka, Mos u (1986); English transl. Geometry of Jet Spaes and Nonlinear Partial Dierential Equations , Gordon and Breah, New York (1986). [28℄ Vinogradov A.M.: From symmetries of partial dierential equations towards seondary (\quantized") alulus , J. Geom. Phys., 14 , pp. 146-194 (1994). homology of dgas. 19 [29℄ Weib el C.A.: An introdution to homologial algebra , Cambridge studies in advaned mathematis, 38 , Cambridge University Press (1994).