Free resolutions via Gr"obner bases

FREE RESOLU TIONS VIA GR ¨ OBNER BASES VLADIMIR DOTSENKO AND ANTON KHOROSHKIN A bstract . For associative algebras in many di ﬀ erent categories, it is po s- sible to develop the machinery of Gr ¨ obner bases. A Gr ¨ obner basis o f deﬁning relations for an algebra of such a category provides a “mono- mial replacement” of this algebra. The main goal of this article is to demonstrate how this machinery ca n be used for t he purposes of h omo- logical algebra. More precisely , our approach goes in t hree steps. First, we deﬁne a combinatorial resolution for the monomial replacement o f an object. Second, we extract from those resolutions expli ci t representatives for h omological class es. F inally , we explain how to “deform” the di ﬀ er- ential to handle the general case. For associative algebras, we recover a well known construction due to Anick. The other case we d iscuss in detail is that of operads, where we disco ver resolutions that haven’t been known previously . W e present various applications, including a proofs of Ho ﬀ beck’s PB W criterion, a proof of Koszulness for a class of o perads coming from commutat ive alg ebras, and a homology computation fo r the operads of Batalin–V ilkovisky algebras and of Rota–Baxter alg e bras. 1. I ntr oduction 1.1. Description of results. Fo r the purpos es of homological and homo- topical algebra, it is often important t o have (quasi-)fr ee resolutions for associative algebras (and their generalisations in various monoidal cate- gories). One r esolution readily availabl e for a generic ass o ciative algebra is obtained by iterating the bar construction (to be more precise, by a cobar- bar construction), however , sometimes it is p referable (and possible) to have a much smaller resolution. The so called minim al r esolution has the homol- ogy of the bar complex as its space of generators ; that homology is also of indep endent interest because it describes highe r sy zygies of the g iven associative algebra (relations, relations between relations etc.). Accor ding to the gener al philosophy of homotopical al gebra [44, 45 ], homology of t he di ﬀ erential induced on the space of indecompo s able e lements of a free r es- olution ( F , d ) → A of the given associative algebra A do e s not depend on the choice of a resolution, a nd this homology coincides with the homology of the d i ﬀ er ential induced on the space o f generators of a r esolution of the trivial A -module by free right A -mod ules. One of most import ant practical results provided (in many d i ﬀ erent frameworks) by Gr ¨ obner bases is that when d ealing with various linear algebra information (ba ses, dimensions etc.) o n e can replace an algebra with complicated r elations by an algebra with monomial r elations without The ﬁrs t author ’s research was supported by the gr ant RF BR-CNRS-07-01-9221 4 and by an IRCSET research fellowship. The second author ’s research was supported by g rants MK-4736.2008 .1, NSh-3035.200 8.2, RFBR-07-01-00526, RFBR 08-01-0 0110, R FBR-CNRS-07- 01-92214 , and by a ETH research fellowship. 1 2 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN losing any information of that so rt. When it comes to questions of homo- logical algebra, things become more subtle, s ince (co)homology may “jump up” for a mo n o mial replacement o f an algebra. Howeve r , the idea of app ly- ing Gr ¨ obner bases to problems of homological algebra is far fr om hopeless. It turns out that f or mo n o mial algebras it is often p ossible t o construct very neat resolutions that can be used for various computations; furthermor e, the data comput ed by these resolutions ca n be used to obtain results i n the general (not necessarily monomial) case. The mail goal of this pape r is to explain this app roac h in detail for computations of the ba r homology . In t he case of usual asso ciative algebras, this approach has been known since the celebrated paper of Anick [1] w h e re in the case of monomial rela tions a minimal right mo d ule resolution of the trivia l module was com- puted, and an explicit way to deform the di ﬀ erential was presente d to handle the gene r al case. Late r , the A nick’s resolution was gene ralized to the case of categories by Malbos [36] who also ask ed whether this work could be e xtended to the case of operads. In this paper , we answer that question, and propose a fram ework that allows to handle ass ociative al- gebras presented via gene rators and r elations in many di ﬀ er ent monoidal categories in a uniform way . Our approac h goes as follows. W e begin with a resolution which is so metimes lar g er than the one o f Anick, but has the advantage of not using much information about the u nderlying mono idal category . It is based on the inclusion–exclusion principle, and is in a sense a version of the cluster method of e numerative combinatorics due to Goulden and Jackson [25]. Once the inclusion–exclusion resolution is constructed , we ﬁnd explicit combinatorial formula s f or ho mological classes of algebras with monomial relations. In the case of algebras, t h is im mediately recovers “chains”, as deﬁned by Anick. T h is is followed, in the sp irit of the Anick’s approac h, of h o w to adapt the d i ﬀ erential of our resolution t o incorporate lower terms of relations and handle arbitrary algebras with known Gr ¨ obner bases. Our main motivating e xample is the case of (sy mmetric) operads. In [11], we introduced new t ype of monoids based o n nons ymmetric collections, shu ﬄ e operads , to develop the machinery of Gr ¨ obner bases for s ymmet- ric operads. Basically , there exists a monoidal st ructur e on nonsy mmetric collections (shu ﬄ e comp o sition) for whic h the for g e tful functor fr om sym- metric collections is monoidal, and this r educes many computations in the symmetric category to thos e in the shu ﬄ e category . The s hu ﬄ e categ ory provides precisely what i s nee d ed t o deﬁne o peradic Gr ¨ obner base s, so ou r approac h applies. Othe r categories where our methods are applied without much c hange a r e c ommutative a ssociative al gebras, a ssociative dial gebras, (shu ﬄ e) coloured op erads, diope rads, 1 2 PROPs et c. W e shall d iscus s details elsewhere. There are various applications of our approac h; some o f them are pre- sented in this p aper . T wo inter e sting theoretica l applica tions are a new short proof of H o ﬀ beck’s PBW criterion for operads [29], and an upper bound on the homology for operads o btained fr om commutative algebras; in particular , we p rove that an operad obtained from a Kos zul commutative algebra is Koszul. S ome interesting concrete examples where all step s of FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 3 our construction can be complete d ar e the case of th e operad RB of Rota– Baxter algebras, its noncommutative analogue ncRB , and, t h e last but not the least , the op erad BV of Batalin–V ilkovisky algebras. Us ing our meth- ods, we were able to compute the bar homolog y of B V and relate it to the gravity operad of Getzler [23]. While preparing o ur paper , we learned that these (and othe r) results on BV we re announced earlier by Drummond-Cole and V al lette (see t h e exte n d ed abstract [14], the slides [51], and the for t h- coming paper [1 3]). The ir a pproach r e lies, on one hand, on on theor ems of Galvez-C arillo, T onks and V allette [21 ] who studied the op erad BV as an op- erad with nonh o mogeneou s (quadratic–linear) relations, and, o n the other hand, on so me new r e sults in ope radic ho mo t opical algebra, in particular , a homoto py transfer the orem for inﬁnity-cooperads . Our methods appear to be completely di ﬀ erent: we t reat BV as an ope r ad with ho mo g eneous rela tions of degrees 2 and 3, and apply the Gr ¨ obner bases approa ch. W e hope that our appr oach to the operad BV is also of independe nt interest as an ill ustration of a rather g eneral method to compute the bar homology . 1.2. Outline of the paper. This paper is or ganised as follows. In S ection 2, we handle (usual) ass o ciative a lgebras with monomial rela- tions. W e construct a free r esolution for such algebras, and then propose a way to choos e explicit representatives of homology classes. It turns out t h at we end up with the celebra ted construction of Anick. W e also explain h o w to “deform” the boundary maps in our resolutions for monomial algebras to obtain resolutions in th e general case of a homog eneous Gr ¨ obner basis. Even though the corresponding homological results for algebras are very well known, we make a point of elaborating on our strategy since it turns out to be fruitful in a ver y gene ral setting. W e also d iscus s an interesting example of a n algebra which exhibits most e ﬀ ects we shall later discover in the operad BV . This algebra also a ppears to provide a counterexample to a theorem of Farkas [18] on the s t ructur e o f the Anick resolution. In Section 3, we demons t rate how to apply our method to shu ﬄ e operads (primarily having in mind applications to symmetric op erads, as mentioned above). W e pr ovide the readers with backgr o und information o n tree mo n o - mials, the replacement for monomials in the case of free shu ﬄ e operads, and explain how t o adapt all t he results previ ously obtained in the case of algebras to the case of operads . Throughout this section, we apply our methods to a particular example, computing dimensions for the low arity homology of the op erad of anti-associativ e algebras, and some low de gree boundary maps in the corresponding r eso lution. In Section 4, we e xhibit applications o f our results outlined above (a new proof o f the PBW criterion, homology estimates for operads coming fr om commutative algebras, and a comp u tation of t he bar homology for the operads RB , ncRB , and BV ). All vector spaces and (co)chain comp lex e s throughout this work are deﬁned over a n arbitrary ﬁeld k of zer o characteristic. 