Koszul duality of operads and homology of partition posets

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• Morphisme de graphes
• Productique
• DGA
• Composition de l'être dans l'Égypte antique
• Monade (informatique)
• Relation de un à un

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• Publié par: Ernoi
• Nombre de lectures: 105
• Langue: English

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Koszul duality of operads and homology of partition posets Benoit Fresse Abstract. Weconsiderpartitionsofasetwith r elementsorderedbyrefine- ¯ment. Weconsiderthesimplicialcomplex K(r)formedbychainsofpartitions which starts at the smallest element and ends at the largest element of the ¯partitionposet. Aclassicaltheoremassertsthat K(r)isequivalenttoawedge of r−1-dimensional spheres. In addition, the poset of partitions is equipped with a natural action of the symmetric group in r letters. Consequently, the associated homology modules are representations of the symmetric groups. ¯One observes that the r−1th homology modules of K(r), where r = 1,2,..., are dualto the Lie representationofthe symmetricgroups. In thisarticle, we would like to point out that this theorem occurs a by-product of the theory of Koszul operads. For that purpose, we improve results of V. Ginzburg and M. Kapranov in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov. Contents Prologue and introduction 0. Conventions 1. Composition products and operad structures 2. Chain complexes of modules over an operad 3. The reduced bar construction 4. Bar constructions with coefficients 5. Koszul duality for operads 6. Epilogue: partition posets References Glossary and notation index Prologue We consider the set of partitions of{1,...,r} equipped with the partial or- der defined by the refinement of partitions (for instance, we have{1,3},{2,4}≤ {1},{3},{2,4}). Thesingleset{1,...,r}formsthesmallestpartitionof{1,...,r} and the collection{1},...,{r} forms the largest partition. We mention that the 1991 Mathematics Subject Classification. 18D50; 18C15; 17B01; 05E25; 05C05. This work was carried out at the Laboratoire Jean-Alexandre Dieudonn´e, Universit´e de Nice-Sophia-Antipolis et CNRS. 16 2 BENOIT FRESSE components of a partition are not assumed to be ordered (for instance, the collec- tions{1,3},{2,4} and{2,4},{1,3} define the same partition). We observe also that the set of partitions is equipped with an action of the symmetric group Σ ,r since a permutation w :{1,...,r}→{1,...,r} maps a partition of{1,...,r} to another partition. ¯We consider the simplicial set K(r) which consists of sequences of partitions λ ≤λ ≤···≤λ such that λ ={1,...,r} and λ ={1},...,{r}. The face d0 1 n 0 n i is given by the omission ofλ . The degeneracys is given by the repetition ofλ .i j j We have the following result: ¯Theorem. We consider the reduced homology of the simplicial setK(r) with coefficients inK=Z,Q orF . This homology is a graded Σ -module since the setp r of partitions is equipped with an action of the symmetric group. We have ( ∨L(r) ⊗sgn , if∗=r−1,r˜ ¯H (K(r),K)=∗ 0, otherwise, whereL(r) denotes the rth component of the Lie operadL. The Lie operadL consists precisely of the sequence of representationsL(r) formed by the “multilinear” components of free Lie algebrasL(Kx ⊕···⊕Kx )1 r (more explicitly, the moduleL(r) is spanned by the Lie monomials of degree 1 in eachvariable). Inthetheorem,weconsidertensorproductsofdualrepresentations ∨L(r) with the classical signature representations sgn .r Here are bibliographical references about this theorem. The character of the ˜ ¯representationH (K(r),K)isdeterminedbyR.Stanleyin[81]andbyP.Hanlonr−1 in [35]. The relationship with the character of the Lie operad is pointed out by A. Joyal in [41]. These results about characters are refined by A. Robinson and S. Whitehouse in [74] (see also S. Whitehouse [86] for the integral version). One can deduce from results of A. Bjorn¨ er that the (reduced) homology of partition posets vanishes in degree∗ = r−1 (cf. [14]). Then, in [5], H. Barcelo ∨˜ ¯defines an isomorphism of representations H (K(r),K)’L(r) ⊗sgn , basedr−1 r on the Lyndon basis of the Lie operad (cf. M. Lothaire [51], C. Reutenauer [71]). This result is improved by P.Hanlonand M. Wachs in [36]. Namely, these authors define a natural morphism (which does not involve the choice of a basis of the Lie operad) from the dual of the Lie operad to the chain complex of the partition poset ∨ ¯L(r) ⊗sgn → C (K(r)).r−1r This morphism fixes a representative of the homology class associated to a given element of the Lie operad. In addition, P. Hanlon and M. Wachs generalize the theorem above and give a relationship between partition posets and structures of Lie algebras with k-ary brackets. On the other hand, a topological proof of the theorem above, based on cal- culations of F. Cohen (cf. [20]), is given by G. Arone and M. Kankaanrinta in [2]. In connection with this result, we should mention that an article of G. Arone and M. Mahowald (cf. [3]) sheds light on the importance of partition posets in homotopy theory. Namely, these