Andre-Quillen cohomology of algebras over an operad

ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS OVER AN OPERAD JOAN MILL ` ES Abstract. W e study the Andr´ e-Quillen cohomology with coeﬃcients of an algebra ov er an operad. Using resolutions of algebras coming from the Koszul dua lity theory , we mak e this cohomology theory explicit and we giv e a Lie theoretic i n terpretation. F or w hi c h op erads i s the associated Andr´ e-Quillen cohomology equal to an Ext-f unctor ? W e give several criteria, based on the cotang ent complex, to charact erize this prop er t y . W e apply i t to homotop y algebras, which gives a new homotop y stable prop erty for algebras ov er coﬁbran t op erads. Keywor ds: Andr´ e-Quillen cohomology; Ext functor; op erad; homotopy algebra. Introduction Ho chsc hild [Ho c45] intro duced a chain co mplex which deﬁnes a cohomolog y theo ry fo r ass o cia- tive algebr as. In 19 4 8, Chev a lle y a nd Eilenberg gav e a deﬁnition of a cohomolo g y theory for Lie algebras . Both cohomo logy theories can b e wr itten as classic al der ived functors (Ext-functors). Later, Quillen [Q ui7 0] deﬁned a co homology theo r y asso ciated to comm utative algebra s with the use of mo del categor y str uctur es. Andr´ e gav e similar deﬁnitions o nly with simplicial metho ds [And74]. This cohomolo gy theory is not equal to an Ext-functor over the env eloping a lgebra in general. Using conceptual mo del category arg uments, we recall the deﬁnition of the Andr ´ e-Quil len c o- homolo gy (for algebr a s over an op er ad) , in the diﬀerential graded setting, from Hinich [Hin97] a nd Go erss a nd Hopkins [GH00 ]. Beca use we work in the diﬀerential gra ded setting, we use known functorial r e solutions of algebr a s to make c hain complexes which compute Andr´ e-Q uillen co homol- ogy explicit. The ﬁrst idea o f this pape r is to use K o szul duality theory of o pe r ads to provide such functorial reso lutio ns. W e can also use the simplicial bar construction, which prov es that cotriple cohomolog y is eq ual to Andr´ e-Quillen co homology . The Andr´ e-Quillen co homology is represented by a n ob ject, called the c otangent c omplex which ther efore plays a crucial ro le in this theor y . The notion of twisting morphism , also ca lle d twisting co chain, coming from alg ebraic top olog y , has bee n extended to (co )op erads a nd to (co)a lg ebras ov er a (co )op e rad b y Getzler and Jones [GJ94]. W e make the diﬀerential o n the c o tangent complex ex plicit using these tw o notions of twisting morphisms all tog ether. When the category of algebr as is mo dele d by a binary K oszul o p e rad, we give a Lie theoretic interpretation of the pr evious construction. In the r eview of [F ra0 1], P irashvili asked the question of characterizing op era ds suc h that the asso ciated Andr ´ e-Quillen co homology of a lgebras is an Ext-functor. This pap er provides a cr iterion to answer that question. When the o p er ad is K oszul, w e describ e the co tangent co mplex a nd the Andr´ e-Q uillen co- homology for the alge bras ov er this op erad using its Kos zul complex. W e recover the classical cohomolog y theories, with their underly ing chain complex e s, like Andr´ e-Quillen cohomolo gy for commutativ e a lgebras, Ho chsc hild cohomo logy for asso c ia tive a lgebras and Chev alley-Eilenberg cohomolog y for Lie algebra s. W e also recov er co homology theor ies which were de ﬁned recently like cohomolog y for Poisson a lgebras [F re06], cohomo logy fo r Le ibniz algebras [LP93], coho mology for pr e -Lie algebra s [Dzh99], cohomolog y for diasso c ia tive a lgebras [F ra0 1] and co homology for Joan Mill` es, Laboratoire J. A. Dieudonn´ e, Universit ´ e de Nice Sophia-Antipolis , Parc V alrose, 06108 Nice Cedex 02, F rance E-mail : joan.milles@mat h.unice.fr URL : http://math. unice.fr/ ∼ jmilles . 1 2 JOAN MILL ` ES Zinbiel algebr as [Bal98]. More generally , Balav oine introduced a chain complex when the op er ad is binary a nd quadratic [Bal9 8]. W e s how that this chain complex deﬁnes Andr ´ e-Quillen cohomol- ogy when the o per ad is Koszul. W e make the new example of Perm algebra s ex plicit. F or any op erad P , we ca n deﬁne a relax version up to homo topy o f the no tion of P -algebra as follows: we call homo topy P -algebra any algebr a ov er a co ﬁbr ant replacement of P (cf. [BV73]). Using the op era dic cobar construction, w e make the cotangent co mplex and the co homology theories for homotopy a lg ebras explicit. F or instance, we recover the ca se of homotopy a sso ciative algebr as [Mar92] a nd the case of homoto p y Lie algebras [HS93]. F or any algebr a A , w e prov e that its Andr ´ e-Quillen cohomology is an additive der ived functor, an E xt-functor, ov er its env eloping algebr a if and only if its cotange nt complex is qua s i-isomorphic to its mo dule of K¨ ahler diﬀerential forms Ω P ( A ). W e deﬁne a functorial c otangent c omplex and a functorial mo dule of K¨ ah ler diﬀer ential forms which dep end only on the o per ad and we reduce the study of the quas i-isomorphisms b etw een the cota ngent co mplex and the mo dule of K ¨ ahler diﬀerential forms for any algebra to the study of the quasi- isomorphisms b etw een the cotang e nt complex and the mo dule of K¨ ahler diﬀerential forms for any c hain co mplex , with trivial a lgebra structure (when P is an PBW-op er ad , tha t is the P -algebras satisfy an analogue of the Poincar´ e- Birkhoﬀ-Witt theo rem and the P -K ¨ ahler diﬀerentials to o). This a llows us to give a uniform treatment for an y alg ebra ov er an oper ad. Assuming that P is a PBW-op erad, w e pr ove that the functorial cota ngent complex is quasi- isomorphic to the functorial mo dule of K ¨ ahler diﬀerential forms (we say so metimes conce ntrated in degree 0 or acy clic), if and o nly if the Andr´ e -Quillen cohomolog y theory for a n y algebr a over this oper ad is an Ext-functor over its env eloping algebra, so this functorial cotang ent complex carries the obstr uctions for the Andr´ e-Quillen cohomo lo gy to be an Ext-functor. F or instance, we prov e that the functorial cotangent complex is acy clic for the op era ds of asso ciative algebra s and Lie alg e bras. In or der to control the map b etw een the functorial cotangent complex and the functorial module of K¨ ahler diﬀeren tial forms, we lo ok at its kernel. This deﬁnes a new chain complex whose homology gro ups can als o b e interpreted as obstructions for the Andr´ e-Q uillen coho mology theory to be an Ext-functor. In this wa y , we give a new, but more conceptua l pro of that the cotangent complex fo r commutativ e algebr as is not alwa ys a cyclic. E q uiv