1.3. Acknowledgements. W e are thankful to Fr ´ ed ´ eric Chapoton, Iai n Gor- don, Ed Gr een, Eric Ho ﬀ beck, Jean–Louis Loday , Dmitri Piontkovsk ii and Emil Sk ¨ oldberg for useful discussions, r emarks on a pr e liminary version of 4 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN this p aper and refer ences to literature. Special thanks are due to Li Guo for drawing our attention to Rota–Baxter algebras as an e xample to whic h our methods could be applied, and espe ciall y to Bruno V allette for ma ny useful discussions a nd in particular for e xplaining t he a pproach to the ope rad B V used in hi s joint work with Gabriel Drummond-Cole. 2. A ssocia tive a lgebras 2.1. Inclusion-exclusi on resolution for monomial algebras . In this s e c- tion, we discus s the case of associative algebras with monomial relations. W e st art with an algebra R = k h x 1 , . . . , x n i / ( g 1 , . . . , g m ) with n g enerators and m r e lations, each of whic h is a monomial in the given generators. W e work under the assumpt ion that none of the monomials g 1 , . . . , g m is divi sible by another; this, for example, is the case when G is the set of leading mono mi- als o f a reduced Gr ¨ obner basis of some algebra for which R is a monomial replac ement. Let us de note by A ( p , q ) the vector space whose basis is for me d by el- ements of th e form x I ⊗ S 1 ∧ S 2 ∧ . . . ∧ S q where I = ( i 1 , . . . , i p ) ∈ [ n ] p , x I = x i 1 ⊗ . . . ⊗ x i p is the correspond ing monomial, and S 1 , . . . , S q ar e (in one-to-one correspondence with) certain divisors x i r x i r + 1 . . . x i s of this mono- mial. Each S i is thought of as a sy mbol of homological degree 1, with the appropriate Kos zul sign rule for wedge products (he re q ≥ 0, so the wedge product might be empty ). As we shall see later , in the classical approac h for associative algebras t here is no ne ed in wed ge products because there exists a natu r al linear o rdering on all divi sors of the given monomial. Ho wever , we introduce t h e we dge notation her e as it becomes cr ucial for the general case (e.g. for operads ). For each p the graded vector space A ( p ) = L q A ( p , q ) is a chain complex with the d i ﬀ erential g iven by t he usual formula with omitted factors: (1) d ( x I ⊗ S 1 ∧ . . . ∧ S q ) = X l ( − 1) l − 1 x I ⊗ S 1 ∧ . . . ∧ ˆ S l ∧ . . . ∧ S q . Moreover , the re exists a natural al gebra structur e on A = L p , q A ( p , q ): (2) ( x I ⊗ S 1 ∧ . . . ∧ S q ) · ( x I ′ ⊗ T 1 ∧ . . . ∧ T q ′ ) = ( x I x I ′ ) ⊗ ı ( S 1 ) ∧ . . . ∧ ı ( S q ) ∧ ı ( T 1 ) ∧ . . . ∧ ı ( T q ′ ) , where ı identiﬁes divisors of x I and x I ′ with the corr esponding divisors of x I x I ′ . The di ﬀ er e ntial d makes A into an a ssociative dg-algebra. It is important to emphasize that the s ymbols S i correspond to d ivisors, i.e. occurr ences of monomials in x I rather than monomials themse lves, so in particular the Ko s zul s ign rule does not imply that elements of our a lgebra square to zer o. The foll owing example s hould make our construction more clear . Example 1. Ass ume that our algebra R has just one ge nerator x . The n the algebra A has an e lement x ⊗ S where S corresponds to the divisor of the monomial x equal to x itself. W e have ( x ⊗ S )( x ⊗ S ) = x 2 ⊗ S 1 ∧ S 2 , FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 5 where S 1 and S 2 indicate the two di ﬀ er e nt d ivisors o f x 2 equal to x (a nd the “naive” result of multiplication g iving x 2 ⊗ S ∧ S = 0 does not make sense at all, because S corresponds t o a divisor of x , not of x 2 ). B asically , wh e n computing products, the S -symbols “remember ” which divi sors o f factors they come fr om. So far we did n o t use the rela tions of our algebra. Let us incorporate rela tions in the picture. W e de n o te by G = { g 1 , . . . , g m } t h e set of r elations of our algebra, and by A G the s ubspace of A s panned by all elements x I ⊗ S 1 ∧ . . . ∧ S q for which the divisor corresponding to S j coincides, for every j , with one of the r elations from G . This subspace is stable under product and di ﬀ erential, i.e . is a dg-subalgebra of A . Example 2. Le t u s consider one of the simplest cases of an associative algebra, t hat of dual numbers : R = k [ x ] / ( x 2 ). Here m = n = 1, g 1 = x 2 . Le t us list all mono mials in the corr esponding algebra A . Such a monomial is of the form x n ⊗ S , whe re S is a wedge pr oduct of s y mbols corr esponding to some of t he divisors of x n equal to x 2 . There ar e n − 1 such divisors, the one covering the ﬁrs t two letters x , t he one covering the second and the third o ne etc. W e denot e t h o se divisors by S ( n ) 1 , S ( n ) 2 ,. . . , S ( n ) n − 1 . Thus , S = S ( n ) i 1 ∧ S ( n ) i 2 ∧ . . . ∧ S ( n ) i r . Fo r instance, fo r n = 1 t he only monomial allowed is x ⊗ 1 , for n = 2 t h e re ar e two monomials, x 2 ⊗ 1 and x 2 ⊗ S (2) 1 , for n = 3 — four monomials x 3 ⊗ 1, x 3 ⊗ S (3) 1 , x 3 ⊗ S (3) 2 , x 3 ⊗ S (3) 1 ∧ S (3) 2 . Products o f those monomials ar e compute d in a s traightforward way , k eeping in mind that the S -symbols control the location of corresponding r elation in the underlying x -monomial. F or instance, ( x ⊗ 1)( x ⊗ 1 ) = x 2 ⊗ 1 , ( x 2 ⊗ S (2) 1 )( x ⊗ 1) = x 3 ⊗ S (3) 1 , ( x ⊗ 1)( x 2 ⊗ S (2) 1 ) = x 3 ⊗ S (3) 2 , ( x 2 ⊗ S (2) 1 )( x 2 ⊗ S (2) 1 ) = x 4 ⊗ S (4) 1 ∧ S (4) 3 and so o n. Theorem 1. The dg-algebr a ( A G , d ) is a free reso lution of the corr esponding algebra with monomial r elations k h x 1 , . . . , x n i / ( g 1 , . . . , g m ) . Pro of. Let u s call a collection of divisors S 1 , . . . , S q of x I indecomposa ble , if each pr o duct x i k x i k + 1 is contained in at least one of the m. The n it is easy to see t hat A i s freely generated by e lements x k ⊗ 1 and x I ⊗ S 1 ∧ . . . ∧ S q where S 1 , . . . , S q is an indecompo sable collection of d ivisors of x I . Similarl y , A G is fr eely g enerated by its basis e lements x k ⊗ 1 and all elements x I ⊗ S 1 ∧ . . . ∧ S q where S 1 , . . . , S q is an indecomposable coll ection of divisors, each of which is a r elation of R . Let us pr ove that A G provides a resolution for R . Since the di ﬀ erential d only omits w edge factors but d oes not change t he monomial, the chain complex A G is isomorphic t o the direct sum of chain complexes A I G spanned by the elemen t s for which the ﬁrs t tens or factor is the g iven monomial x I ∈ k h x 1 , . . . , x n i . If x I is not divisibl e by any relation, the complex A I G 6 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN is concentrate d in d egree 0 and is spanned by x I ⊗ 1. Thus , to prove the theorem, we should show that A I G is ac yclic whenever x I is divi sible by some r elation g i . Assume that the re are exactly k divisors of x I which are relations of R . W e immediately s ee that the complex A I G is isomorp hic to the inclusion– exclusion complex for the set [ k ] (3) 0 ← ∅ ← k M i = 1 { i } ← M 1 ≤ i < j ≤ k { i , j } ← . . . ← { 1 , . . . , k } ← 0 (with the us ual di ﬀ erential omitting eleme nts). The latter one is acyclic whenever k > 0, whic h completes the pr o of.  Remark 1. A s imila r construction works for commutative algebras a s w ell, producing the corr e sponding homology groups. W e shall not discuss it in d etail here; the main idea is t hat one can make the symmetric groups act on the fr ee algebras A and A G fr om this section in such a way that the subalgebra of inva riants is a fr ee (super)commutative dg-algebra who se cohomology is the given monomial commutative algebra (acycl icity o f the corresponding resolution can be derived from the acycli city in the asso cia- tive case, a s subcomplexes of invariants of symmetric g roups acting on the acyclic complexe s have to be acyclic by the Maschke’s th e orem). All further constructions o f the paper apply as well; we do not discuss any details or consequences here. It wo u ld be interesting to compare thus constructed resolutions with other known resolutions for monomial commutative alge- bras [5, 3 0]. 2.2. Right mo dule resolution for monomial algebras. Re sults of the pre- vious section compute the bar homology of R , since it coincides with the homology of t he di ﬀ erential induced on the space of generators of a free resolution. This homology also coincides with the homology T or R q ( k , k ) o f R , w hich is t he homology of the complex of gene rators of a fr e e R -module resolution of k . Le t us explain how to materialize this st ate ment via a free module resolution. W e denote by V 0 the linear span of all x k ⊗ 1, and by V q , q > 0, the linear span of all elements x I ⊗ S 1 ∧ . . . ∧ S q as above. W e shall construct a free r e solution of the form (4) . . . → V q ⊗ R → V q − 1 ⊗ R → . . . → V 1 ⊗ R → V 0 ⊗ R → R → k → 0 . It is enough to deﬁne boun d ary maps on th e fr ee mod ule gene r ato rs V q , since boundary maps ar e morphisms o f R -modules. First of all, we let d 0 : V 0 ⊗ R → R be de ﬁned as d 0 ( x k ⊗ 1) = x k . As sume that q > 0 and let x I ⊗ S 1 ∧ . . . ∧ S q ∈ V q . In the fr e e algebra A G , the di ﬀ erential d maps this element to a sum of elements corresponding t o all possible omissions of S j . If afte r the o miss ion of S j we st ill have an indecompo sable cov- ering, this summand survives in the di ﬀ er ential. Otherwise, if after the omission of S j the resulting covering is decomposable, and there exists a decomposition x I = x J x K so that S 1 , . . . , ˆ S j , . . . , S q form an indecomposable covering o f x J , then the corr espond ing summand of the di ﬀ erential becomes ( − 1) j − 1 ( x J ⊗ S 1 ∧ . . . ∧ ˆ S j ∧ . . . ∧ S q ) ⊗ x K ∈ V q − 1 ⊗ R . The following proposition is quite easy to p rove, and we omit the details. FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 7 Proposition 2.1. T he construction above provi des a r esolution of the trivial module by fr ee R -modules. Readers familia r with the machinery of twisting cochains [6] may see our construction of t h e free right module r esolution fr o m the fr e e dg -algebra res- olution as a vari ation of the twisting cochain const ruction. More preci sely , the di ﬀ erential of a gene rator in ou r fr ee algebra resolution is a su m of prod- ucts of g e nerators; this provides the sp ace of g e nerators wi th a s tructure of an ∞ -coalgebra, and the twisting cochain met hod applies. See [34, 43] for details in t he case of algebras, and [13] for de tails in the op erad case . 2.3. Homology classes for monomial algebras. The inclusion-exclusion resolutions constructed above are in general not minimal. In this section, we give a de scription of generato rs of a minimal resolution for any monomial algebra. Our construction is a natur al reﬁnement of t he inclusion-exclusion construction; we also ident ify it with the construction of Anick [1], thus explaining one possible way to invent his results. Let R = k h x 1 , . . . , x n i / ( g 1 , . . . , g m ) be an algebra with n generators and m monomial relations. For the ge nerators of a minimal resolution for the trivial module, one can take homology classes for t h e di ﬀ erential induced on the s pace o f ge nerators ( A G ) ab = ( A G ) + / ( A G ) 2 + of the algebra A G . W e s hall now give one pos s ible s et of representatives for t hese ho mology classes. Let us restrict ourselves to one of the chain complexes ( A I G ) ab correspond- ing to a particular monomial x I . W e ﬁx some ordering of the set of all rela tion divisors of f ; let thos e r elations be S 1 < S 2 < . . . < S m . T hen on the acyclic complex A I G we have m anticommuting di ﬀ erentials ∂ 1 , . . . , ∂ m where ∂ p = ∂ ∂ S p , and t h e corresponding contracting homotopies ı p ( · ) = S p ∧ · . Let us denote ∂ ( p ) = ∂ 1 + . . . + ∂ p , so that the di ﬀ erential d is nothing but ∂ ( m ) . Our key observation is that for each p ≤ m , H (( A G ) I ab , ∂ ( p ) ) = H ( . . . ( H (( A G ) I ab , ∂ 1 ) , . . . ) , ∂ p ) , in o ther words, the homo log y of the to t al complex coincides with the iter - ated homology . This can be easily proved by indu ction on p , since on each step the spectral sequ ence of the corr espond ing bicomplex degenerate s at its page E 1 (since computing ∂ − 1 ( p ) commutes with ∂ p + 1 ). This observation results in the following statement, that can also be proved by induction on p . Proposition 2.2. For each p ≤ m , the homology H (( A G ) I ab , ∂ ( p ) ) has for r eprese n- tatives of all classes all monomials M = x I ⊗ S i 1 ∧ . . . ∧ S i q that satisfy the following pr operties: (i) for all j ≤ p, the monomial ∂ j ( M ) is either decomposable or equal to 0 . (ii) for each q ′ ≤ p such that ∂ q ′ ( M ) = 0 , there exist s q < q ′ for which ı q ′ ∂ q ( M ) , 0 in ( A I G ) ab (i.e., th is monomial is i ndecomposabl e). Setting p = m in this result, we ge t the following de scription o f the homology of ( A G ) I ab , that is the generator s of the minimal resolution: Theorem 2. The homolog y H (( A G ) I ab , d ) has for repr esentatives of all classes all monomials M = f ⊗ S i 1 ∧ . . . ∧ S i q that sati sfy the following pr operties : 8 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN (i) for all j, the monomial ∂ j ( M ) is either decomposable or equal to 0 . (ii) for each q ′ such that ∂ q ′ ( M ) = 0 , there exists q < q ′ for which ı q ′ ∂ q ( M ) , 0 in ( A G ) I ab (i.e., this monomial i s indecompo sable). There is a natural way to order all divisors of a given monomial x I , listing them according to t heir st arting point. It turns out that for this ordering there is anothe r eleg ant description for the repr e sentatives of homolog y classes. L et us r ecall the deﬁnition of Anick chains [1, 50]. Every chain is a monomial of the free algebra k h x 1 , . . . , x n i . For q ≥ 0, q -chains and the ir tails ar e deﬁned inductively a s follows: - each generator x i is a 0 -chain; it coinci des with its tail; - each q -chain is a monomial m equal to a product nst wher e t is the tail of m , and ns is a ( q − 1)-chain whose tail is s ; - in the above decomposition, the product st h as e xactly one divisor which is a relation of R ; this divisor is a right divisor of st . In oth e r w ords, a q -chain is a monomial formed by linking one after another q r elations so that only neighbouring r elations ar e linked, the ﬁrs t ( q − 1) of them form a ( q − 1)-chain, and no proper left divisor is a q -chai n. I n o ur nota- tion above, s uch a monomial m corresponds to the g enerator m ⊗ S 1 ∧ . . . ∧ S q where S 1 , . . . , S q ar e the r e lations we li nked. Proposition 2.3. For the above ordering of divisors, the re pr esentati ves for homol- ogy class es fro m Theor em 2 ar e exactly Anick chains. Pro of. Indee d, condition (i) means t hat only ne ighbours are linked, and condition (ii) means that no proper b eginning of a q -chain forms a q -chain.  