authors prove that the Goodwillie tower of the identity functor on topological spaces is precisely determined by partition posets. Introduction In this article, we would like to point out that the theorem of the prologue occurs as a by-product of the theory of Koszul operads. For that purpose, weKOSZUL DUALITY OF OPERADS 3 improve results of V. Ginzburg and M. Kapranov (cf. [31]) in several directions. More particularly, we extend the Koszul duality of operads to operads defined over a field of positive characteristic (or over a ring). In addition, we obtain more conceptual proofs of theorems of V. Ginzburg and M. Kapranov. We recall that an operad is an algebraic structure which parametrizes a set of operations. Originally, the notion of an operad is introduced by P. May in a topological context in order to model the stucture of an iterated loop space (cf. [61]). To be precise, the definition of an operad arises on one hand from the notion of an algebraic theory (cf. J. Boardman and R. Vogt [15], F. Lawvere [48], S. Mac Lane [53]) and on the other hand from the structure of a monad (also called triple, cf. J. Beck [10], S. Eilenberg and C. Moore [23]). In particular, we should mention that J. Boardman and R. Vogt consider certain prop structures, which are equivalent to operads, in their work on iterated loop spaces (cf. [15]). We refer to Adams’s book [1] for a nice survey of this subject. In this article, we are concerned with operads in a category of modules over a ground ringK. In this context, an operad P consists of a sequence of representa- tions P(r) of the symmetric groups Σ together with composition productsr P(r)⊗P(n )⊗···⊗P(n )→ P(n +···+n ).1 r 1 r In general, an operadP is associated to a category of algebras, calledP-algebras, forwhichtheelementsofP(r)representmultilinearoperationsonr variables. The composition products of P determine the composites of these operations. For in- stance,thereareoperadsdenotedbyC,respectivelyL,associatedtothecategoryof commutativeandassociativealgebras, respectivelytothecategoryofLiealgebras. (To be precise, in this article, we consider commutative algebras without units.) TheK-moduleC(r)isgeneratedbythesinglemonomialp(x ,...,x )=x ·····x1 r 1 r inr commutative variable which has degree 1 in each variable. Consequently, this K-moduleC(r) is the trivial representation of the symmetric group Σ . The K-r moduleL(r) is generated by the Lie monomials in r variables which have degree 1 in each variable. Classically, a sequence of representations of the symmetric groups M(r) is equivalent to a functor S(M):KMod→ KMod whose expansion ∞M ⊗rS(M)(V)= (M(r)⊗V )Σr r=0 ⊗rconsists of coinvariant modules S (M)(V) = (M(r)⊗ V ) . For instance,r Σr the functor associated to the commutative operad S(C) : KMod → KMod can be identified with the augmentation ideal of the symmetric algebra S(C)(V) = L∞ ⊗r(V ) . Similarly,thefunctorassociatedtotheLieoperadS(L):KMod→Σr=1 r KMod is given by the structure of a free Lie algebra. In general, the structure formed by a sequence of representations of the symmetric groupsM(r) is called a Σ -module (or a symmetric module in plain English).∗ Oneobservesthatclassicalcomplexesassociatedtocommutativealgebrasand Lie algebras are functors determined by Σ -modules. On one hand, we consider∗ the cotriple construction of a commutative algebra A which is a chain complex cotriple C (A) such that∗ cotripleC (A)=S(C)◦···◦S(C)(A)∗ | {z } ∗−14 BENOIT FRESSE (cf. M. Barr [6], J. Beck [10]). We prove precisely that the cotriple construction ¯is the functor associated to the chain complex of the partition posets C (K(r)).∗ Lcotriple ∞ ⊗r¯We have explicitly C (A)= (C (K(r))⊗A ) .∗ ∗ Σrr=0 HarrOn the other hand, we consider the Harrison complex C (A) (cf. D. Har-∗ rison, [37]) which has the coinvariant module Harr ∨ ⊗rC (A)=(L(r) ⊗sgn ⊗A )Σr−1 r r in degree r−1 (see paragraph 6.6 for more precisions). We define a comparison morphism Harr cotriple C (A)→ C (A).∗ ∗ In positive characteristic, the cotriple homology differs from the Harrison homol- cotriple ogy. Inparticular,thecotriplehomologyofasymmetricalgebraH (S(C)(V))∗ Harrvanishes for general reasons while the Harrison homology H (S(C)(V)) does∗ not(cf. M.Barr[6],D.Harrison[37],S.Whitehouse[86]). Consequently,thecom- parison morphism above does not define a quasi-isomorphism of chain complexes. Nevertheless, we deduce from the Koszul duality of operads that this compari- ∼∨son morphism is induced by a quasi-isomorphism of Σ -modulesL(r) ⊗sgn −→∗ r ¯C (K(r)). (In fact, we recover exactly the chain equivalence of M. Wachs and P.∗ Hanlon.) The theorem of the prologue follows from this assertion. Similarly, for Lie algebras, we have, on one hand, the cotriple complex cotripleC (G)=S(L)◦···◦S(L)(G)∗ | {z } ∗−1 CEand, on the other hand, the Chevalley-Eilenberg complex C (G), which can be∗ defined by the formula CE ∨ ⊗rC (G)=(C(r) ⊗sgn ⊗G )r−1 r Σr (cf. M. Barr [7], J.-L. Koszul [45]). One can deduce from classical arguments that the cotriple homology agrees with the Chevalley-Eilenberg homology (cf. M. Barr [7]). We make this assertion more p