alen tly , it means that there exist commutativ e algebra s such that their Andr´ e-Quillen coho mology is not an E xt-functor ov er their env eloping algebra . With the s ame metho d, we show the sa me result for Perm a lg ebras. W e can summarize all these prop erties in the following theor em (Section 4 and 5). Theorem A. L e t P b e an op er ad. The fol lowing t wo pr op erties ar e e quivalent: ( P 0 ) the Andr´ e -Qu il len c ohomolo gy is an Ext -functor over the enveloping algebr a A ⊗ P K for any P -algebr a A ; ( P 1 ) the c otangent c omplex is qu asi-isomorphi c to the mo dule of K¨ ah ler diﬀer ential forms for any P -algebr a A . They mor e over imply the fol lowing e quivalent pr op erties: ( P 2 ) the functorial c otangent c omplex L P is quasi-isomorphi c to t he functorial mo dule of K¨ a hler diﬀer entia l forms Ω P ; ( P 3 ) the mo dule of obstructions O P is acyclic. When P b e a PBW-op er ad, the pr evious implic ation is an e quivalenc e, that is ( P 0 ) ⇔ ( P 1 ) ⇔ ( P 2 ) ⇔ ( P 3 ) . In the case o f homotopy algebra s , w e prov e that the obstructions for the cohomolog y to b e an Ext-functor v anish. Mo reov er, any P -algebra is also a homo to py P -algebra . Thus we can compute its Andr´ e-Quillen cohomo lo gy in tw o diﬀerent wa ys. W e show that the t wo coincide. Hence we get the following theo rem. Theorem B. L et A b e a P -algebr a and let M b e an A -mo dule over the Koszu l op er ad P . We have H • P ( A, M ) ∼ = Ext • A ⊗ P ∞ K (Ω P ∞ ( A ) , M ) . ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 3 Therefore, even if the Andr´ e- Q uillen co homology of commutativ e and Perm algebras cannot a l- wa ys b e written as an Ext-functor ov er the env eloping algebra A ⊗ P K , it is alwa ys an Ext-funtor ov er the env eloping alg ebra A ⊗ P ∞ K . The pap er b egins with ﬁr st deﬁnitions and prop erties of diﬀerential graded (co)op er ads, (co)al- gebras, mo dules a nd free mo dules ov er an alg ebra (over an o pe rad). In Section 1, we recall the deﬁnition o f the Andr ´ e-Quillen co homology theory for dg algebras over a dg op era d, from Hinic h and Go erss- Hopkins. W e intro duce functor ial reso lutions fo r algebra s ov er an op er a d, which allow us to make the cotang en t complex and the cohomolo gy theories explicit. Then, in Sectio n 2, we give a Lie interpretation of the chain complex deﬁning the Andr´ e-Q uillen cohomology . Using the notion of twisting mo r phism o n the level of (co)algebr a ov er a (co)op erad, we make the diﬀerential on the cotang ent co mplex explicit (Theorem 2.4.2). Section 3 is devoted to applicatio ns a nd examples. In Section 4, w e prove that the cotangent complex is qua si-isomor phic to the module of K¨ ahler diﬀerential forms for any algebr a if and o nly if the Andr´ e-Quillen cohomo lo gy theor y is a n Ext-functor ov er the env eloping alg ebra for any algebr a. Mor eov er, we study the Andr´ e-Q uillen cohomolog y theory for op erads. In Section 5, we introduce the functorial co ta ngent co mplex and the functoria l mo dule of K¨ ahler diﬀerential forms a nd we ﬁnish to prove Theorem A. In Section 6, we study the Andr´ e-Quillen cohomo lo gy for ho motopy alg ebras and we prove Theorem B. Contents Int ro duction 1 Notation and preliminary 3 1. Andr´ e-Quillen cohomolo gy of alg ebras ov er an oper a d 8 2. Lie theo retic descr iption 13 3. Applications and new ex amples of cohomology theor ie s 18 4. The co tangent complex and the module of K¨ ahler diﬀeren tial forms 22 5. The functor ial cotang ent complex 24 6. Is Andr´ e - Quillen cohomolog y a n Ext-functor ? 30 Ac knowledgment s 32 References 32 Not a tion and preliminar y W e recall the classic al notation for S -mo dule, comp o sition pro duct, (co)op erad, (co)alg ebra ov er a (co)op era d and mo dule ov er a n a lg ebra over a n oper ad. W e refer to [GK9 4] a nd [GJ 94] for a complete exp ositio n and [F re04] for a more modern treatment. W e also refer to the b o oks [L V] and [MSS02]. In the whole pap er , we w ork ov er a ﬁeld K o f characteristic 0 . In the sequel, the gro und categor y is the categ ory of g r aded mo dules, o r g-mo dules . F or a mor phism f : O 1 → O 2 betw een diﬀere ntial graded mo dules, the nota tio n ∂ ( f ) s tands fo r the der iv ative d O 2 ◦ f − ( − 1) | f | f ◦ d O 1 . Here f is a map of gr aded mo dules a nd ∂ ( f ) = 0 if a nd only if f is a map of dg -mo dules. Moreov er, for an other morphism g : O ′ 1 → O ′ 2 , w e deﬁne a morphism f ⊗ g : O 1 ⊗ O ′ 1 → O ′ 2 ⊗ O ′ 2 using the Koszul- Quillen conv ention: ( f ⊗ g )( o 1 ⊗ o 2 ) := ( − 1) | g | | o 1 | f ( o 1 ) ⊗ g ( o 2 ), where | e | denotes the degr ee of the element e . W e denote by g M od K the categ ory who s e ob jects a re diﬀerential gr aded K -mo dules (and not only graded K -mo dules) a nd mor phisms are maps of gra de d mo dules. W e have to b e careful with this deﬁnition b eca us e it is not usual. How ever, we denote as usual b y dg M od K the category of diﬀerential graded K -mo dules. In this pap er, the mo dules ar e all diﬀeren tial graded, except e xplicitly stated. 0.1. Diﬀerential graded S - mo dules . A dg S -mo dule (or S -mo dule for short) M is a colle c tio n { M ( n ) } n ≥ 0 of dg mo dules ov er the s ymmetric group S n . A morphism of d g S -mo dules is a 4 JOAN M ILL ` ES collection of equiv ariant morphisms of chain complexes { f n : M ( n ) → N ( n ) } n ≥ 0 , with resp ect to the ac tio n of S n . W e deﬁne a monoidal product on the category of dg S -mo dules b y ( M ◦ N )( n ) := M k ≥ 0 M ( k ) ⊗ S k M i 1 + ··· + i k = n I n d S n S i 1 ×···× S i k ( N ( i 1 ) ⊗ · · · ⊗ N ( i k )) ! . The unit for the monoidal pro duct is I := (0 , K , 0 , . . . ). Let M , N a nd N ′ be dg S -mo dules . W e deﬁne the right line a r ana log M ◦ ( N , N ′ ) of the comp osition product by the following formula [ M ◦ ( N , N ′ )]( n ) := M k ≥ 0 M ( k ) ⊗ S k    M i 1 + ··· + i k = n k M j =1 I n d S n S i 1 ×···× S i k ( N ( i 1 ) ⊗ · · · ⊗ N ′ ( i j ) | {z } j th p osition ⊗ · · · ⊗ N ( i k ))    . Let f : M → M ′ and g : N → N ′ be morphisms of dg S -mo dules. W e denote by ◦ ′ the inﬁn itesimal c omp osite of morphisms f ◦ ′ g : M ◦ N → M ′ ◦ ( N , N ′ ) deﬁned by k X j =1 f ⊗ ( id N ⊗ · · · ⊗ g |{z} j th p osition ⊗ · · · ⊗ id N ) . Let ( M , d M ) and ( N , d N ) b e t wo dg S -mo dules. W e deﬁne a grading on M ◦ N by ( M ◦ N ) g ( n ) := M k ≥ 0 e + g 1 + ··· + g k = g M e ( k ) ⊗ S k M i 1 + ··· + i k = n I n d S n S i 1 ×···× S i k ( N g 1 ( i 1 ) ⊗ · · · ⊗ N g k ( i k )) ! . The diﬀere ntial on M ◦ N is given by d M ◦ N := d M ◦ id N + id M ◦ ′ d N . The diﬀere ntial on M ◦ ( N , N ′ ) is given by d M ◦ ( N , N ′ ) := d M ◦ ( id N , i d N ′ ) + id M ◦ ′ ( d N , id N ′ ) + id M ◦ ( id N , d N ′ ) . Moreov er, for any dg S -mo dules M , N , we deno te by M ◦ (1) N the dg S -mo dule M ◦ ( I , N ). When f : M → M ′ and g : N → N ′ , the map f ◦ ( id I , g ) : M ◦ (1) N → M ′ ◦ (1) N ′ is denoted by f ◦ (1) g . 