2.4. Resolutions for g eneral relations. I n this se ction, we shal l explain the machinery that transforms o u r resolution for a monomial replac ement of the given algebra into a r esolution for the original algebra. Let e R = k h x 1 , . . . , x n i / ( e G ) be an algebra, and let R = k h x 1 , . . . , x n i / ( G ) be its monomial version, that is, e G = { ˜ g 1 , . . . , ˜ g m } is a Gr ¨ obner basis of relations, and G = { g 1 , . . . , g m } are the correspond ing leading monomials. W e have a fr ee resolution ( A G , d ) for R , so t h at H q ( A G , d ) ≃ R . Let π (resp., e π ) be t h e algebra homomorphism fr o m A G to R (r e sp., e R ) that kills all g enerators of positive homological degree, and o n elements o f homological degree 0 is the canonical projection from k h x 1 , . . . , x n i to its quotient. De n o te by h the contracting homotopy f or this r e s olution, so th at ( dh ) | ker d = Id − π . Theorem 3. Ther e exists a “deformed” di ﬀ er ential D on A G and a homotopy H : ker D → A G such that H q ( A G , D ) ≃ e R, and ( DH ) | ker D = Id − e π . Pro of. W e shall construct D and H simultaneously by induction. Let us introduce the following partial ordering of monomials in A G : f ⊗ S 1 ∧ . . . ∧ S q is, by deﬁnition, les s than f ′ ⊗ S ′ 1 ∧ . . . ∧ S ′ q ′ if t h e monomial f is less t h an f ′ in the free algebra. This partia l ordering sugges ts the foll owing de ﬁn ition: for an element u ∈ A G , its leading t e rm ˆ u is the part of the e xpansion of u as a combination of basis elements w h e re we k e ep only basis eleme n t s f ⊗ S 1 ∧ . . . ∧ S q with ma ximal possible f . FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 9 If L is a h o mogeneou s linear operato r on A G of so me ﬁxed (homologica l) degree of ho mogeneity (like D , H , d , h ), we denote by L k the op erator L acting on e lements of ho mological de gree k . W e shall de ﬁne the operators D and H by indu ction: we deﬁne the p air ( D k + 1 , H k ) ass uming that all previous pairs ar e d eﬁned. At each step, we shall also be pr oving that D ( x ) = d ( ˆ x ) + l ower terms , H ( x ) = h ( ˆ x ) + l ower terms , where t he words “lower te rms” mean a linear combination of basis elements whose und e rlying monomial is smaller than t he und erlying monomial of ˆ x . Basis of induction: k = 0, so we have to de ﬁn e D 1 and H 0 (note that D 0 = 0 because there ar e no elements of negative homological degrees). In general, to deﬁne D l , we should only consider the case when our element is a generator of A G , since in a dg-algebra the di ﬀ er e ntial is deﬁned by images of gene r ato rs. For l = 1, this means that we should consider the case where our g e nerator corresponds to a leading monomial f = lt( g ) of some rela tion g , and is of t he form f ⊗ S wher e S corr e sponds to t he only divisor of m whic h is a leading te r m, that is f its e lf. W e put D 1 ( f ⊗ S ) = 1 c g g , where c g is the leading coe ﬃ cient of g . W e see that D 1 ( f ⊗ S ) = f + lower te rms , as requir ed. T o deﬁne H 0 , we use a yet another inductive argument, d e- cr easing t he monomials on which we want to deﬁne H 0 . First of all, if a monomial f is not divisible by any of the leading te rms of relations, we put H 0 ( f ) = 0. Assume that f is divisible by s ome leading terms of rela- tions, and S 1 , . . . , S p ar e the corresponding divisors. Then on A f G we can use S 1 ∧ · as a homotopy , s o h 0 ( f ) = f ⊗ S 1 . W e put H 0 ( f ) = h 0 ( f ) + H 0 ( f − D 1 h 0 ( f )) . Here the leading term of f − D 1 h 0 ( f ) is smaller than f (since we alr eady know that the leading term of D 1 h 0 ( f ) is d 1 h 0 ( f ) = f ), so induction on the leading term applies. Note that by induction the leading te rm of H 0 ( f ) is h 0 ( f ). Suppose that k > 0, that we kn o w the pairs ( D l + 1 , H l ) for all l < k , and that in t hese degrees D ( x ) = d ( ˆ x ) + l ower terms , H ( x ) = h ( ˆ x ) + l ower terms . T o deﬁne D k + 1 , we should, as above, only consider the case of ge nerators. In this case, we put D k + 1 ( x ) = d k + 1 ( x ) − H k − 1 D k d k + 1 ( x ) . The property D k + 1 ( x ) = d k + 1 ( ˆ x ) + lower terms now easily follows by in- duction. T o deﬁne H k , w e proceed in a way very similar to what we did for the induction basis. Assume that u ∈ k e r D k , and that we k n o w H k on all elements of ker D k whose leading term is less than ˆ u . Since 10 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN D k ( u ) = d k ( ˆ u ) + lower terms, we see that u ∈ ker D k implies ˆ u ∈ ker d k . Then h k ( ˆ u ) is deﬁned, and we put H k ( u ) = h k ( ˆ u ) + H k ( u − D k + 1 h k ( ˆ u )) . Here u − D k + 1 h k ( ˆ u ) ∈ ker D k and its leading term is s maller than ˆ u , so induc- tion on the leading t erm applies (and it is easy to check that by induction H k + 1 ( x ) = h k + 1 ( ˆ x ) + lower terms). Let us check that the mappings D and H d eﬁned by these formulas satisfy , for each k > 0, D k D k + 1 = 0 and ( D k + 1 H k ) | ker D k = Id − e π . A comput ation checking that is somewhat similar to the way D and H were constructed. Let us pr ove both st atements simultaneously by induction. If k = 0, the ﬁrst state me nt is obvious. Let us p rove t he second o ne and e s tablish that D 1 H 0 ( f ) = (Id − e π )( f ) for each monomial f . Slightly rephrasing that, we shall p rove that for each monomial f we have D 1 H 0 ( f ) = f − f whe re f is the residue of f modulo G [11]. W e shall prove this state ment by induction on f . If the monomial f is not divisibl e by any leading ter ms of r elations, we have H 0 ( f ) = 0 = f − f . Let f be divisib le by leading terms f 1 , . . . , f p , and let S 1 , . . . , S p be the corresponding divisors. W e ha ve H 0 ( f ) = h 0 ( f ) + H 0 ( f − D 1 h 0 ( f )), so D 1 H 0 ( f ) = D 1 h 0 ( f ) + D 1 H 0 ( f − D 1 h 0 ( f )) . By induction, we may assume that D 1 H 0 ( f − D 1 h 0 ( f )) = f − D 1 h 0 ( f ) − ( f − D 1 h 0 ( f )) . Also, D 1 h 0 ( f ) = D 1 ( f ⊗ S 1 ) = 1 c g f ′ , where g is the rela tion with the leading monomial f 1 , and 1 c g f ′ = 1 c g m f , f 1 ( f ) = f − r g ( f ) is the (normali zed) result of substitut ion of g into f in the place des cribed by S 1 . Consequently , D 1 H 0 ( f ) = f − r g ( f ) +  ( f − D 1 h 0 ( f )) − ( f − D 1 h 0 ( f ))  = = f − r g ( f ) + ( r g ( f ) − r g ( f )) = f − r g ( f ) = f − f , since the residue does not depend on a particular c hoice of r e ductions. Assume that k > 0, and that our statement is true for a ll l < k . W e have D k D k + 1 ( x ) = 0 since D k D k + 1 ( x ) = D k ( d k + 1 ( x ) − H k − 1 D k d k + 1 ( x )) = = D k d k + 1 ( x ) − D k H k − 1 D k d k + 1 ( x ) = D k d k + 1 ( x ) − D k d k + 1 ( x ) = 0 , because D k d k + 1 k ∈ ke r D k − 1 , and so D k H k − 1 ( D k ( y )) = D k ( y ) by induction. Also, for u ∈ ker D k we have D k + 1 H k ( u ) = D k + 1 h k ( ˆ u ) + D k + 1 H k ( u − D k + 1 h k ( ˆ u )) , FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 11 and by the induction on ˆ u we may assume that D k + 1 H k ( u − D k + 1 h k ( ˆ u )) = u − D k + 1 h k ( ˆ u ) (on elements of pos itive homologica l degree, e π = 0), so D k + 1 H k ( u ) = D k + 1 h k ( ˆ u ) + u − D k + 1 h k ( ˆ u ) = u , which is exactly what we need.  Remark 2. • The above const ruction w o rks without a p robl em for ev- ery ﬁnite ly gener ate d algebra with a Gr ¨ obner basis of relations, pro- vided that the monomials in t he free algebra for m a well-order e d se t ; in that case one can be sure that i nductive deﬁnitions provi de well- deﬁned objects . In t he case when the relations are ho mo g eneous , the resolution that we obtain comes with an add itional inte rnal g rading corresponding to the grading on the quo t ient algebra that we are trying to resolve. • A similar construction for the case of free resolution of the trivial module over the given augmented algebra was described by An- ick [1 ] and Kobayashi [32] (se e also Lambe [33]). There are several other ways to con s truct fr e e r e solutions, see e.g. [7, 9, 30, 48] wher e the idea is to s tart fr om the bar complex, select there candidates that we want to be the ge nerators of a smaller fr ee resolution, and construct a contraction of the bar complex o n that subcomplex. W e shall discuss analogues of these constructions beyond the ca se of algebras (e.g. for operads) elsewhere. 2.5. An instructional example: an algebra which resembles th e B V op- erad. In this section, we shall consider a particular e xample of an ass ociative algebra which we ﬁnd quite useful. Deﬁnition 1. T he algebra bv i s an ass ociative algebra with t wo generators x , y and two relations y 2 = 0 , x 2 y + x yx + yx 2 = 0 . From the Gr ¨ obner bases viewpoint, this algebra shar es ma ny similar f ea- tures with the op e rad of Batalin–V ilkovisky algebras (which we shall con - sider in Se ction 4.4): it h as r elations of degr ees 2 and 3 , and a Gr ¨ obner basis is obtained from the m by adjoining a rela tion of deg ree 4, the r esolution for a monomial replacement of this algebra is automatically minimal , but deformation of the di ﬀ erential to incorporate lowe r te r ms leads to many cancellati ons e tc. However , the relevant computations here are les s de- manding, so we hop e it would be easier for our readers to ge t a ﬂavour of our appr oach fr om this example. The fol lowing r esult can be check e d by a d irect computation. 12 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN Proposition 2.4. L et x > y. For the lexico graphi c order ing of monomials, th e Gr ¨ obner ba sis for the a lgebra bv is given by y 2 = 0 , x 2 y + x yx + yx 2 = 0 , x yx y − yx yx = 0 . This immediately tranlates into results for the monomial version of our algebra, that is the algebra with r e lations y 2 = 0 , x 2 y = 0 , x yx y = 0 . Proposition 2.5. The homology of the monomial version of bv is r epr esented by the classe s x , y ∈ T or 1 , y k + 2 ∈ T or k + 2 ( k ≥ 0) , x 2 y k + 1 ∈ T o r k + 2 ( k ≥ 0) , ( x y ) l + 1 x y k + 1 ∈ T or k + l + 2 ( k , l ≥ 0) , x ( x y ) l + 1 x y k + 1 ∈ T o r k + l + 3 ( k , l ≥ 0) . Pro of. These elements ar e precisely t he g enerators o f the free resolution, which in this case is minima l because only ne ighbour r elations overlap in them.  Theorem 4. The homology T or bv q ( k , k ) of the algebra b v is r epres ented by the classes x , y ∈ T or 1 , y k + 2 ∈ T or k + 2 ( k ≥ 0) , x ( x y ) k + 1 ∈ T or k + 2 ( k ≥ 0) . Pro of. One can check that for the deformed di ﬀ er ential D we have D ( x 2 y k + 2 ) = x yx y k + 1 + lower terms , D ( x ( x y ) l + 1 x y k + 2 ) = ( x y ) l + 2 x y k + 1 + lower terms , which leaves from the cocycles d escribed in Proposition 2.5 only y k + 2 for all k ≥ 0, x 2 y k + 1 for k = 0 (which coincides w ith x ( x y ) k + 1 for k = 0), and x ( x y ) l + 1 x y k + 1 for k = 0 (whic h coincides with x ( x y ) k + 1 for k = l + 1). The im- ages of these elemen t s under the di ﬀ er ential ar e killed by the augmentation, and the t heorem follows.  In fact, it is po ssible to exhibit a minimal resolution of bv . Let us denote by a k the homology class repr e sented by y k , and by b k the homology class repr e sented by x ( x y ) k − 1 ; the se cla sses form a basis of the two-dimensional FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 13 space V k = T or bv k ( k , k ). W e leave i t to the r eade r to verify that the f ollowing formulas deﬁne a resolution of k by fr ee bv -modules V k ⊗ bv : D ( a k ) = a k − 1 ⊗ y D ( b k ) =              b k − 1 ⊗ x y − a k − 1 ⊗ x k , k ≡ 0 (mod 3) , b k − 1 ⊗ yx + a k − 1 ⊗ x k , k ≡ 1 (mod 3) , b k − 1 ⊗ ( x y + yx ) + a k − 1 ⊗ x k , k ≡ 2 (mod 3) . Remark 3. Anot h e r inte resting featur e of the algebra bv we considered he re is that it provides a counterexample to a result of Farkas [18] who gave a very s imple formula for di ﬀ erentials in the A nick resolution in terms of the Gr ¨ obner basis of b v . One can easily check that if we apply the F arkas’ di ﬀ erential to the chain x 2 y 2 once, we get the element x 2 y ⊗ y , and if we apply the di ﬀ erential once again, we get x ⊗ yx y − y ⊗ x yx − y ⊗ yx 2 , a nonzero element of the r esolution. Therefore, Farkas’ formula for the di ﬀ erential is generally incorrect, and ther e seem to be no formula that is as simple as the one he sugg ested . For the cont rast, there ar e various algorithmic approac hes to computing maps in the Anick r eso lution, see e .g. [1 0, 26]. 3. O perads 3.1. Shu ﬄ e o p erads. For information o n symmetr ic and nonsymmetric op- erads, we refer the reader to the monograph [41], for information on shu ﬄ e operads and Gr ¨ obner bases for operads — to our paper [11]. Throughout this paper by an ope rad we mean a shu ﬄ e operad, u nless ot h e rwise spec- iﬁed; there is no machinery of Gr ¨ obner bases availabl e in t he s ymmetric case, so we have to sacriﬁce the symmetric gr oups action. H owever , once we move to the shu ﬄ e category , all cons tructions of the previous section work perfectly ﬁne (in fact, the cons truction for algebras is a particular case of the cons truction below , applied to algebras thought of as op erads concentrated in a rity 1). For the monoidal catego ry of shu ﬄ e operads, it is possible to deﬁne the bar complex of an augme nted ope rad O . The bar complex B q ( O ) is a dg- cooperad fr eely g enerated by the degree shift O + [1] of the augmentation ideal of O ; the d i ﬀ er ential comes fr om operadic compositions in O . Sim- ilarly , for a cooperad Q , it is p ossible to deﬁne the cobar complex Ω q ( Q ), which is a dg-ope rad freely generated by Q + [ − 1], w ith the appropriate dif- ferential . The bar-cobar construction Ω q ( B q ( O )) gives a free resolution of O . This can be proved in a rathe r standard way , similarly to known pr oofs in the case of operads, pr operads etc. [24, 19, 52]. The gene ral homotopical algebra philosophy mentioned in t h e introduction is applicable in the case of op erads as we ll; various checks and justiﬁcations neede d to ensu re that ar e quite s t andar d and similar to t he ones available in t he literatur e; we refer the r eader to [ 4, 20, 2 8, 39, 46, 49] where symmetric op erads ar e handled. It is important to recall here that the for getful functo r f : P → P f fr om the category of s ymmetric operads to t he category of shu ﬄ e op erads is monoidal [11], which easily impli es that for a symmetric operad P , we have B q ( P ) f ≃ B q ( P f ) , 14 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN that is the (symmetric) bar complex of P is naturally identiﬁed, as a shu ﬄ e dg-cooperad, with the (s h u ﬄ e) bar complex of P f . Thus, our approach would enable us to compute the ho mology even in the symmetric case, only without i nformation on the s y mmetric gr ou p s action. 3.2. T ree mo nomials. Let us reca ll tree combinatorics used to describe monomials in s hu ﬄ e operads. Se e [ 11] for mor e details. Basis elements of t he free operad ar e represented by (decorated ) trees. A (rooted) tree is a non-empty connecte d directed graph T o f genus 0 for which each v ertex has at least one incoming edge and exactly one outgoing edge. So me edge s of a tree might be bounded by a vertex at one end only . Such edges ar e called external . E ach tree s hould have exactly one outgoing external edge, its output . The endpoint of this ed ge which is a vertex of our tree is called t he r oot o f the tree. The endpoints of incoming external edges which ar e not vertices of our tr ee are c alled leav es . Each tree with n leaves should be (bijectively) labelled by [ n ]. Fo r each vertex v of a tree, the ed ges going in and ou t of v will be referr ed to as inpu ts and outputs at v . A tr ee with a single vertex is called a cor olla . There is also a tree with a single input and no vertices called t he degenerate tree. T rees ar e originally cons ide red as abstr act graphs but to work with them we would need some partic ular r epresentatives that we now going to d escribe. For a tree with labelled leaves, its canonical planar repr esentative is deﬁned as follows. In general, an embedding of a (rooted) tree in the plane is determined by an ordering of inputs for e ach vertex. T o compar e two inputs of a vertex v , we ﬁnd the minimal leaves that o ne can reach fr om v via the corresponding input. The input for which t he minimal leaf is smaller is considered to be less than the othe r one . Note that t his choice of a representative is ess e ntially the same one as we already made when we identiﬁed symmetric compositions with shu ﬄ e compositions. Let us intr oduce an e xplicit r ealisation of the free operad generate d by a collection M . The basis of this operad will be indexed by planar r epre- sentative of trees with decorations of all vertices. First o f all, t h e simplest possible t ree is the dege nerate tree; it correspond s to the unit of our operad. The secon d s implest type of trees is given by cor ollas. W e shall ﬁx a ba- sis B M of M and decorate the vertex of e ach cor olla with a basis element; for a cor olla with n inputs, the corr esponding element should belong to the basis of V ( n ). The basis for w hole fr ee oper ad consist s of all planar repr e sentatives of trees built from these corolla s (explicitly , one starts with this collection of corollas, deﬁnes compositions of trees in terms of graft- ing, and then cons iders all tr ees obtained from cor ollas by iterated shu ﬄ e compositions). W e shall r efer to e lements of this basis as tr ee monomials . There ar e two s tandard ways to think of elements of an operad deﬁned by generators and r elations: us ing either tree monomials o r ope rations. Our approac h is s o mewhere in the midd le: we prefer (and strongly encou rage the reader) to think of tr ee monomials, but to write formulas r equired for deﬁnitions and proofs we prefer the language of o perations since it makes things mor e compact. Let us give an example of how to translate betwee n thes e two languages . Let O = F M be the free op erad for which the only nonzero component of FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 15 M is M (2), and the basis of M (2) is given by 2 1 2 1 , . . . , s 2 1 Then the ba sis of F M (3) is given by the tr ee monomials i j 1 2 3 , i j 1 3 2 , and j i j 3 2 1 with 1 ≤ i , j ≤ s . If we ass u me that the j th coroll a corresponds to the operation µ j : a 1 , a 2 7→ µ j ( a 1 , a 2 ) , then the above tree monomial s correspond to operations µ j ( µ i ( a 1 , a 2 ) , a 3 ) , µ j ( µ i ( a 1 , a 3 ) , a 2 ) , and µ j ( a 1 , µ i ( a 2 , a 3 )) respectively . T ake a t ree monomial α ∈ F M . If we forget the labels o f its vertices and its leaves, we get a planar t ree. W e shall refer to this planar t ree as the un derlying tr ee of α . Divisors of α in the fr ee ope rad cor respond to a sp ecial kind of subgraphs of its underlying tree. A llowed subg r aphs contain, tog ether with each vertex, all its incoming and outgoing edges (but not necessarily other endpoints of these edges). Thr oughout this paper we consider only this kind of subgraphs, and we refer t o them as subtrees hoping th at it does not lead to any confusion. Clearly , a subtree T ′ of every t ree T is a tr ee itself. Let us d eﬁne the tree monomial α ′ corresponding to T ′ . T o lab el vertices of T ′ , we r ecall the la bels of its vertices in α . W e immediately observe that these la bels match th e r estriction labels of a tree monomial s h o uld have: each vertex has the same number of inputs as it had in the original t ree, so for a vertex with n inputs its label does belong to th e basis of M ( n ). T o label leaves of T ′ , not e that each su ch leaf is either a leaf of T , or is an outp ut of some vertex of T . This allows us to assign to each leaf l ′ of T ′ a leaf l of T : if l ′ is a leaf of T , put l = l ′ , otherwise let l be the small est leaf of T that can be r eached t h rough l ′ . W e then number the leaves according to these “smallest descend ants ”: the leaf wi th the smallest possible des cendant gets the label 1, t he s econd smallest — th e label 2 etc. Example 3. L e t us conside r the tree mono mial f g f f f f 1 2 3 4 5 6 7 16 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN in th e free ope rad with two binary ge nerators labelled a and b ; in the language of operations, it corresponds to the expression f ( g ( f ( a 1 , a 3 ) , a 5 ) , f ( f ( a 2 , a 7 ) , f ( a 4 , a 6 ))) . One of its s u btrees (indicated by bold lines in the ﬁg u re) pr o duces the tree monomial f g 1 f 2 4 5 ≃ f g 1 f 2 3 4 (in the language of op erations f ( g ( a 1 , a 5 ) , f ( a 2 , a 4 )) ≃ f ( g ( a 1 , a 4 ) , f ( a 2 , a 3 ))), where the lab els on t he l eft come from minimal leaves, as explained above. Note that eve n though in t his example t he subtree shares t he root vertex with the o riginal tr ee, in g eneral it is not requir e d. For two tr e e monomials α , β in t he free operad F M , we s ay that α is divisi ble by β , if there exists a su btree of the underlying tr ee o f α for which the corr esponding tr e e monomial α ′ is equal to β . Let us introduce an example of an o perad wh ich will be used to illustrate methods of t his se ction. It w as deﬁne d and s t udied by Markl and Re mm in [40]. Deﬁnition 2. The anti-associativ e operad f As is the nonsy mmetric operad with one g enerator f (- , -) ∈ f As(2) and one relation (5) f ( f (- , -) , -) + f (- , f (- , -)) = 0 . For the path-lexicographic ordering, the