0.2. (Co)op e rad. An op er ad is a mono id in the monoida l catego ry of dg S -mo dules with resp ect to the mo no idal pr o duct ◦ . A morphism of op er a ds is a morphism of dg S -mo dules commuting with the o pe r ads structur e. The notion of c o op er a d is the dual version, i.e. a comono id in the category o f dg S -mo dules. How ever, we use the inv aria nts for the diagonal actions in the deﬁnition of the monoidal pro duct instead of the coinv a riants, that is, M k ≥ 0   M ( k ) ⊗ M i 1 + ··· + i k = n ( N ( i 1 ) ⊗ · · · ⊗ N ( i k )) ⊗ K [ S n ] ! S i 1 ×···× S i k   S k . Since we work ov er a ﬁeld of character istic 0, the inv ar iants ar e in one-to- one corresp o ndence with the coinv ar iants a nd both deﬁnitions a r e e quiv alen t. The deﬁnition with the inv aria nt s a llows to deﬁne pr op erly the s igns. The unit of an op erad P is denoted b y ι P : I → P and the c ounit of a co op era d C is deno ted by η C : C → I . Mor eov er when ( P , γ ) is a n op erad, w e deﬁne the p artial pr o duct γ p by P ◦ (1) P ֌ P ◦ P γ − → P and when ( C , ∆) is a co op erad, we deﬁne the p a rtial c opr o duct ∆ p by C ∆ − → C ◦ C ։ C ◦ (1) C . ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 5 Example. Let V be a dg K - mo dule. The dg S -mo dule E nd ( V ) := { Hom( V ⊗ n , V ) } n ≥ 0 , endow ed with the comp o sition of maps, is an op era d. 0.3. Mo dule o v er an op e rad and relativ e com p ositi on pro duct. A right P -mo dule ( L , ρ ) is a n dg S -mo dule endow ed with a ma p ρ : L ◦ P → L compatible with the pr o duct and the unit of the op erad P . W e deﬁne simila rly the no tion o f left P -mo dule . W e deﬁne the r elative c omp osition pr o duct L ◦ P R b etw een a right P -mo dule ( L , ρ ) and a left P -mo dule ( R , λ ) by the co equa lizer diagra m L ◦ P ◦ R ρ ◦ id R / / id L ◦ λ / / L ◦ R / / / / L ◦ P R . 0.4. Algebra o v er an op erad. Let P b e an o per ad. An algebr a over the op er ad P , or a P - algebr a , is a dg K -mo dule V endow ed with a morphism of o pe rads P → E nd ( V ). Equiv alen tly , a P -alge br a structur e is given by a ma p γ V : P ( V ) → V whic h is compatible with the co mp os ition pro duct and the unit y , wher e P ( V ) := P ◦ ( V , 0 , 0 , · · · ) = M n ≥ 0 P ( n ) ⊗ S n V ⊗ n . 0.5. Coalgebra ov er a co op erad. Dually , let C b e a co op era d. A c o algebr a over t he c o op er ad C , or a C -c o algebr a , is a dg K -mo dule V endow ed with a map δ : V → C ( V ) = ⊕ n ≥ 0 ( C ( n ) ⊗ V ⊗ n ) S n which s atisﬁes compatibility pr op erties. The no tation ( − ) S n stands for the spa ce of inv aria nt elements. 0.6. Mo dule o v er a P -alge bra. Let P b e a dg S -mo dule and let A b e a dg vector spa ce. F or a dg vector space M , we deﬁne the vector s pace P ( A, M ) by the formula P ( A, M ) := P ◦ ( A, M ) = M n P ( n ) ⊗ S n ( n ⊕ j =1 A ⊗ · · · ⊗ M |{z} j th p osition ⊗ · · · ⊗ A ) . Let ( P , γ ) b e a n op er ad and let ( A, γ A ) be a P -algebr a. An A -m o dule ( M , γ M , ι M ), or A -mo dule over P , is a vector space M endow ed w ith t wo maps γ M : P ( A, M ) → M and ι M : M → P ( A, M ) such that the following diag rams commute P ( P ( A ) , P ( A, M )) id P ( γ A , γ M ) / / ∼ =   P ( A, M ) γ M   M and ( P ◦ P )( A, M ) γ ( id A , id M ) / / P ( A, M ) γ M O O M ι M / / = $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ P ( A, M ) γ M   M . ( Associativ ity ) ( U ni tari ty ) The category of A -mo dules over the op era d P is denoted b y M P A . The ob jects in M P A are diﬀerential graded A -mo dules ov er P . Ho w ever, the morphisms in M P A are only maps o f graded A -mo dules over P . Examples. • The op era d P = A ss enco des as s o ciative alg ebras (not neces s arily with unit). Then the map γ n : A ss ( n ) ⊗ S n A ⊗ n → A stands for the a sso ciative pro duct of n elements, where A ss ( n ) = K [ S n ]. W e r epresent an element in A ss ( n ) by a coro lla with n entries. Then, 6 JOAN M ILL ` ES an element in A ss ( A, M ) can b e represented by a 1 a 2 · · · m · · · a n ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❈ ❈ ❈ ❈ ④ ④ ④ ④ ♠ ♠ ♠ ♠ ♠ ♠ ♠ . How ever, a 1 · · · a k m a k +1 · · · a n ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❆ ❆ ❆ ❆ ✉ ✉ ✉ ✉ ✉ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ = γ ◦ γ ◦ γ   a 1 · · · a k m a k +1 · · · a n ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ■ ■ ■ ■ ❍ ❍ q q q q q q q ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧   , then by s everal uses o f the asso ciativity diag ram of γ M , we get that an A -mo dule ov er the op erad A ss is g iven by tw o morphisms A ⊗ M → M and M ⊗ A → M . Finally , we get the cla s sical notion o f dg A -bimodule. • The op erad P = C om enco des classical a sso ciative and commutativ e a lgebras. W e have C om ( n ) = K and an element in C om ( A, M ) can b e represe n ted by a 1 a 2 · · · m · · · a n ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ❈ ❈ ❈ ❈ ④ ④ ④ ④ ♠ ♠ ♠ ♠ ♠ ♠ ♠ where the corolla is non-plana r. Like befor e, an A -mo dule structure over the o pe rad C om is given b y a morphism A ⊗ M → M . Hence, w e get the classical notion of dg A -mo dule. • The op erad P = L ie enco des the Lie alg ebras. In this case, an A -mo dule ov er the op erad L ie is a ctually a classical dg L ie mo dule or equiv alently a classical asso ciative mo dule ov er the universal env eloping algebr a of the Lie alg e br a A . Go erss and Hopkins deﬁned in [GH00] a free A -module. W e recall her e the deﬁnition. 0.6.1. Prop ositi o n (Pr op osition 1 . 10 o f [GH00 ]) . The for getful functor U : M P A → g M od K has a left adjoint, denote d by N 7→ A ⊗ P N . That is we have an isomorphism of dg mo d ules (Hom M P A ( A ⊗ P N , M ) , ∂ ) ∼ = (Hom g M od K ( N , U M ) , ∂ ) for al l N ∈ g M od K and M ∈ M P A . A descr iption of A ⊗ P N is g iven by the follo wing co equalizer diagram in dg M od K P ( P ( A ) , N ) c 0 / / c 1 / / P ( A, N ) / / / / A ⊗ P N , where the t wo ﬁrst maps are given by the op erad pr o duct P ( P ( A ) , N ) ֌ ( P ◦ P )( A, N ) γ ( id A , id N ) − − − − − − − → P ( A, N ) and the P - algebra structure P ( P ( A ) , N ) id P ( γ A , id N ) − − − − − − − − → P ( A, N ) . Remark. W e have to make note of the fact tha t the symbol ⊗ P is just a notation a nd not a classical tenso r pro duct (except in the case P = C om ), as we will s ee in the following examples. Examples. • When P = A ss , we ca n write a 1 · · · a l n a l +1 · · · a k ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❱ ❱ ❱ ❱ ❱ ❱ ❏ ❏ ❏ ♠ ♠ ♠ ♠ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ = c 0   a 1 · · · a l n a l +1 · · · a k ❖ ❖ ❖ ❖ ♣ ♣ ♣ ♣ ❙ ❙ ❙ ❙ ❙ ♦ ♦ ♦ ♦ ❱ ❱ ❱ ❱ ❱ ❱ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢   = c 1   a 1 · · · a l n a l +1 · · · a k ❖ ❖ ❖ ❖ ♣ ♣ ♣ ♣ ❙ ❙ ❙ ❙ ❙ ♦ ♦ ♦ ♦ ❱ ❱ ❱ ❱ ❱ ❱ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢   = a 1 · · · a l n a l +1 · · · a k ❯ ❯ ❯ ❯ ❯ ❣ ❣ ❣ ❣ ❣ ❣ , then we get A ⊗ A ss N ∼ = N ⊕ A N ❆ ❆ ❆ ③ ③ ③ ⊕ N A ❉ ❉ ❉ ⑥ ⑥ ⑥ ⊕ A N A ▲ ▲ ▲ r r r ∼ = N ⊕ A ⊗ N ⊕ N ⊗ A ⊕ A ⊗ N ⊗ A ∼ = ( K ⊕ A ) ⊗ N ⊗ ( K ⊕ A ) as mo dules o ver K . ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 7 • When P = C om , we get A ⊗ C om N ∼ = N ⊕ A ⊗ N ∼ = ( K ⊕ A ) ⊗ N as modules ov er K . • When P = L ie , we get A ⊗ L ie N ∼ = U e ( A ) ⊗ N as mo dules ov er K , where U e ( A ) is the env eloping algebra o f the Lie algebra A . These examples lead to the study of the A -mo dule A ⊗ P K which is the enveloping algebr a of the P -algebr a A (deﬁned in [HS93, GJ94]). It has a m ultiplication g iven by ( A ⊗ P K ) ⊗ ( A ⊗ P K ) ∼ = A ⊗ P ( A ⊗ P K ) → A ⊗ P K , where the a rrow is induced by the co mpo sition γ of the o p e r ad (indeed, the kernel of the map P ( A, P ( A, K )) ։ A ⊗ P ( A ⊗ P K ) is sent to 0 by the map P ( A, P ( A, K )) ֌ ( P ◦P )( A, K ) γ ( id A , id K ) − − − − − − − → P ( A, K ) ։ A ⊗ P K ). This multiplication is asso ciative a nd has a unit K → A ⊗ P K . 0.6.2. Prop ositi on (Prop osition 1 . 14 o f [GH00 ]) . The c ate gory M P A of A -mo dules over P is isomorphi c to the c ate gory of left unitary A ⊗ P K -mo dules g M od A ⊗ P K . Remark. W e work in a diﬀerential g raded setting. The diﬀerential on A ⊗ P K is induced by the diﬀerential on P ( A, K ). It is easy to see that the iso morphism is compatible with the graded diﬀerential fra mework. Given a map of P -algebra s B f − → A , there exists a forgetful functor f ∗ : M P A → M P B , whos e left adjoint g ives the notion of free A -mo dule on a B -mo dule. 0.6.3. Prop osi tion (Lemma 1 . 16 of [GH00]) . The for getful functor f ∗ : M P A → M P B has a left adjoint denote d by N 7→ f ! ( N ) := A ⊗ P B N . That is we have an isomorphism of dg mo d ules Hom M P A ( f ! ( N ) , M ) ∼ = Hom M P B ( N , f ∗ ( M )) for al l M ∈ M P A and N ∈ M P B . It is also p ossible to make explicit the A -mo dule A ⊗ P B N as the following co e qualizer A ⊗ P ( B ⊗ P N ) / / / / A ⊗ P N / / / / A ⊗ P B N . The module A ⊗ P ( B ⊗ P N ) is a quo tient of P ( A, P ( B , N )), then we deﬁne on P ( A, P ( B , N )) the co mp os ite P ( A, P ( B , N )) id P ( id A , id P ( f , id N )) − − − − − − − − − − − − − − → P ( A, P ( A, N )) ֌ ( P ◦ P )( A, N ) γ ( id A , id N ) − − − − − − − → P ( A, N ) ։ A ⊗ P N . This ma p induced the ﬁrst arrow A ⊗ P ( B ⊗ P N ) → A ⊗ P N . Similarly , the second map is induced b y the comp osite P ( A, P ( B , N )) id P ( id A , γ N ) − − − − − − − − → P ( A, N ) ։ A ⊗ P N , where γ N enco des the B -mo dule structure o n N . Remark. The A -module A ⊗ P B N is a quo tient of the free A -mo dule A ⊗ P N . As for the notation ⊗ P , w e hav e to b e ca reful abo ut the notation ⊗ P B which is not a classical tensor pro duct ov er B (except for P = C om ), as w e see in the follo wing e x amples. Examples. Pr ovided a mo rphism of algebras B f − → A , we hav e the dg K -mo dules isomorphisms • A ⊗ A ss B N ∼ = ( K ⊕ A ) ⊗ B N ⊗ B ( K ⊕ A ), where the map B → K ⊕ A is given by f , • A ⊗ C om B N ∼ = ( K ⊕ A ) ⊗ B N , where the map B → K ⊕ A is given by f , • A ⊗ L ie B N ∼ = U e ( A ) ⊗ B N , where U e ( A ) is the env eloping alg e bra of the Lie alg ebra A . In a ll these exa mples, the notation ⊗ B stands for the usual tensor pro duct ov er B . 8 JOAN M ILL ` ES 1. And r ´ e-Quillen cohomolo gy of algebras over an operad First we rec all the conceptual deﬁnition of Andr´ e - Quillen cohomo logy with co eﬃcients of a n algebra ov er a n op era d fro m [Hin97, GH00]. Then w e recall the cons tructions and theor ems of Koszul duality theory of op er ads [GK 94]. Finally , we recall the deﬁnition o f t wisting mo r phism given by [GJ 94]. This s ection contains no new res ult but we will use these three theories througho ut the text. W e only wan t to emphas ize that oper adic resolutions fro m Kosz ul duality theory deﬁne functorial co ﬁbrant reso lutions o n the le vel of algebr as and then pr ovide explicit c hain complexes which compute Andr´ e-Quillen coho mology . W e w ork with the co ﬁbrantly g enerated mo del catego ry of a lgebras over a n op erad and of mo d- ules over an ope rad given in [GJ9 4], [Hin97] and [BM03]. 1.1. Deriv ation and cotangent complex. T o study the structure of the P -a lgebra A , we der ive the functor of P -der iv ations from A to M in the Quillen sense (non-ab elian setting). 1.1.1. Algebras ov e r a P - al gebra. Let A b e a P -algebr a. A P -alg ebra B endow ed with a n augmentation, that is a ma p of P - a lgebras B f − → A , is called a P -algebr a over A . W e denote by P - A l g / A the categ ory o f dg P -alg e bras ov er A (the morphis ms are given by the mor phisms of graded alg e bras which comm ute with the a ugmentation maps). 1.1.2. Deriv at ion. L e t B b e a P -a lgebra ov er A and let M be an A -mo dule. An A -derivation fr om B t o M is a linear ma p d : B → M such that the following diagra m comm utes P ( B ) = P ◦ B γ B   id P ◦ ′ d / / P ( B , M ) id P ◦ ( f , id M ) / / P ( A, M ) γ M   B d / / M , where the inﬁnitesimal comp osite of morphisms ◦ ′ was deﬁned in 0.1. W e denote by Der A ( B , M ) the set of A -deriv ations fro m B to M . This functor is representable on the rig h t b y the ab elian extension of A by M and on the left by the B -mo dule Ω P B of K¨ ahler diﬀeren tial forms as follows. 1.1.3. Ab eli an extensio n. Let A be a P -algebr a and let M b e an A -mo dule. The ab elian extension of A by M , denoted by A ⋉ M , is the P -algebr a ov er A whose under ly ing s pace is A ⊕ M and whose algebra structure is given by P ( A ⊕ M ) ։ P ( A ) ⊕ P ( A, M ) γ A + γ M − − − − − → A ⊕ M . The mo r phism A ⋉ M → A is just the pro jection o n the ﬁrst summand. 1.1.4. Lemma (Deﬁnition 2 . 1 o f [GH00]) . L et A b e a P -algebr a and M b e an A -mo dule. Then ther e is an isomorphi sm of dg mo dules Der A ( B , M ) ∼ = Hom P - A lg/A ( B , A ⋉ M ) . Proof. Any morphism of P -a lgebras g : B → A ⋉ M is the sum of the augmentation B → A and a deriv ation d : B → M and vice versa.  1.1.5. Lemm a (Lemma 2 . 