element f ( f ( f (- , -) , -) , -) is a small common multiple of the leading monomial with itself, and t he correspond- ing S-polynomial is equ al t o 2 f (- , f (- , f (- , -))). These relations togethe r al- ready imply that A ( k ) = 0 for k ≥ 4, so we have the following Proposition 3.1. T wo r elatio ns f ( f ( - , - ) , - ) + f ( - , f ( - , - )) and f ( - , f ( - , f ( - , - ))) form a Gr ¨ obner ba sis of r elations that deﬁne f As . The corresponding op erad with monomial relations is deﬁned by r e la- tions f ( f (- , -) , -) = 0 and f (- , f (- , f (- , -))) = 0, and has a monomial basis { id , f , g : = f (- , f (- , -)) } . Operads in the di ﬀ erential graded setting. The above de scription of the fr ee shu ﬄ e o perad wo rks almost literally when we work with o perads whose comp o nents are chain complexes (as opposed to vector spaces), and the symmetric monoidal structure on the corr es ponding category involves signs. The only di ﬀ erence is that every tree monomial should carry an ordering o f its internal vert ices, so that two di ﬀ erent orderings contribute appropriate signs. In this section, we give an e xamples of a shu ﬄ e dg - operad that should help a reader to und erstand the graded case bet t er; it is very close to t h e (ungraded ) anti-associative operad which we discuss throughout the paper . Lie the anti-associative operad, it is also introduced in [40]. Deﬁnition 3. The o d d (2 k + 1)-associative operad is a nonsymmetric ope r ad with one gene rator µ of arity n = 2 k + 1 and homologica l degree 2 l + 1, and rela tions µ ◦ p µ = µ ◦ 2 k + 1 µ for all p ≤ 2 k . FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 17 Let us sho w that the B uchberger algorithm for ope r ads from [11] dis- covers a cubic relation in the Gr ¨ obner basis for this operad, thus sho wing that this operad fails to be PBW . W e use the path-lexicographic or dering of monomials. From the common multiple ( µ ◦ 1 µ ) ◦ 1 µ o f the leading term µ ◦ 1 µ with itself, we compu te the S-polynomial ( µ ◦ n µ ) ◦ 1 µ − µ ◦ 1 ( µ ◦ n µ ) . W e can perform t he following chain of reductions (with leading mono mials underlined): ( µ ◦ n µ ) ◦ 1 µ − µ ◦ 1 ( µ ◦ n µ ) = ( µ ◦ n µ ) ◦ 1 µ − ( µ ◦ 1 µ ) ◦ n µ 7→ 7→ ( µ ◦ n µ ) ◦ 1 µ − ( µ ◦ n µ ) ◦ n µ = − ( µ ◦ 1 µ ) ◦ 2 n − 1 µ − ( µ ◦ n µ ) ◦ n µ 7→ 7→ − ( µ ◦ n µ ) ◦ 2 n − 1 µ − ( µ ◦ n µ ) ◦ n µ = − ( µ ◦ n µ ) ◦ 2 n − 1 µ − µ ◦ n ( µ ◦ 1 µ ) 7→ 7→ − ( µ ◦ n µ ) ◦ 2 n − 1 µ − µ ◦ n ( µ ◦ n µ ) = − 2( µ ◦ n µ ) ◦ 2 n − 1 µ. Note that w e u s ed the f ormula ( µ ◦ n µ ) ◦ 1 µ = − ( µ ◦ 1 µ ) ◦ 2 n − 1 µ which reﬂect the fact that t he o peration µ i s of odd homologica l degree. The monomial ( µ ◦ n µ ) ◦ 2 n − 1 µ cannot be reduced furt h e r , and we r e cover the relati on ( µ ◦ n µ ) ◦ 2 n − 1 µ = 0 discove red in [40 ]. Furthe rmore, we arrive at the following proposition (note t he similarity with the computation of the Gr ¨ obner basis for the operad AntiCom in [11]). Proposition 3.2. Elements µ ◦ p µ − µ ◦ 2 k + 1 µ with 1 ≤ p ≤ 2 k and ( µ ◦ n µ ) ◦ 2 n − 1 µ form a G r ¨ obner basis for the ope rad of odd (2 k + 1) -associa tive algebras. 3.3. Inclusion-exclusi on resolution. Let us construct a free r e solution for an arbitrary op e rad with mono mial relations. Assume that the op e rad O is generate d by a collection of ﬁnite se ts M = { M ( n ) } , with m mon o mial rela tions g 1 , . . . , g m (this means that every t ree monomial divisible by any o f the relations is equal to zero), O = F M / ( g 1 , . . . , g m ). W e deno te by A ( T , q ) the vector space with the basis consisting of all elements T ⊗ S 1 ∧ S 2 ∧ . . . ∧ S q where T is a tr e e monomial fr om the free shu ﬄ e operad F M , and S 1 , . . . , S q , q ≥ 0, ar e tree divisors o f T . The di ﬀ er ent ial d wi th (6) d ( T ⊗ S 1 ∧ . . . ∧ S q ) = X l ( − 1) l − 1 T ⊗ S 1 ∧ . . . ∧ ˆ S l ∧ . . . ∧ S q makes the graded vector space (7) A ( n ) = M T with n leaves M q A ( T , q ) into a chain complex. There is also a natural o perad structure on the collection A = { A ( n ) } ; the operadic compo sition compos es the t rees, and computes the wedge product of divisors (using, as in the case of algebras the identiﬁcation ı of tr ee divisors of T and T ′ with the corresponding d ivisors of their composition). Overall, we deﬁned a shu ﬄ e dg-operad. Let G = { g 1 , . . . , g m } be the set of relations of our operad. The dg-operad ( A G , d ) is spanned by the elements T ⊗ S 1 ∧ . . . ∧ S k where for each j the d ivisor of T corresponding to S j is a relation. The di ﬀ erential d is the restriction of 18 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN the di ﬀ erential d eﬁned above. Informally , a n element of th e operad A G is a tree with some distinguished divisors that are r elations from the g iven set. The following th e orem is pr oved ana logously to its counterpart for a sso- ciative algebras, Theor em 1 above. Theorem 5. The dg-opera d ( A G , d ) is a free reso lution of the corresp onding operad with monomial r elations O = F M / ( G ) . 3.4. Right mo dule resolution for mon omial operads. Similarly to how it is done i n Section 2.2, it is possible to pr ove the following Theorem 6. L et O be a mon omial operad, and let us denote by V q the collectio n of the generators of the free reso lution fr om the prev ious sect ion of homologica l degr ee q. Then ther e exists an ex act sequence of co llections (8) . . . → V q ◦ O → V q − 1 ◦ O → . . . → V 1 ◦ O → V 0 ◦ O → O → k → 0 . Let us describe so me low degree maps of the resolution for t he monomial version of the anti-associative operad. W e use the following not ation for the low degree basis elements: α ∈ V 0 (2) is the eleme n t corresponding to f , β ∈ V 1 (3) and γ ∈ V 1 (4) are the elements corr esponding to f ( f (- , -) , -) and f (- , f (- , f (- , -))) respectively , and ω ∈ V 2 (4) correspond s to t he small common multiple we d iscussed earlier (the o ve r lap of two cop ies of f ( f (- , -) , -)); t here ar e other elements in V 2 , but we shall use only this one in our example. The following pr o p osition is straightforwar d; we encourage o u r r eaders to per fo r m the compu tations the ms elves to ge t familiar with our a pproach. Proposition 3.3. We ha ve d 0 ( α ( - , - )) = f ( - , - ) , h 0 ( f ( - , - )) = α ( - , - ) , d 1 ( β ( - , - , - )) = α ( f ( - , - ) , - ) , d 1 ( γ ( - , - , - , - )) = α ( - , g ( - , - , - )) , h 1 ( α ( - , g ( - , - , - ))) = γ ( - , - , - , - ) , h 1 ( α ( f ( - , - ) , f ( - , - ))) = β ( - , - , f ( - , - )) , h 1 ( α ( g ( - , - , - ) , - )) = β ( - , f ( - , - ) , - ) , d 2 ( ω ( - , - , - , - )) = β ( f ( - , - ) , - , - ) . 3.5. Homology cla sses. Also, one can obtain r epresentatives for ho mo l- ogy classes in exactly the same way as for associative algebras. L et us choose a tr e e monomial T , and work with the inclusion-exclusion complex A G ( T ) : = L q A G ( T , q ). W e ﬁx some or dering of the set o f al l r elation divi- sors o f T ; let those relations be S 1 < S 2 < . . . < S m . On the acyclic complex A G ( T ) we have m anticommuting d i ﬀ erential s ∂ 1 , . . . , ∂ m where ∂ p = ∂ ∂ S p , and the corresponding contracting homotopies ı p ( · ) = S p ∧ · . Let us denote ∂ ( p ) = ∂ 1 + . . . + ∂ p , so t hat the d i ﬀ er ential d is nothing but ∂ ( m ) . The following th e orem is pr oved ana logously to its counterpart for a sso- ciative algebras, Theor em 2 above. Theorem 7. The homology H ( A G ( T ) ab , d ) has for r epr esentatives of all clas ses monomials M = T ⊗ S i 1 ∧ . . . ∧ S i q that satisfy the following prop erties: FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 19 (i) for all j, the monomial ∂ j ( M ) is either decomposable or equal to 0 . (ii) for each q ′ such that ∂ q ′ ( M ) = 0 , there exists q < q ′ for which ı q ′ ∂ q ( M ) , 0 in A G ( T ) ab (i.e., th is monomial is i ndecomposabl e). Unlike the s ituation for monomials in ass ociative algebras, t h e re is no natural o rdering of divisors for a tree monomial in t he free shu ﬄ e operad (since “trees grow in several di ﬀ erent directions”), so there is no descrip- tion of repr esentatives in a manner as eleg ant as it is in the case of A nick resolution. However , in some cases it i s possible to make use of it. W e shall discuss some a pplications below , when dealing with particul ar examples. Using r esults of this section, we can compute r e presentatives for low degree ho mo log y classes of t he monomial version of the anti-associative operad. Our results on dimensions of component s for t he corresponding minimal r esolution R ar e summarised in t he following Proposition 3.4. We ha ve dim R (2) 0 = 1 , dim R (3) 1 = 1 , dim R (4) 1 = dim R (4) 2 = 1 , dim R (5) 2 = 5 , dim R (5) 3 = 1 , dim R (6) 3 = 15 , d im R (6) 4 = 1 , dim R (7) 3 = 4 , dim R (7) 4 = 35 , d im R (7) 5 = 1 . 3.6. Resolutions for general relations. Let e O = F M / ( e G ) be an operad, and let O = F M / ( G ) be its mono mial version, that is, e G is a Gr ¨ obner basis of rela tions, and G cons ists of all leading monomials of e G . In Section 3.3, we deﬁned a fr ee r e solution ( A G , d ) for O , so that H q ( A G , d ) ≃ O . Let π (resp., e π ) be the canonical homomorp hism fr om A G to O that k ills all generators of positive homolog ical deg ree, and on elements of ho mo log ical degree 0 is the canonical projection from F M to its quotient. Denot e by h the cont racting homotopy for this resolution, so that ( dh ) | ker d = Id − π . Similarl y to how it is p roved in Section 2.4, we o btain the fo llowing Theorem 8. T her e exists a “deformed” di ﬀ er ential D on A G and a homotopy H : ker D → A G such that H q ( A G , D ) ≃ e O , and ( DH ) | ker D = Id − e π . Let us compute some low degree maps of t he r esolution for the anti- associative ope r ad. W e shall cheat a little bit and deform the right mod ule resolution, not t he dg -operad resolution, as th e former is smaller and s o the amount of computations is not excessive. R e sults of Pr oposition 3.3 can be used to c ompute the deformed ma ps as foll ows. 20 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN Proposition 3.5. We ha ve D 0 ( α ( - , - )) = f ( - , - ) , H 0 ( f ( - , - )) = α ( - , - ) , D 1 ( β ( - , - , - )) = α ( f ( - , - ) , - ) + α ( - , f ( - , - )) , D 1 ( γ ( - , - , - , - )) = α ( - , g ( - , - , - )) , H 1 ( α ( - , g ( - , - , - ))) = γ ( - , - , - , - ) , H 1 ( α ( f ( - , - ) , f ( - , - ))) = β ( - , - , f ( - , - )) − γ ( - , - , - , - ) , H 1 ( α ( g ( - , - , - ) , - )) = β ( - , f ( - , - ) , - ) + γ ( - , - , - , - ) , D 2 ( ω ( - , - , - , - )) = β ( f ( - , - ) , - , - ) + β ( - , f ( - , - ) , - ) − β ( - , - , f ( - , - )) + 2 γ ( - , - , - , - ) . Pro of. Formulas for D 0 and H 0 ar e obvious. For D 1 and H 1 , the compu t ation goes as foll ows: D 1 ( β (- , - , -)) = α ( f (- , -) , -) − H 0 D 0 ( α ( f (- , -) , - )) = = α ( f (- , -) , -) − H 0 ( f ( f (- , -) , - )) = α ( f (- , -) , -) + H 0 ( f (- , f (- , -))) = = α ( f (- , -) , -) + α (- , f (- , -)) , D 1 ( γ (- , - , - , -)) = α (- , g (- , - , -)) − H 0 D 0 ( α (- , g (- , - , -))) = α (- , g (- , - , -)) , H 1 ( α (- , g (- , - , -))) = γ (- , - , - , -) + H 1 ( α (- , g (- , - , - )) − D 1 γ (- , - , - , -)) = γ (- , - , - , -) , H 1 ( α ( f (- , -) , f (- , - ))) = = β (- , - , f (- , -)) + H 1 ( α ( f (- , -) , f (- , - )) − D 1 ( β (- , - , f (- , -)))) = = β (- , - , f (- , -)) − H 1 ( α (- , g (- , - , -))) = β (- , - , f (- , -)) − γ (- , - , - , -) , H 1 ( α ( g (- , - , -) , -)) = β (- , f (- , -) , -) + H 1 ( α ( g (- , - , -) , -) − D 1 ( β (- , f (- , -) , - ))) = = β (- , f (- , - ) , -) + H 1 ( − α (- , − g (- , - , -))) = β (- , f (- , -) , -) + γ (- , - , - , -) . For D 2 , we may use the formulas we already obtained, getting D 2 ( ω (- , - , - , -)) = β ( f (- , -) , - , -) − H 1 D 1 ( β ( f (- , -) , - , -)) = = β ( f (- , -) , - , -) − H 1 ( α ( f ( f (- , -) , -) , -) + α ( f (- , -) , f (- , -))) = = β ( f (- , -) , - , -) − H 1 ( − α ( g ( - , - , -) , -) + α ( f (- , -) , f (- , -))) = = β ( f (- , -) , - , -) + ( β (- , f (- , -) , -) + γ (- , - , - , -)) − ( β (- , - , f (- , -)) − γ (- , - , - , -)) = = β ( f (- , -) , - , -) + β (- , f (- , - ) , -) − β (- , - , f (- , -)) + 2 γ (- , - , - , -) .  In particular , whe n we use our resolution t o comput e T or f As q ( k , k ), all summands killed by the augmentation vanish, and we ge t d 1 β = d 1 γ = 0 , d 2 ω = 2 γ, so we see once again that T or f As 2 ( k , k ) is one-dimensional. (This r e sult is not surprising: the second term of the bar homolog y encodes rela tions, and in our case the space of relations is one-dimensional, and β is the leading t erm of that r elation.) Moreover , using r e sults of S e ction 3.5, and computing th e “deformed” di ﬀ erential , it is easy to check that in all arities less than 8 the homology is FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 21 concentrated in o n e homological degree. Dimensions ar e summarised in the following Proposition 3.6. We ha ve dim T or f As 1 ( k , k )(2) = 1 , dim T or f As 2 ( k , k )(3) = 1 , dim T or f As 3 ( k , k )(5) = 4 , dim T or f As 4 ( k , k )(6) = 14 , dim T or f As 5 ( k , k )(7) = 30 . 4. A pplica ti ons 4.1. Another proof of the PBW crit erion for Koszulness. The goal of this section is to prove the following statement (which brings to the common ground the PBW criterion of Priddy [42] for associative algebras and the PBW criterion of Ho ﬀ beck [29] for operads). Theorem 9. An associative algebra (commu tative algebra, ope rad et c.) with a quadratic Gr ¨ obner basis is Koszul. Pro of. First of all, it is e n o ugh to p rove it in the monomial case, since it gives an upper bound on the homolog y: for the deformed di ﬀ erential, the cohomology may only decrease. In t he monomial case, it follows from an easy obse r vation that the fr ee right module resolution we const ructed actually coincides with t he Koszul complex o f t he corresponding algebra (commutative algebra, operad e tc.), and acycli city of the Koszul c omplex is one of t he equivalent c riteria of Koszulness.  4.2. Operads and co m mutative algebras. R ecall a construction o f an op- erad fr om a graded commutative al gebra described in [3 1]. Let A be a g raded commutative algebra. Deﬁne an o p erad O A as follows. W e put O A ( n ) : = A n − 1 , and l et the partial comp o sition map ◦ i : O A ( k ) ⊗ O A ( l ) = A k − 1 ⊗ A l − 1 → A k + l − 2 = O A ( k + l − 1) be the p roduct in A . As we remarked in [11], a basis of the algebra A leads to a basis of the operad O A : product of gene rators of the polynomial algebra is r ep laced by the iterated composition of the corresponding gene rators o f the free operad where each composition is s ubstitution into t he last slot of an operation. Assume that we know a Gr ¨ obner basis for the algebra A (as an associative algebra). It leads to a Gr ¨ obner basis for the operad O A as follows: we ﬁrst impose the quadratic r elations d eﬁning t he o perad O k [ x 1 ,..., x n ] coming fr om the polynomial algebra (stating that t he result of a compos ition depend s only on t he operations compos e d, not o n the order in which w e compose operations), and then use the identiﬁcation of rela tions in the po lynomial algebra with element s of the corresponding operad, as above. Our next g oal is to explain how to use the Anick resolution of the trivial module for A to construct a s mall resolution o f the trivial module f or O A . 22 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN Theorem 10 . Ther e exists a fr ee right module resolutio n R ◦ O A → k → 0 of the trivial O A -module. Here R is the ope rad generated by all bar homology classes of A, wher e we put classes of internal degr ee k − 1 in ar ity k, with rel ations c 1 ◦ k c 2 = 0 , for all classes c 1 , c 2 wher e c 1 has arity k. In other words, in R all compositi ons are all owed except for th ose using th e last slot of a n ope ration. Pro of. This statement is almost immediate from o ur previous results. In- deed, we k n o w ho w to o bt ain a Gr ¨ obner basis for O A fr om a Gr ¨ obner b asis of A . If we apply the result of Th e orem 7, it is clear that the minimal r eso- lution over the monomial replacement of O A is of the same f orm as deﬁned above, but we should st art with the operad generated by chains, not by the homology . T o obtai n a r esolution over O A , let us look carefully into the general r econst ruction scheme fr o m the previ ous section. It recovers lower terms of di ﬀ er entials and homot opies by r ecalling lower terms of elements of the Gr ¨ obner basis. L e t us do t he reconstruction in two step s . At ﬁrst, we shall recal l all lower ter ms of relations except for tho se starting with α ( β (- , -) , -); th e latter are s till assumed to vanish. On the next step we shall reca ll all lower terms of those quadratic r elations. No te that after the ﬁrst step we model many copies of the associative algebra resolution and the di ﬀ erential the re; so w e can compute the homology e xplicitly . At the next step, a di ﬀ er ential will be induced on this homology we c omputed, and we end up with a resolution o f the r equ ired type.  In so me cases the exist e nce of such a resolution is e nough to comput e t he bar homology of O A ; for example, it is so when the algebra A is Koszul, as we shall s ee now . In general, the di ﬀ erential of this r es olution incorp o rates lots of information, including the higher ope rations (Massey (c o)products) on the homology of A . Recall that if the algebra A is qu adratic, then the operad O A is quadratic as we ll. In [11], we p roved t hat if the algebra A is PB W , then the operad O A is PBW as well, and he nce is Ko szul. Now we shall prove the following substantial generalisation of this s tatement (substantially simplifying t he proof of this statement given in [31]). Theorem 11 . If the a lgebra A is Koszul, then the oper ad O A is Koszul as well. Pro of. Koszulness of ou r algebra implies that the homology of the bar res- olution is concentrated on the diagonal. Cons equently , the o perad R con- structed above is automatically concentrated o n the diagonal, and so is its homology , which completes t he pr oof.  It is wor t h mentioning t h at the same can be applied to dioperads. For a commutative graded algebra A , let us put D A ( m , n ) : = A m + n − 2 , and let the partial composition ma p ◦ i , j : D A ( m , n ) ⊗ D A ( p , q ) = A m + n − 2 ⊗ A p + q − 2 → → A m + n + p + q − 4 = D A ( m + p − 1 , n + q − 1) be the product in A . The bi-collection { D A ( m , n ) } forms a dioperad, which is quadratic whenever the algebra A is, and, as it turns out, is Koszul wheneve r FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 23 the algebra A is. This ca n be proved similarly to how the previous theo rem is pr oved. 4.3. The op erads of Rota– Baxter algebras. T h e mai n goal of this s e ction is to compute the bar homology for t h e operad of Rota–Baxter algebras, and the op e rad of noncommutative Rota–Baxter algebras. Those are among th e simplest examples of ope r ads which ar e not covered by the Koszul duality theory , be ing operads with nonho mogeneous r e lations. 4.3.1. The op erads RB and n cRB, and th eir Gr ¨ obner bases. Deﬁnition 4. A commutative Rota–Baxter algebra of weight λ is a vector space with an associative commutative p roduct a , b 7→ a · b and a unary operator P w hich satisfy the following identity: (9) P ( a ) · P ( b ) = P ( P ( a ) · b + a · P ( b ) + λ a · b ) . W e de note by RB the operad of Rota–Baxter algebras. W e view it as a shu ﬄ e operad with one binary and one unary generator . Commutative R o ta–Baxter algebras were deﬁned by Rota [47] who was inspired by work of Baxter [3] in probabi lity the o ry . V arious constructions of free commutative Rota–Baxter algebras we re g iven by R ota, Cartier [8] and, more recently , Guo and Ke igher [27]. The latter paper also contains extensive bibliography and information on various applications of those algebras. Deﬁnition 5. A noncommutative Rota–Baxter algebra of weight λ is a vector space w ith an asso ciative pr od uct a , b 7→ a · b a nd a unary operator P which satisfy the same identity as above: P ( a ) · P ( b ) = P ( P ( a ) · b + a · P ( b ) + λ a · b ) . W e denot e by n cRB the operad of noncommutative Rota–Baxter algebras. Somehow , it is a bit simpler than the operad in the commutative case, because it can be viewed as a nonsymmetric operad with one binary and one unary generator . Noncommutative Rota–Baxter algebras has been ext e nsively studied in the past years. W e r e fer the r e ader to the paper o f Ebrahimi –Far d and Guo [16] for an exte nsive discussion of applications and occurrences of those algebras in various areas of mathe matics, and a combinatorial construction of the correspond ing free al gebras. Let us conside r the p ath-lexicographic or dering of the free operad; we assume that P > · . Proposition 4.1. The deﬁning r elations fo r operads R B and ncRB form a Gr ¨ obner basis. Pro of. Here we present a proof for the case of n cRB , the proof for RB is es- sentially the same, with the only e x ce p tion that there are two S-po lynomials to be reduced, as opposed to one S-polynomial in the case o f ncRB (whic h, as we pointed above, i s easier b ecause we are dealing w ith a no nsymmetric operad). 24 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN For the ass ociative suboperad o f ncRB , t he deﬁning relations form a Gr ¨ obner basis, so the S-polyno mials coming from the small common multi- ples the leading term of the ass ociativity relation has with itself clearly can be reduced to zero. The leading term of the Rota–Baxter r elation is P ( P ( a 1 ) a 2 ). This term only has a nontrivial overlap with itself, not with the leading term of t he associativity rela tion, and that overlap is P ( P ( P ( a 1 ) · a 2 ) · a 3 ). From t h is overlap, we compu t e the S-polynomial − P ( P ( a 1 · P ( a 2 )) · a 3 ) − λ P ( P ( a 1 · a 2 ) · a 3 ) + P (( P ( a 1 ) · P ( a 2 )) · a 3 ) + + P (( P ( a 1 ) · a 2 ) · P ( a 3 )) + λ P (( P ( a 1 ) · a 2 ) · a 3 ) − P ( P ( a 1 ) · a 2 ) · P ( a 3 ) , and it can be reduced to zero by a leng thy sequen ce of reductions which we omit he re (but which in fact can be read from the formula for d ν 3 in Proposition 4 below). By Diamond Lemma [11], our relations form a Gr ¨ obner basis.  