3 o f [GH00]) . L et B b e a P -algebr a over A and M b e an A -mo dule. Ther e is a B -mo dule Ω P B and an isomorphism of dg mo dules Der A ( B , M ) ∼ = Hom M P B (Ω P B , f ∗ ( M )) , wher e the for getful funct or f ∗ endows M with a B -mo dule struct u r e. Mor e o ver, when B = P ( V ) is a fr e e algebr a, we get Ω P B ∼ = B ⊗ P V . ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 9 The second part of the lemma is given b y the fact that Der A ( P ( V ) , M ) ∼ = Hom g M od K ( V , M ), that is an y der iv ation from a free P -alg e br a is c hara cterized by the images of its ge nerators. The B -mo dule Ω P B is called the mo dule of K¨ ahler diﬀer ential forms . It can be made explicit by the co equalizer diag ram B ⊗ P P ( B ) / / / / B ⊗ P B / / / / Ω P B , where the ﬁrst arr ow is B ⊗ P γ B and the map P ( B , P ( B )) ֌ ( P ◦ P )( B , B ) γ ( id B , id B ) − − − − − − − → P ( B , B ) ։ B ⊗ P B factors thr ough B ⊗ P P ( B ) to give the sec ond arr ow. 1.1.6. Co rol lary . L et B b e a P -algebr a over A and M b e an A -mo dule. Ther e is an isomorphi sm of dg mo dules Der A ( B , M ) ∼ = Hom M P A ( A ⊗ P B Ω P B , M ) . Proof. W e use Lemma 1.1 .5 and the fact that A ⊗ P B − is left adjoint to the forgetful functor f ∗ (Prop osition 0 .6 .3).  Finally , we get a pair o f adjoint functors A ⊗ P − Ω P − : P - A lg / A ⇋ M P A : A ⋉ − . F rom now on, we work ov er a g round ﬁeld K of characteris tic 0. W e recall the mo del categ ory structure s on P - A lg / A a nd M P A given in [Hin9 7]. It is obtained by the following transfer principle (see als o [GJ94] and [BM03]). Let D b e a coﬁbrantly g e nerated mo del ca tegory and let E b e a categor y with small co limits a nd ﬁnite limits. Assume that F : D ⇋ E : G is an adjunction with left adjoint F . Then the catego ry E inherits a coﬁbrantly generated mo del categor y structure from D , provided that G pr eserves ﬁltered colimits and that Quillen’s small o b ject (o r Quillen’s path-ob ject) argument is veriﬁed. In this mo del categor y structure, a map f in E is a weak equiv alence (resp. ﬁbration) if and only if G ( f ) is a weak equiv alence (resp. ﬁbration) in D . In [Hin97], Hinich tr ansfers the mo del ca tegory structure of the categor y o f chain complexes ov er K to the category o f P -a lgebras (see Theor em 4 . 1 . 1 of [Hin97], every op era d is Σ-s plit since K is of characteristic 0). Finally , we obtain a mo del categ ory structur e on P - A lg / A in which g : B → B ′ is a weak equiv alence (resp. a ﬁbration) when the under lying map b etw een diﬀer ent ial graded mo dules is a qua s i-isomorphis m (r esp. sur jection). The ca tegory M P A of A -mo dules is isomorphic to the category g M od A ⊗ P K of diﬀere n tial graded mo dule ov er the enveloping algebra A ⊗ P K (Prop ositio n 0.6.2). Then the catego ry M P A inherits a mo del categor y struc tur e in w hich g : M → M ′ is a weak equiv alence (resp. a ﬁbr a tion) when g is a quas i- isomorphism (res p. surjection) of A ⊗ P K -mo dules. 1.1.7. Prop os ition. The p air of adjoi nt functors A ⊗ P − Ω P − : P - A lg / A ⇋ M P A : A ⋉ − forms a Quil len adjunction. Proof. B y Lemma 1 . 3 . 4 o f [Hov99], it is enough to prov e that A ⋉ − pr eserves ﬁbratio ns a nd acyclic ﬁbra tions. Let g : M ։ M ′ be a ﬁbr ation (r esp. acyclic ﬁbration) b etw een A -mo dules. Then g is a surjection (resp. a surjectiv e quasi-isomor phism). The image of the map g under the functor A ⋉ − is id A ⊕ g : A ⋉ M → A ⋉ M ′ , deno ted by id A ⋉ g . It follows that id A ⋉ g is surjective (resp. surjective and a quasi-isomor phism), w hich completes the pro o f.  Thu s, w e consider the der ived functors and we ge t the following adjunction b etw een the homo- topy ca tegories L ( A ⊗ P − Ω P − ) : Ho( P - A l g / A ) ⇋ Ho( M P A ) : R ( A ⋉ − ) . It follows that the cohomolog y of Hom Ho( M P A ) ( A ⊗ P R Ω P R, M ) ∼ = Der A ( R, M ) ∼ = Hom Ho( P - A lg/ A ) ( R, A ⋉ M ) 10 JOAN M ILL ` ES is indep endent o f the c hoice of the co ﬁbr ant resolution R of A in the mo del catego ry of P - algebras ov er A . 1.1.8. Andr´ e-Quill en (co)hom ology and cotangent com plex. Let R ∼ − → A b e a coﬁbrant resolution of A . The c otangent c omplex is the total (left) derived functor of the previous a djunction and a representation of it is given by L R/ A := A ⊗ P R Ω P R ∈ Ho( M P A ) . The Andr´ e-Quil len c ohomo lo gy of the P - algebr a A with c o eﬃcients in an A -mo dule M is deﬁned by H • P ( A, M ) := H • (Hom Ho( M P A ) ( L R/ A , M )) . The Andr ´ e-Quil len homol o gy of the P -algebr a A with c o eﬃci ents in an A -mo dule M is deﬁned by H P • ( A, M ) := H • ( M ⊗ A ⊗ P K L R/ A ) . The study of the Andr´ e-Quillen homo logy with co eﬃcients is analo gous to the study of the Andr´ e - Quillen cohomolog y with c o eﬃcients. In this pap er, we only work with Andr´ e-Q uillen co homology with co eﬃcients. Remark. W e use the left de r ived functor of the a djunction to deﬁne the Andr´ e -Quillen coho- mology . It is equiv alent to deﬁne the Andr´ e-Quillen coho mology by means of the rig ht derived functor. W e make this choice her e b eca us e we are in terested in co nsidering ho momorphisms in a mo dules ca tegory . 1.2. Bar construction of an op erad and Koszul op e rad. T o ma ke this cohomolo gy theor y explicit, we need a c o ﬁbrant resolution for algebra s ov er an op erad. In the mo del categor y of algebras over an op erad, a co ﬁbrant ob ject is a retra ct o f a quasi-free algebra endow ed with a go o d ﬁltration (for example, a no n- negatively g raded algebra). So we lo ok for quasi-free resolutions o f algebras . Op er a dic reso lutions provide such functoria l coﬁbra nt reso lutions for alg ebras. Ther e are mainly three op eradic res olutions: the simplicial bar cons tr uction which induces a Go dement type resolution for a lg ebras, the (co )a ugmented (co )bar construction on the level of (co)op era ds and the Koszul complex for op era ds . This last o ne induces the bar-coba r resolution (o r B oardman-V o gt resolution [BV73, BM06]) on the level o f algebra s. The a im of the tw o ne x t subsections is to reca ll the op er adic res o lutions. Here, we brieﬂy recall the (co )bar cons truction of a (co)op er a d a nd the notion of K oszul op erad. W e refer to [GK94, GJ94, F r e04] fo r a complete exp ositio n. 1.2.1. Bar construction. Let P b e an aug ment ed op erad. W e denote by sV the susp ensio n of V (that-is-to -say ( sV ) d := V d − 1 ). The b ar c onstruct ion of P is the quasi-free co op erad B ( P ) := ( F c ( s P ) , d B ( P ) := d 1 − d 2 ) , where the map d 1 is induced by the internal diﬀerential of the op era d ( d s P := id K s ⊗ d P ) and the comp onent d 2 is induced by the pro duct of the op er a d by F (2) ( s P ) ∼ = M 2-v ertic es tr e es K s ⊗ P ⊗ K s ⊗ P id K s ⊗ τ ⊗ id P − − − − − − − − → M 2-v ertic es trees K s ⊗ K s ⊗ P ⊗ P Π s ⊗ γ P − − − − − → K s ⊗ P , where τ : P ⊗ K s → K s ⊗ P is the symm et ry isomorphism giv en explicitly by τ ( o 1 ⊗ o 2 ) := ( − 1) | o 1 || o 2 | o 2 ⊗ o 1 and Π s : K s ⊗ K s → K s is the morphism of deg ree − 1 induced by Π s ( s ⊗ s ) := s . Remark. Assume that P is weigh t graded. Then the ba r construction is bigraded by the nu mber ( w ) of non- trivial indexed vertices a nd by the to tal weigh t ( ρ ) B ( w ) ( P ) := ⊕ ρ ∈ N B ( w ) ( P ) ( ρ ) . Dually , we deﬁne the c ob ar c onstruction of a c o augment e d c o o p er a d C by Ω( C ) := ( F ( s − 1 C ) , d 1 − d 2 ) . F rom now on, we assume that P is an augmented op erad a nd C is a c o augmented co o p e rad. ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 11 1.2.2. Quadratic op erad. A op erad P is quadr atic whe n P = F ( V ) / ( R ), where V is the S - mo dule of g enerator s , F ( V ) is the free ope r ad and the space of re la tions R lives in F (2) ( V ), the set of trees w ith t wo vertices. W e endow F ( V ) with a weight grading , which diﬀer s fr o m the homolo gical degree, g iven by the num b e r o f vertices, this induces a weight gr ading on each quadratic op era d. In this pap er, we c o nsider only no n-negatively weight g raded o per ad and we say that a weight gr aded dg op erad P is c o nnected when P = K ⊕ P (1) ⊕ P (2) ⊕ · · · , wher e P (0) = K is concentrated in homological degree 0. 1.2.3. Koszul op erad. W e deﬁne the Koszul dual c o op er ad of P b y the weigh t graded dg S - mo dule P ¡ ( ρ ) := H ρ ( B ( • ) ( P ) ( ρ ) , d 2 ) . An op era d is ca lled a Koszul op erad when the injection P ¡ ֌ B ( P ) is a quasi-iso morphism. When P is of ﬁnite type, that is P ( n ) is ﬁnite dimensional for each n , we can dualize linearly the co op erad P ¡ to get the Koszu l dual op er ad of P , denoted b y P ! . F o r any S n -mo dule V , we denote by V ∨ the S n -mo dule V ∗ ⊗ ( sg n n ), where ( sg n n ) is the one- dimensional sig nature representation of S n . W e deﬁne P ! ( n ) := P ¡ ( n ) ∨ . The pro duct on P ! is given by t ∆ P ¡ ◦ ω where ω : P ¡ ∨ ◦ P ¡ ∨ → ( P ¡ ◦ P ¡ ) ∨ . 1.2.4. Algebras up to homotop y. Let P be a Koszul op er a d. W e deﬁne P ∞ := Ω( P ¡ ). A P ∞ -algebra is called an algebr a up to homotopy o r homotopy P -algebr a (see [GK9 4]). The notion of P ∞ -algebra s is a la x version o f the notion o f P -a lgebras. Examples. • When P = A ss , we ge t the notion of A ∞ -algebra s; • when P = L ie , we get the notion of L ∞ -algebra s; • when P = C om , we g e t the notion of C ∞ -algebra s. 1.3. Op e radic t wisting mo rphism. W e refer to [GJ94, MV09] for a gener al a nd complete treatment. Let α , β : C → P b e morphisms of S -modules. W e deﬁne the c o nv olution pro duct α ⋆ β : C ∆ p − − → C ◦ (1) C α ◦ (1) β − − − − → P ◦ (1) P γ p − → P . The S -mo dule Hom( C , P ) is endow ed with an op era d str ucture. Mo reov er, the co nvolution pr o duct is a pre-Lie pro duct on Hom( C , P ), that is, it s atisﬁes the relation ( α ⋆ β ) ⋆ γ − α ⋆ ( β ⋆ γ ) = ( − 1) | β || γ | [( α ⋆ γ ) ⋆ β − α ⋆ ( γ ⋆ β )] for all α , β and γ in Hom( C , P ) . 1.3.1. Deﬁnition. An op er adic twisting morphism is a map α : C → P o f degree − 1 sa tisfying the Maur er-Cartan e quation ∂ ( α ) + α ⋆ α = 0 . W e denote the set o f op er a dic t wisting morphisms fro m C to P by Tw( C , P ). In the weight g raded ca se, we assume that the twisting mor phisms and the internal diﬀerentials preserve the weigh t. 1.3.2. Theorem (Theorem 2 . 17 of [GJ94]) . The fun ctors Ω and B form a p air of adjoint fun ctors b etwe en the c ate gory of c onne ct e d c o augmente d c o op er ads and augmente d op er ad s. The natur al bije ct ions ar e given by the set of op er ad ic twisting m orphisms: Hom dg − Op (Ω( C ) , P ) ∼ = Tw( C , P ) ∼ = Hom dg − C o op ( C , B ( P )) . Examples. W e give examples of op era dic t wisting morphisms. • When C = B ( P ) is the bar construction o n P , the prev io us theorem gives a natural op eradic twisting mor phism π : B ( P ) = F c ( s P ) ։ s P s − 1 − − → P ֌ P . This morphism is 12 JOAN M ILL ` ES universal in the sens e that each twisting morphism α : C → P fac to rizes uniquely thro ugh the map π C α / / f α " " ❉ ❉ ❉ ❉ P B ( P ) , π < < ② ② ② ② ② ② ② ② where f α is a morphism of dg co op er a ds. • When C = P ¡ is the K oszul dual co op erad of a qua dratic op erad P , the ma p κ : P ¡ ֌ B ( P ) π − → P is a n op eradic twisting morphism (the prec o mpo sition of an op era dic t wisting morphism by a ma p of dg co op erads is an oper adic t wisting mo r phism). Actually we hav e P ¡ ֌ F c ( sV ) and the map κ is given b y P ¡ ։ P ¡ (1) ∼ = sV s − 1 − − → V ֌ P . • When P = Ω( C ) is the coba r constructio n o n C , the previous theorem gives a natur al op eradic twisting morphism ι : C ։ C s − 1 − − → s − 1 C ֌ Ω( C ) = F ( s − 1 C ). This morphism is universal in the sens e that each twisting morphism α : C → P fac to rizes uniquely thro ugh the map ι Ω( C ) g α ! ! ❈ ❈ ❈ ❈ C α / / ι > > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ P , where g α is a morphism of dg op era ds. 1.3.3. Twisted comp osi tion pro duct. Let P be a dg oper ad and le t C b e a dg co op erad. Let α : C → P b e an op era dic twisting mo rphism. The twiste d c omp osition pr o duct P ◦ α C is the S -mo dule P ◦ C endowed with a diﬀerent ial d α := d P ◦C − δ l α , where δ l α is deﬁned by the co mpo site δ l α : P ◦ C id P ◦ ′ ∆ C − − − − − → P ◦ ( C , C ◦ C ) id P ◦ ( id C , α ◦ id C ) − − − − − − − − − − − → P ◦ ( C , P ◦ C ) ֌ ( P ◦ P ) ◦ C γ ◦ id C − − − − → P ◦ C . Since α is an op eradic twisting morphism, d α is a diﬀerential. When A is a P -a lgebra, w e denote b y C ◦ α A the chain complex ( C ( A ) , d α := d C ( A ) + δ r α ), where δ r α is the compo s ite C ( A ) ∆ p ◦ id A − − − − − → ( C ◦ (1) C )( A ) id C ◦ (1) α ◦ id A − − − − − − − − → ( C ◦ (1) P )( A ) id C ◦ γ A − − − − → C ( A ) . Finally , we denote b y P ◦ α C ◦ α A the vector spac e P ◦ C ( A ) endow ed with the diﬀeren tial d α := d P ◦C ( A ) − δ l α ◦ id A + id P ◦ ′ δ r α = d P ◦ ( C ◦ α A ) − δ l α ◦ id A . The nota tion d α stands for diﬀerent diﬀeren tials. The diﬀerential is g iven witho ut ambiguit y by the context. 1.3.4. Op eradic resolutions . In [GJ9 4], Getzler a nd Jones pro duced functor ial res olutions of algebras g iven by the follo wing theorems. 1.3.5. Theorem (Theo rem 2 . 1 9 of [GJ 94]) . The augmente d b ar c onstruct ion gives a r esolution P ◦ π B ( P ) ◦ π A ∼ / / / / A. 1.3.6. Theorem (Theorem 2 . 