In the case of the operad ncRB , our compu tation immediately provides bases for free noncommutative R ota–Baxter algebras. Inde e d, s ince our op- erad is no nsymmetric, the degree n component of t he free noncommutative Rota–Baxter algebra g enerated by the se t B is nothing but ncRB ( n ) ⊗ V ⊗ n , where V = span( B ), so we c an use the a bove Gr ¨ obner basis to de scribe that component. More precisely , we ﬁrst deﬁne the set o f admissible e xpressions on a s et B r ecursively as follows: • elemen t s o f B a re admiss ible expr essions; • if b is an admissible expression, then P ( b ) is an admissible exp ression; • if b 1 , . . . , b k ar e admissible expr essions, and for each i e ither b i is an element o f B or b i = P ( b ′ i ) with b ′ i an admissible expression, then t heir associative pr oduct b 1 · b 2 · . . . · b k is an admi ssible expression. Based on this deﬁnition, we shall call s ome of admissible expressions the Rota–Baxter monomials, t r acing the construction o f an admissible expres- sion and putting some restrictions. Namely , • elemen t s o f B a re R ota–Baxter monomia ls; • if b is a Ro ta–Baxter monomial , which, as an admi ssible expression, is either b = P ( b ′ ) or b = b 1 · b 2 · . . . · b k with b 1 ∈ B , then P ( b ) is a Rota–Baxter monomia l; • if b 1 , . . . , b k ar e Rota–Baxter monomials, and for each i either b i ∈ B or b i = P ( b ′ i ) f or some b ′ i , then their a ssociative pr oduct b 1 · b 2 · . . . · b k is a Ro t a–Baxter monomial . Our pr evious discussion means that the following result holds: Theorem 12. The set of all Rota–Baxter monomials forms a basis in the fr ee noncommutative Rota– Baxter algebra generate d by the set B. It would be interesting to compare this basis with the basis fr o m [17]. 4.3.2. Bar homology. In this section, we compute t he bar homology for both operads RB a nd ncRB . Proposition 4.2. For each of the operads RB and ncRB, the r esolution for its monomial version from Sec.4.2 is minimal, that is the di ﬀ er ential induced on the space of ge nerators is z er o. FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 25 Pro of. In the case of the operad RB , the overlaps obtained fr om the leading monomials ( a 1 · a 2 ) · a 3 , ( a 1 · a 3 ) · a 2 , and P ( P ( a 1 ) · a 2 ) ar e , in arity n , (( . . . (( a 1 · a i 2 ) · a i 3 ) · . . . ) · a i n and P ( P ( P ( . . . P ( P ( P ( a 1 ) · a i 2 ) · a i 3 ) . . . ) · a i n − 1 ) · a i n ) , for all permutations i 2 , i 3 . . . , i n of inte gers 2 , 3 , . . . , n . It is easy to see that for each o f them there exists o n ly one indecomposable covering by rela tions, so the d i ﬀ erential maps such a generator to the space of decomposable elements, and t he statement f ollows. Similarl y , i n t he case of the op erad ncRB , t he only overlaps obt ained from the leading mon o mials ( a 1 · a 2 ) · a 3 and P ( P ( a 1 ) · a 2 ) ar e , in arity n , (( . . . (( a 1 · a 2 ) · a 3 ) . . . ) · a n − 1 ) · a n and P ( P ( P ( . . . P ( P ( P ( a 1 ) · a 2 ) · a 3 ) . . . ) · a n − 1 ) · a n ) . It is easy to see that for each of them ther e exists only one inde composable covering by r elations, so t he di ﬀ erential maps such a generator t o the sp ace of decomposable elements, and the statement fol lows.  Theorem 13 . We have • dim H l ( B ( RB ))( k ) =      ( k − 1)! , l = k ≥ 1 , ( k − 1)! , l = k + 1 ≥ 2 . • dim H l ( B ( ncRB ))( k ) =      1 , l = k ≥ 1 , 1 , l = k + 1 ≥ 2 . Pro of. In both cases, the subspace of generato rs of the free resolution s plits into tw o parts : the part obtained as overlaps of the leading terms o f t he associativity r elations, and the part obtained as ove r laps of the leading term o f the Rota–Baxter relation with itself. In arity k , the former ar e all of homological d egree k − 1, while the latter — of homological degree k . This means that when we compute the homolog y of the di ﬀ er ential o f our resolution r estricted to the space of generators, the only cancellations can happen if s ome of the elements r esolving t he associativity relation appear as di ﬀ erential s of some elements resolving t he Rota–Baxter r elation. However , it is clear that all the terms appearing in the formulas for the latter ar e of degree at least 1 in P , so no cancellations ar e impossible.  In add ition to the bar homology computation, one can ask for explicit formulas for di ﬀ erential s in the free r esolutions. It is not di ﬃ cult to write down formulas for small arities (see t he example below), but in general compact formulas are ye t to be found . W e expe ct that they incorp o rate the Spitzer ’s identity and its noncommutative analogue [15]. However , the following statement is easy to check. Corollary 1. • The minimal model RB ∞ for the opera d RB is a quasi-fr ee op- erad whose space of generators has a ( k − 1) ! -dimensional space of genera tors of homological degr ee ( k − 2) in each arity k ≥ 2 , and a ( k − 1)! -dimensi onal space of generators of homologi cal degre e k − 1 in each ari ty k ≥ 1 . • The min imal model n cRB ∞ for the opera d ncRB is a quasi-fr ee operad generate d by operatio ns µ k ( k ≥ 2) of homologi cal degr ee k − 2 , and ν l ( l ≥ 1) of homo logical degr ee l − 1 . 26 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN Let us conclude this section with f ormulas for low arities di ﬀ erential s in ncRB ∞ , to give the reader a ﬂ avour of w h at so r t of formulas to exp ect. Example 4. W e have d µ 2 = 0 , d µ 3 = µ 2 ( µ 2 (- , -) , -) − µ 2 (- , µ 2 (- , -)) , d ν 1 = 0 , d ν 2 = P ( µ 2 ( P (-) , -)) + P ( µ 2 (- , P (-))) − µ 2 ( P (-) , P (-)) + λ P ( µ 2 (- , -)) , d ν 3 = ν 2 ( µ 2 ( P (-) , -) , -) − ν 2 (- , µ 2 ( P (-) , -)) + ν 2 ( µ 2 (- , P (-)) , -) − ν 2 (- , µ 2 (- , P (-))) + + P ( µ 2 ( ν 2 (- , -) , -) + µ 2 (- , ν 2 (- , -))) + µ 2 ( ν 2 (- , -) , P (-)) − µ 2 ( P (-) , ν 2 (- , -)) + + µ 3 ( P (-) , P (-) , P (-)) + µ 3 ( P (-) , P (-) , -) − P ( µ 3 ( P (-) , - , P (-)) + µ 3 (- , P (-) , P (-))) + + λ  ν 2 ( µ 2 (- , -) , -) − ν 2 (- , µ 2 (- , -)) − − P ( µ 3 ( P (-) , - , -) + µ 3 (- , P (-) , -) + µ 3 (- , - , P (-)))  − − λ 2 P ( µ 3 (- , - , -)) 4.4. The operad BV and h ypercommutative algebras. The main goal o f this section is to explain how our results can be used to study the op - erad BV of Batalin–V ilkovisky algebras. The k ey result below (Theorem 16) is among those announced by Drumm ond-Cole and V all ette earl ier [1 4, 5 1] (see also [13]); our proofs ar e based on me t hods entirely di ﬀ erent fr om theirs. 4.4.1. The operad BV and its Gr ¨ obner basis. B atalin–V ilkovisky algebras show up in various ques t ions of mathematical p hysics. In [21], a coﬁbrant resolu- tion for t he corresponding op erad was presented . Ho wever , t hat resolution is a little bit mor e that minimal. In this s ection, we p resent a minimal res- olution for this operad in the shu ﬄ e categ o ry . The ope rad B V , as deﬁned in most sources, is an operad with quadratic–linear relations: the odd Lie bracket can be exp ressed in terms of the product and the unary operator . However , alternatively one can say that a BV -a lgebra is a dg-commutative algebra with a unary operator ∆ w h ich is a di ﬀ er ential ope r ato r of o rder at most 2. This deﬁnition of a BV -algebra is certainly not new , see, e. g., [22]. W ith this presentation, the corr esponding operad becomes an operad with homog eneous relations (of de grees 2 and 3). Our choice of de grees and signs is taken from [ 21] wher e it i s e xplained how to translate betwe e n this convention and ot her popular d eﬁnitions of B V -algebras. Deﬁnition 6 (Batalin– V ilkovisky algebras with homogeneous relations) . A Batalin-V ilkov isky algebr a , or B V -algebra for short, is a d i ﬀ erential graded vector space ( A , d A ) endowed wi th - a symmetric bi nary product • of degree 0, - a unary operator ∆ of degree + 1, such that ( A , d A , ∆ ) is a bic omplex, d A is a derivation with respect t o the product, and such that - the pr oduct • is asso ciative, - the operator ∆ s atisﬁes ∆ 2 = 0, FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 27 - the operations sati sfy the cubi c identity (10) ∆ ( - • - • - ) = (( ∆ ( - • -) • - ) − ( ∆ ( - ) • - • - )) . (1 + (123) + ( 132)) , In what follows, it is very helpful to have in mind the computations of S ection 2.5: phenomena discovered the re for an associative algebra k h x , y | y 2 , x 2 y + x yx + yx 2 i ar e very similar to the phenomena we shall observe for the op erad BV . Informally , one should think of the generator y of the abovementioned algebra as o f an analogue of the Batalin–V ilkovisky operator ∆ , and of the generato r x as o f an analogue of the binary product - • -. Let us consider the o rdering of th e fr ee operad where we ﬁrst compare lexicographically the operations on the paths from the root to leaves, and then the p lanar permutations of leaves; we assume that ∆ > • . Proposition 4.3. The above r elations toget her with the degr ee 4 r elation (11) ( ∆ ( - • ∆ ( - • - )) − ∆ ( ∆ ( - ) • - • - )) . (1 + (123) + ( 132)) = 0 form a G r ¨ obner basis of r elations for the operad of BV -algebra s. Pro of. Here and below we use the language of operations , as opposed the language of tree monomials; our operations reﬂect the structure of the corresponding tree monomials in the free shu ﬄ e operad. Fo r each i , the ar gument a i of an operation corr esponds t o the leaf i of the corresponding tree mono mial. W ith respect to ou r ordering, the leading monomials of o ur original rela tions are ( a 1 • a 2 ) • a 3 , ( a 1 • a 3 ) • a 2 , ∆ 2 ( a 1 ), and ∆ ( a 1 • ( a 2 • a 3 )). The only small common multiple of ∆ 2 ( a 1 ) and ∆ ( a 1 • ( a 2 • a 3 )) gives a nontrivi al S-polynomial which, is pr e cisely the r e lation (11). The leading term of that rela tion is ∆ ( ∆ ( a 1 • a 2 ) • a 3 ). It is well know n th at dim B V ( n ) = 2 n n ! [22], s o to verify t hat o u r relations form a Gr ¨ obner basis, it is su ﬃ cient t o sho w that the restrictions imposed by these leading monomials are str ong enough, that is that t he number of arity n tree monomials that a re not divi sible by any of the se is equal to 2 n n !. Moreover it is su ﬃ cient to check that for n ≤ 4, since all S-polynomials of our r e lations wil l be e leme nts of arity at most 4. This can be easily checked by hand, or by a computer pr o g ram [12].  4.4.2. Bar homology of the operad BV . Le t us denote by G th e Gr ¨ o bner basis fr om the pr evious section. Proposition 4.4. For the monomial versio n of BV , the resol ution A G fr om Sec- tion 3.3 is minimal, that is the di ﬀ er ential induced on the space of generators is zer o. Pro of. Let us describe explicitly the sp ace of generato rs, that is po ssible indecomposable coverings of monomials by leading terms o f relations (all monomials below are chosen fr o m t he basis of the free shu ﬄ e operad, so the corr ect or d ering of s ubtrees is assumed). These ar e - all monomials ∆ k ( a 1 ), k ≥ 2 (covered by s everal copies of ∆ 2 ( a 1 )), - all “Lie m onomials” (12) λ = ( . . . (( a 1 • a k 2 ) • a k 3 ) • . . . ) • a k n 28 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN where ( k 2 , . . . , k n ) is a permutation of numbers 2, . . . , n (only t he leading terms ( a 1 • a 2 ) • a 3 and ( a 1 • a 3 ) • a 2 ar e u s ed in t h e covering), - all the m onomials (13) ∆ k ( ∆ ( λ 1 • ( λ 2 • λ 3 ))) where k ≥ 1, each λ i is a Lie monomial as described above (several copies of ∆ 2 , the leading term of degree 3, and se veral Lie monomials ar e used), - all monomials (14) ∆ k ( ∆ ( . . . ∆ ( ∆ ( λ 1 • λ 2 ) • λ 3 ) • . . . ) • λ n ) where k ≥ 0, n ≥ 3, and λ i ar e Lie monomials (several copies of all leading terms ar e use d, including at least o ne copy o f t h e de gree 4 leading term). This is a complete list of tr ee monomials T for which A T G is nonzer o in positive homological d egrees. It is easy to se e that for each of the m there exists only o n e indecompo sable covering by rela tions, that is only o ne generator of A G of shape T . Conseque ntly , the di ﬀ erential maps s u ch a generator to ( A G ) 2 + , so the di ﬀ erential induced on generators is iden t ically zero.  