25 of [GJ 94]) . When t he op er ad P is Koszul, ther e is a smal ler r esolution of A given by the Koszul c omplex P ◦ κ P ¡ ◦ κ A ∼ / / / / A. The augmented bar r esolution a dmits a dual v ersion. 1.3.7. Theorem (Theorem 4 . 18 o f [V al07]) . F o r every weight gr ade d c o augmente d c o op er ad C , ther e is an isomorphi sm Ω( C ) ◦ ι C ∼ − → I . This gives, for all Ω( C )-algebr a A , a quasi-isomor phism Ω( C ) ◦ ι C ◦ ι A ∼ − → A . ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 13 1.4. Description of the cotangen t complex. Thanks to these reso lutions, we can describ e the underlying vector space of the cotangent complex. 1.4.1. Quasi-free resol ution. Let A b e a P - a lgebra, let C be a C - coalgebr a endow ed with a ﬁltration F p C such that F − 1 C = { 0 } and let α : C → P be an op eradic twisting morphism. W e denote by P ◦ α C the complex ( P ( C ) , d α := d P ◦ C − δ l α ). The diﬀerential δ l α on P ( C ) is giv en by δ l α : P ( C ) id P ◦ ′ ∆ − − − − − → P ◦ ( C, C ( C )) id P ◦ ( id C , α ◦ id C ) − − − − − − − − − − − → P ◦ ( C, P ( C )) ֌ P ◦ P ( C ) γ ◦ id c − − − → P ( C ) . A qu asi-fr e e r esolution of A is a co mplex P ◦ α C such that P ◦ α C ∼ − → A and δ l α | F p C ⊂ P ( F p − 1 C ). Except the no rmalized cotriple co ns truction, all the previous reso lutions are of this form when A is non-negatively g r aded. With this resolution, w e make the co tangent complex explicit. 1.4.2. Theorem . L et P ( C ) b e a quasi-fr e e r esolution of the P -algebr a A . With t his r esolution, the c otangent c omplex has the form L P ( C ) / A ∼ = A ⊗ P C. Proof. The cotangent co mplex is isomorphic to A ⊗ P R Ω P R = A ⊗ P P ( C ) Ω P ( P ( C )) ∼ = A ⊗ P P ( C ) ( P ( C ) ⊗ P C ) (Lemma 1.1.5) ∼ = A ⊗ P C (Prop ositions 0 .6.1 and 0 .6.3) .  When we use the a ugmented bar constr uc tio n, we get the cotangent complex for a ny a lgebra ov er any op erad. How ever this co mplex may b e huge and it can b e useful to work with smaller resolutions. When we use the Koszul res o lution, we can use the Koszul complex and we g et the cotangent co mplex o f an algebra ov er a Koszul op er a d. F or homotopy algebra s, we use the coaug- men ted cobar cons truction. In this pa p er , we consider only res olutions coming from op era dic resolutions. In [Mil10], we w ork with even smaller reso lutions, but whic h are not functorial with resp ect to the alg ebra. T o describ e completely the co tangent complex, we have to make its diﬀerential explicit. In the next se ction, we will trace the bounda ry map on Der A ( R, M ) thro ugh the v arious isomo rphisms. 2. Lie theoretic description W e endow the chain complex deﬁning the Andr´ e- Q uillen co homology with a structure of Lie algebra. The notion of twisting morphism (o r twisting co chain) ﬁr st app eared in [Bro5 9] and in [Mo o71] (see also [HMS74]). It is a par ticular kind of maps betw een a coasso ciative co algebra and an asso ciative algebr a . Getzler a nd Jones extend this deﬁnition to (co)algebra s ov er (co)oper ads (see 2 . 3 of [GJ94]). W e show that the diﬀerential on the cotangent complex A ⊗ P C is obta ine d by twisting the in ternal diﬀeren tial by a twisting mor phis m. In the sequel, let ( P , γ ) deno te a n op erad, ( C , ∆) denote a co op era d and ( C , ∆ C ) denote a C -coa lgebra. 2.1. A Lie algebra structure. Let α : C → P b e an op er adic twisting morphism. Let C be a C -coa lgebra and let A b e a P -a lgebra. Let M b e a n A -module. F o r all ϕ in Hom g M od K ( C, A ) and g in Hom g M od K ( C, M ), we deﬁne α [ ϕ, g ] := P n ≥ 1 α [ ϕ, g ] n , wher e α [ ϕ, g ] n is the compo s ite C ∆ C − − → C ( C ) ։ ( C ( n ) ⊗ C ⊗ n ) S n α ⊗ ϕ ⊗ n − 1 ⊗ g − − − − − − − − → P ( n ) ⊗ A ⊗ n − 1 ⊗ M ։ P ( A, M ) γ M − − → M . The no tation ⊗ H stands for the Hadamard pr o duct: for any S - mo dules M a nd N , ( M ⊗ H N )( n ) := M ( n ) ⊗ N ( n ). Let E nd s − 1 K be the c o op erad deﬁned by E nd s − 1 K ( n ) := Hom(( s − 1 K ) ⊗ n , s − 1 K ) 14 JOAN M ILL ` ES endow ed with the natur al action of S n . When ( C, ∆ C ) is a C -coalgebr a, we endow s − 1 C := s − 1 K ⊗ C with a structure of E nd s − 1 K ⊗ H C -coa lgebra g iven by ∆ s − 1 C : s − 1 C ∆ C ( n ) − − − − → s n − 1 s − n ( C ( n ) ⊗ C ⊗ n ) S n τ n − →  ( E nd s − 1 K ( n ) ⊗ C ( n )) ⊗ ( s − 1 C ) ⊗ n  S n , where ∆ C ( n ) is the comp osite C ∆ C − − → C ( C ) ։ ( C ( n ) ⊗ C ⊗ n ) S n and τ n is a map which p er m utes comp onents and is induced b y comp ositions of τ (se e n in Section 1 . 2 . 1). The diﬀerential o n s − 1 C is given by d s − 1 C := id s − 1 K ⊗ d C . In the fo llowing r esults, the oper a d P is quadratic and binar y a nd the co op erad C = P ¡ is the Koszul dual co o per ad of P . The t wisting mor phism κ : P ¡ → P is deﬁned in the examples after Section 1 .3.2. 2.1.1. Theorem. L et P b e a quadr atic binary op er ad and let C = P ¡ b e the Koszul dual c o op er ad of P . L et A b e a P -algebr a and C b e a P ¡ -c o algebr a. The chain c omple x (Hom g M od K ( C, A ) , κ [ − , − ] , ∂ ) forms a dg Lie algebr a whose br acke t κ [ − , − ] is of de gr e e − 1 , that is κ [ ϕ, ψ ] = − ( − 1) ( | ϕ |− 1)( | ψ |− 1) κ [ ψ , ϕ ] . Proof. Ther e is a n isomorphis m o f chain complexes Hom • g M od K ( C, A ) ≃ − → Hom • +1 g M od K ( s − 1 C, A ) ϕ 7→ ( ¯ ϕ : s − 1 c 7→ ϕ ( c )) , since ∂ ( ϕ ) = d A ◦ ϕ − ( − 1) | ϕ | ϕ ◦ d C = d A ◦ ¯ ϕ − ( − 1) | ϕ |− 1 ¯ ϕ ◦ d s − 1 C = ∂ ( ¯ ϕ ). Moreov er, we have the equa lit y κ [ ϕ, ψ ] = ( − 1 ) | ¯ ϕ | ¯ κ [ ¯ ϕ, ¯ ψ ], where ¯ κ ( s n − 1 µ c ) := κ ( µ c ) is not a map of S n -mo dules. W e show now that the dg mo dule (Hom • g M od K ( s − 1 C, A ) , ( − 1 ) | ¯ ϕ | ¯ κ [ ¯ ϕ, ¯ ψ ] , ∂ ) forms a Lie a lgebra. Since C is a P ¡ -coalge br a, we get that ( s − 1 C ) ∗ ∼ = sC ∗ is a P ! -algebra . That is, there is a mor phism of o pe rads P ! → E nd ( sC ∗ ). Hence, we obtain a mo rphism P ! ⊗ H P → E n d ( sC ∗ ) ⊗ H E nd ( A ) ∼ = E nd ( sC ∗ ⊗ A ). W e apply Theorem 29 of [V a l08] a nd we get that Hom g M od K ( s − 1 C, A ) ∼ = sC ∗ ⊗ A is a Lie a lg ebra. The Lie alg ebra structure is given by ( − 1) | ¯ ϕ | ¯ κ [ ¯ ϕ, ¯ ψ ], which is of deg r ee 0 s ince κ is non-zero o nly on P ¡ (2). Therefore Hom • +1 g M od K ( s − 1 C, A ) is a Lie algebr a with br a ck et o f degr ee 0.  