The resolution of the ope rad BV which one can de rive by our methods fr om this one is quite small (in particular , smaller than t h e one of [21]) but still not minimal . Howe ver , we now have enough information to compute the bar homology of the operad BV . Theorem 14 . The basi s of H ( B ( BV )) is formed by monomia ls ∆ k ( a 1 ) , k ≥ 1 , and all monomial s of the fo rm (15) ∆ ( . . . ∆ ( ∆ ( | {z } n − 1 times λ 1 • λ 2 ) • . . . ) • ( λ n • a j )) , n ≥ 1 fr om the monomial r esolution discussed above. Here all λ i ar e Lie monomials . Pro of. First of a ll, let us notice that since Ω ( B ( BV )), a free operad ge nerated by B ( BV )[ − 1], p rovi des a resolution for B V , the sp ace H ( B ( B V ))[ − 1] is the space of g enerators of the minimal free r e solution, and we shall st udy the resolution p rovi ded by our meth o ds. Similarl y to how things work for the operad f As in Section 3.6, it is easy to check that the e lement ∆ ( ∆ ( a 1 • a 2 ) • a 3 ) that corresponds to the leading term of the only contributing S-polynomial will be killed by the di ﬀ er en t ial of the element ∆ 2 ( a 1 • ( a 2 • a 3 )) (cover ed by two leading terms ∆ 2 ( a 1 ) and ∆ ( a 1 • ( a 2 • a 3 ))) in the de formed resolution. This obse r vation goe s much further , namely we ha ve for k ≥ 1 (16) D ( ∆ k ( ∆ ( . . . ∆ ( ∆ ( λ 1 • λ 2 ) • λ 3 ) • . . . ) • ( λ n • a j )) = = ∆ k − 1 (( ∆ ( . . . ∆ ( ∆ ( λ 1 • λ 2 ) • λ 3 ) • . . . ) • λ n ) • a j ) + lowe r terms FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 29 in th e sens e of the partial ordering we discussed earlier). So, if we retain only leading t erms of the di ﬀ erential, the resulting homolog y classes are repr e sented by all the monomials of a rity m (17) ∆ ( . . . ∆ ( ∆ ( λ 1 • λ 2 ) • . . . ) • λ n ) with λ n having at least two leaves. They all have the same homological degree m − 2 in the resolution, and so there are no further cancellations.  So far we hav e not been able to describe a minimal r esolution of the operad BV by relatively compact c losed formulas, even though in principle our pr oof, once pr ocessed by a version o f Br own’s machinery [7, 9], would clearly yield such a resolution ( in the shu ﬄ e catego ry). 4.4.3. Operad s H y com and Grav . The operads Hycom and its Koszul dual Grav we re originally deﬁn e d in terms of mod u li spaces of curves of gen u s 0 with marked points M 0 , n + 1 [23, 24]. Howe ver , we are interested in the algebraic aspects of the story , and we us e the following descriptions of these operads as quadratic algebraic operads [23]. An al gebra over H ycom is a cha in complex A with a sequence of graded sy mmetric pr od ucts ( x 1 , . . . , x n ) : A ⊗ n → A of d egree 2( n − 2), which satisfy t he following relations (her e a , b , c , x 1 , . . . , x n , n ≥ 0, ar e eleme nts of A ): (18) X S 1 ∐ S 2 = { 1 ,..., n } ± (( a , b , x S 1 ) , c , x S 2 ) = X S 1 ∐ S 2 = { 1 ,..., n } ± ( a , ( b , c , x S 1 ) , x S 2 ) . Here, for a ﬁnite set S = { s 1 , . . . , s k } , x S denote s for x s 1 , . . . , x s k , and ± means the Koszul s ign rule. An algebra over Grav is a chain complex with graded antisymmetric products [ x 1 , . . . , x n ] : A ⊗ n → A of degree 2 − n , which satisfy the r elations: (19) X 1 ≤ i < j ≤ k ± [[ a i , a j ] , a 1 , . . . , b a i , . . . , b a j , . . . , a k , b 1 , . . . , b ℓ ] = =      [[ a 1 , . . . , a k ] , b 1 , . . . , b l ] , l > 0 , 0 , l = 0 , for all k > 2, l ≥ 0, and a 1 , . . . , a k , b 1 , . . . , b l ∈ A . For example, setting k = 3 and l = 0, we obtain the Jacobi relation for [ a , b ]. (Similarly , the ﬁrst relation for Hycom is the associativity of the p roduct ( a , b ).) Let us deﬁne an admissible or dering of the free operad whose quotient is Grav as follows. W e introduce an add itional weight grading, putting the weight o f the corolla c orresponding to the bina ry bracket equal t o 0, all other weights of cor ollas e qual to 1, and extending it to compositions by additivity of weight. T o compare t wo mono mials, we ﬁrst compare their weights, then the root cor o llas, and then path seque nces [11] according to the reverse p ath-lexicographic o rder . For both of the latter st eps, we ne ed an ordering of corollas; we assume that corollas of larger arity are smaller . Th e n for the r elation ( k , l ) in (19) (written in the shu ﬄ e notation with variabl es 30 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN in the pr oper order), its leading monomial is equal to the monomial in the right hand side for l > 0, and t o t h e mono mial [ a 1 , . . . , a n − 2 , [ a n − 1 , a n ]] for l = 0. The following theorem, togeth e r with the PBW cri terion, implies t h at the operads Grav and Hycom are Koszul, the fact ﬁrst pr o ved by Get zler [22]. Theorem 15. F or our ordering, the rel ations of Grav form a Gr ¨ obner basis of r elations. Pro of. The tr ee mono mials that are not divisible by leading terms of r elations ar e preci sely (20) [ λ 1 , λ 2 , . . . , λ n − 1 , a j ] , where all λ i , 1 ≤ i ≤ ( n − 1) are Lie monomials as in (12) (but made from brackets, not pr o ducts). Lemma 4.1. T he gr aded characte r of the space of such elements of ari ty n is (21) (2 + t − 1 )(3 + t − 1 ) . . . ( n − 1 + t − 1 ) . Pro of. T o compute the number of basis elements w here the top d egree coroll a is of arity k + 1 (or , equivalently , degree 1 − k ), k ≥ 1, let us no- tice that this number is equal to the number of basis elements [ λ 1 , λ 2 , . . . , λ k ] where the arity of λ k is at least 2 (a s imple bijection: join λ n − 1 and a j into [ λ n − 1 , a j ]). T he latter nu mber is equal to (22) X m 1 + ... + m k = n , m i ≥ 1 , m k ≥ 2 ( m 1 − 1)!( m 2 − 1)! . . . ( m k − 1)! m 1 m 2 · . . . · m k ( m 1 + m 2 + . . . + m k )( m 2 + . . . + m k ) · . . . · m k m 1 + . . . + m k m 1 , m 2 , . . . , m k ! where each factor ( m i − 1)! count s the number of Lie monomials of arity m i , and the r e maining factor is t he number of shu ﬄ e p ermutations of the ty pe ( m 1 , . . . , m k ) ([12]). This can be r ew ritten in the form X m 1 + ... + m k = n , m i ≥ 1 , m k ≥ 2 ( m 1 + . . . + m k − 1)! ( m 2 + . . . + m k )( m 3 + . . . + m k ) · . . . · m k and if we introduce new v ariabl es p i = m i + . . . + m k , it ta kes the form X 2 ≤ p k − 1 <...< p 1 ≤ n − 1 ( n − 1)! p 2 . . . p k , which clearl y is the coe ﬃ cient of t 1 − k in the pr od uct (23) ( n − 1)!  1 + 1 2 t   1 + 1 3 t  · . . . · 1 + 1 ( n − 1) t ! = =  2 + t − 1   3 + t − 1  . . .  n − 1 + t − 1  .  FREE RE S OLUTIONS VIA GR ¨ OBNER BA SES 31 Since the graded character of Grav is g iven by the same formula [23], we inde ed se e that the leading t erms of deﬁning relations give an up p er bound on dimensions ho mogeneou s components of Grav t hat coincides with the actual dimensions, so t h e re is no room for furt her Gr ¨ o bner basis elements.  4.4.4. BV ∞ and hyp er commutative algebras . Theorem 16 . On the level of collectio ns of graded vector sp aces, we hav e (24) H ( B ( BV ))[ − 1] ≃ Grav ∗ ⊗ End k [1] ⊕ δ k [ δ ] , wher e Grav ∗ is the cooperad dual to Grav , and δ k [ δ ] is a cofree coalgebra generated by an el ement δ o f degr ee 2 . Pro of. As above, ins t ead of looking at the bar complex, we shall study the basis of the sp ace of ge nerators of the minimal resolution obtained in Theorem 14. In ari ty 1, the element δ k (of degree 2 k ) corr espond s to ∆ k ( a 1 )[ − 1] (of degr ee k + ( k − 1) + 1 = 2 k , the ﬁrst summand coming from the fact that ∆ is of degree 1, the second from the fact that ∆ k is an overlap of k − 1 relations, and the la st one is the degree shift). The case of elements of internal degree 0 (which in both cases ar e Lie monomials) is also obvious; a L ie mono mial o f a rity n in the space of g e nerators of the f ree resolution is of homo log ical d egree n − 2 + 1 = n − 1, the se cond summand coming from the degree shift, and this matches the degree shift given by End k [1] ( n ). For elements of internal d e gree k − 1, let us extract fr om a typical monomial ∆ ( . . . ∆ ( ∆ ( | {z } k − 1 times λ 1 • λ 2 ) • . . . ) • ( λ k • a j )) , of this d egree and o f arity n the Lie monomials λ 1 , λ 2 , . . . , λ k − 1 , λ k , a j , and assign to this th e elemen t of Grav ∗ ⊗ End k [1] corresponding to the dual element of the mono mial [ λ 1 , λ 2 , . . . , λ k − 1 , λ k , a j ] in the g ravity operad. This establishes a de gree-pr eserving bi jection, because if arities of λ 1 , . . . , λ k ar e n 1 , . . . , n k , the t otal (internal plus homolog ical) degree o f the former e leme nt is ( k − 1) + ( k − 2 + 1 + ( n 1 − 1) + . . . + ( n k − 1)) + 1 = n + k − 2 (where we add up the ∆ de gree, the overlap degree, and the degree shift by 1), and the total degree o f the latter one is ( k − 1) + ( n − 1) = n + k − 2.  W e conclude this p aragraph wi th a d iscussion on how our results ma tch those of B arannikov and Kont s evich ([2], see also [35, 37]) who pr oved in a rather indirect way that for a dg BV -al gebra that satisﬁes the “ ∂ − ∂ -lemma”, there exists a Hycom-algebra structure on its cohomology . Their result hints that o ur isomorphism ( 24) exist s not j ust on the level of graded vector spaces, but rather has some de ep operadic s tructure behind it. For precise statements and m ore de tails we r efer the r eader to [1 3]. From Theorem 15, it follows that the ope rads Grav and Hy com are Koszul, so Ω (Grav ∗ ⊗ End k [1] ) is a minimal model for Hycom. More pre- cisely , we shall show that the di ﬀ erential of BV ∞ on generators coming fr om Grav ∗ deforms the di ﬀ er ential of H ycom ∞ in the foll owing sense. Let D and d deno te the di ﬀ erential s of BV ∞ and Hycom ∞ respectively . W e can decompose D = D 2 + D 3 + . . . (respectively d = d 2 + d 3 + . . . ) accor d ing to 32 VLADIMIR DOTSENKO AND ANTON KHOROSHKIN the ∞ -cooperad structure it provides on the space of gene r ato rs. Also, let m ∗ denote the obvious coalgebra structure on δ k [ δ ]. W e sh all call a tr ee monomial in B V ∞ mixed , if it contains both cor ollas fr om Grav ∗ ⊗ End k [1] and fr om ( δ k [ δ ]). Then we have (25) D 2 = d 2 + m ∗ , while for k ≥ 3 the co-operation D k is zero on the gen e rators δ k [ δ ], and maps generators from Grav ∗ into linear combinations of mixed tree monomials. Indeed , the result of Barannikov and Konts evich [2] ess entially implies that there exists a mapping fr om Hycom to the homotopy quotient BV / ∆ . In fact, it is an isomorphism, which can be proved in several d i ﬀ erent ways, bot h us ing Gr ¨ obner bases and geomet rically; see [38 ] for a short geometric argument proving t hat. This means that the following maps exist (the vertical arrows ar e quasiisomorphisms betwee n the operads and their mini mal models): BV ∞     π # # # # G G G G G G G G G Hycom ∞     BV BV / ∆ Hycom ] o o Lifting π : BV ∞ → B V / ∆ ≃ Hycom to the minimal mo d el Hycom ∞ of Hycom, we obtai n the commutative d iagram BV ∞ ψ / / / /     π # # # # G G G G G G G G G Hycom ∞     BV BV / ∆ Hycom ] o o so ther e exists a map of dg-operads (and not just graded vector spaces, as it follows from ou r previous computations) between BV ∞ and Hy com ∞ . Commutativity of our diagram t o gether with simple degree considerations yields what w e need. 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S chool of M a thema tics , T rinity C ollege , D ublin 2, I reland E-mail address : vdots@maths.tcd.ie D ep artement M a tema tik , ETH, R ¨ amistrasse 10 1, 8092 Z uri ch , S witz erland and ITEP , B olsha y a C heremushkinska y a 2 5, 117259, M oscow , R ussia E-mail address : anton.khoroshkin@m ath.ethz.ch

Free resolutions via Gr"obner bases

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