2.1.2. Theorem. L et P b e a quadr atic binary op er ad and take C = P ¡ . Le t A b e a P -algebr a, let C b e a C -c o algebr a and let M b e an A -mo dule. Then the dg mo dule (Hom g M od K ( C, M ) , κ [ − , − ] , ∂ ) is a dg Lie mo dule over (Hom g M od K ( C, A ) , κ [ − , − ] , ∂ ) . Proof. The pr o of is analo guous to the pro of of Theorem 2.1 .1 in the following wa y . A A - mo dule structure ov er the op erad P is equiv alen t to a map of o per ads P → E nd A ( M ), where E nd A ( M ) := E nd ( A ) ⊕ E nd ( A, M ) with E nd ( A, M )( n ) := n M j =1 Hom( A ⊗ · · · ⊗ A | {z } j − 1 time s ⊗ M ⊗ A ⊗ · · · ⊗ A | {z } n − j times , M ) . The compo sition product is given b y the co mpo s ition o f maps when p ossible and z ero otherwise. W e get Hom g M od K ( s − 1 C, M ) ∼ = sC ∗ ⊗ M a nd ther e is a ma p of op era ds L ie → P ! ⊗ H P → E nd ( sC ∗ ) ⊗ E nd A ( M ) ∼ = E nd sC ∗ ⊗ A ( sC ∗ ⊗ M ). Therefore, Hom g M od K ( C, M ) is a dg Lie mo dule ov er Hom g M od K ( C, A ).  ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 15 2.2. Algebraic t wisting morphism . In this section, we deﬁne the notion of twisting morphism on the level of (co)algebra s introduced in 2 . 3 of [GJ 94]. Assume now that α : C → P is an o pe r adic t wisting morphism. Let A b e a P -a lgebra and let C b e a C -coalgebra . F or all ϕ in Hom M od K ( C, A ), we deﬁne the maps ⋆ α ( ϕ ) : C ∆ C − − → C ( C ) α ◦ ϕ − − → P ( A ) γ A − − → A. An algebr aic twisting morphism with r esp e ct to α is a map ϕ : C → A of deg r ee 0 satisfying the Maurer-Ca rtan equation ∂ ( ϕ ) + ⋆ α ( ϕ ) = 0 . W e denote by Tw α ( C, A ) the set o f algebra ic t wisting mo rphisms with resp ect to α . Examples. W e consider the t w o e x amples of Section 1.3 once aga in. • The map η B ( P ) ( A ) := η B ( P ) ◦ i d A : B ( P )( A ) ։ I ◦ A ∼ = A is an algebra ic t wisting morphism with re s pec t to π . F or s implicity , a ssume d A = 0. W e get ∂ ( η B ( P ) ( A )) = d A ◦ η B ( P ) ( A ) − η B ( P ) ( A ) ◦ d r π = − η B ( P ) ( A ) ◦ ( d B ( P ) ◦ id A + δ r π ) = − η B ( P ) ( A ) ◦ δ r π since d B ( P ) = 0 on F (0) ( s P ). Then ∂ ( η B ( P ) ( A ))( e ) is non-z e r o if a nd only if e = sµ ⊗ ( a 1 ⊗ · · · ⊗ a n ) ∈ F (1) ( s P )( A ) and is equal to − µ ( a 1 , · · · , a n ) in this case. Moreover, ⋆ π ( η B ( P ) ( A )) sa tisﬁes the same prop er ties. So the assertion is pr ov ed. • The ma p η P ¡ ( A ) : P ¡ ( A ) ֌ B ( P )( A ) ։ A is an algebr aic twisting morphism with resp ect to κ . Let us now mak e explicit the maps κ a nd η P ¡ ( A ) in the cas es P = A ss , C om and L ie . W e refer to [V al08] for the categorica l deﬁnition of the Koszul dual coo per ad. • When P = A ss , the Kos z ul dual A ss ¡ is a co op erad cogener ated b y the element s s ❄ ❄ ⑧ ⑧ , that is the e lemen ts ❄ ❄ ⑧ ⑧ ∈ A ss (2) s uspe nded by an s of de g ree 1, with co r elations s ❄ ❄ ⑧ ⑧ ⊗ ( s ❄ ❄ ⑧ ⑧ ⊗ ) − s ❄ ❄ ⑧ ⑧ ⊗ ( ⊗ s ❄ ❄ ⑧ ⑧ ), that we can represent b y s 2 ( ❄ ❄ ❄ ⑧ ⑧ ⑧ ⑧ ⑧ − ❄ ❄ ❄ ❄ ❄ ⑧ ⑧ ⑧ ). The map κ : A ss ¡ → A ss sends s ❄ ❄ ⑧ ⑧ onto ❄ ❄ ⑧ ⑧ and is zero elsewhere. The map η A ss ¡ ( A ) sends A onto A and is zer o elsewhere. • When P = C om , the map κ sends the cogenera tor of C om ¡ on the gener ator o f C om and is zero outside C om ¡ (2). The map η C om ¡ ( A ) is just the pro jection o nto A . • When P = L ie , the map κ sends the c ogenerato r of L ie ¡ on the gener ator of L ie and is zero outside L ie ¡ (2) a nd the ma p η L ie ¡ ( A ) is just the pro jection o nto A . When P is a binary quadr a tic op er ad, C = P ¡ is its K o szul dua l co op erad a nd α = κ , then algebraic twisting morphisms with resp ect to κ ar e in o ne-to-one cor resp ondence with so lutions of the Maur er-Car tan equa tion in the dg Lie algebra introduced in Theo rem 2.1.1. 2.3. Twisted diﬀe rential. Let α : C → P b e an o per adic twisting mor phism and let ϕ : C → A be an algebra ic twisting mor phism with r esp ect to α . W e asso ciate to α and ϕ a twiste d diﬀer ential ∂ α, ϕ , deno ted simply by ∂ ϕ , o n Hom g M od K ( C, M ) b y the formu la ∂ ϕ ( g ) := ∂ ( g ) + α [ ϕ, g ] . 2.3.1. Lemma. If α ∈ Tw( C , P ) and ϕ ∈ Tw α (C, A) , then ∂ ϕ 2 = 0 . Proof. W e recall that | α | = − 1 and | ϕ | = 0. Let us mo dify a little bit the op erator α [ ϕ, g ] n . W e deﬁne for all ψ in Hom g M od K ( C, A ) and g in Ho m g M od K ( C, M ) the op erator α [ ϕ, ( ψ, g )] n to b e the co mp os ite: C ∆ C, n − − − → ( C ( n ) ⊗ C ⊗ n ) S n P j α ⊗ ϕ j − 1 ⊗ ψ ⊗ ϕ n − j − 1 ⊗ g − − − − − − − − − − − − − − − − − − → P ( n ) ⊗ A ⊗ n − 1 ⊗ M ։ P ( A, M ) γ M − − → M . W e deﬁne α [ ϕ, ( ψ , g )] := P n ≥ 2 α [ ϕ, ( ψ , g )] n . (W e have to pay attention to the fact that sign ( − 1 ) | ψ || g | may appear . The elements of C ( C ) are inv ar ia nt under the action of the symmetr ic groups , so they a re of the form P σ ∈ S n ε σ µ c · σ ⊗ c σ − 1 (1) ⊗ · · · ⊗ c σ − 1 ( n ) , wher e ε σ depe nds o n ( − 1) | c i || c j | . F or example, ε (12) = ( − 1) | c 1 || c 2 | , 16 JOAN M ILL ` ES ε (123) = ( − 1) | c 1 || c 3 | + | c 2 || c 3 | and ε (132) = ( − 1) | c 1 || c 2 | + | c 1 || c 3 | . Moreover, the co in v ariant elements in P ( A, M ) satis fy µ ⊗ S n ( a 1 ⊗ · · · ⊗ a n ) = ε σ µ · σ ⊗ S n ( a σ − 1 (1) ⊗ · · · ⊗ a σ − 1 ( n ) ). The imag e of ( − 1) | c 1 || c 2 | µ c · (1 2) ⊗ c 2 ⊗ c 1 under γ M ◦ ( α ⊗ ψ ⊗ g ) in M is ( − 1) | c 1 || c 2 | + | µ c | ( | ψ | + | g | )+ | g || c 2 | α ( µ c )( ψ ( c 2 ) , g ( c 1 )) = ( − 1) | c 1 || c 2 | + | µ c | ( | ψ | + | g | )+ | g || c 2 | ( − 1) | ψ ( c 2 ) || g ( c 1 ) | µ ( g ( c 1 ) , ψ ( c 2 )) = ( − 1) | ψ || g | ( − 1) | µ c | ( | ψ | + | g | )+ | ψ || c 1 | µ ( g ( c 1 ) , ψ ( c 2 )) . Therefore, the ope r ator α [ ϕ, ( ψ , g )] can be understo o d a s follows α [ ϕ, ( ψ , g )] = X    ϕ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ϕ ❖ ❖ ❖ ❖ ❖ ❖ ψ ❄ ❄ ❄ ϕ ϕ ⑧ ⑧ ⑧ g ♦ ♦ ♦ ♦ ♦ ♦ ϕ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ α + ( − 1 ) | ψ || g | ϕ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ϕ ◆ ◆ ◆ ◆ ◆ ◆ g ❃ ❃ ❃ ϕ ϕ ⑧ ⑧ ⑧ ψ ♦ ♦ ♦ ♦ ♦ ♦ ϕ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ α    . ) The maps ∆ C and γ M are ma ps of dg modules and we hav e the equa lity ∂ ( α ⊗ ϕ ⊗ n − 1 ⊗ ψ ) = ∂ ( α ) ⊗ ϕ ⊗ n − 1 ⊗ ψ + ( − 1 ) | α | α ⊗ ∂ ( ϕ ⊗ n − 1 ) ⊗ ψ + ( − 1 ) | α | α ⊗ ϕ ⊗ n − 1 ⊗ ∂ ( ψ ) , where ∂ ( ϕ ⊗ n − 1 ) = P j ϕ j − 1 ⊗ ∂ ( ϕ ) ⊗ ϕ n − j − 1 . Therefor e we get ∂ ( α [ ϕ, g ]) = ∂ ( α )[ ϕ, g ] + ( − 1) | α | α [ ϕ, ( ∂ ( ϕ ) , g )] + ( − 1 ) | α | α [ ϕ, ∂ ( g )] . It follows that ∂ ϕ 2 ( g ) = ∂ ϕ ( ∂ ( g ) + α [ ϕ, g ]) = ∂ 2 ( g ) + ∂ ( α [ ϕ, g ]) + α [ ϕ, ∂ ( g )] + α [ ϕ, α [ ϕ, g ]] = ∂ ( α )[ ϕ, g ] + ( − 1 ) | α | α [ ϕ, ( ∂ ( ϕ ) , g )] + ( − 1 ) | α | α [ ϕ, ∂ ( g )] + α [ ϕ, ∂ ( g )] + α [ ϕ, α [ ϕ, g ]] = ∂ ( α )[ ϕ, g ] − α [ ϕ, ( ∂ ( ϕ ) , g )] + α [ ϕ, α [ ϕ, g ]] . The following picture P ϕ ❂ ❂ g ϕ ✁ ✁ ϕ ❯ ❯ ❯ ❯ ❯ ❯ ϕ ❖ ❖ ❖ ❖ α ϕ ♦ ♦ ♦ ♦ ϕ ✐ ✐ ✐ ✐ ✐ ✐ α = P    ϕ g ϕ ❂ ❂ ϕ ϕ ✁ ✁ ϕ ϕ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❖ ❖ ❖ ❖ α ♦ ♦ ♦ ♦ ✐ ✐ ✐ ✐ ✐ ✐ ✐ α + ϕ ϕ ϕ ❂ ❂ g ϕ ✁ ✁ ϕ ϕ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❖ ❖ ❖ ❖ α ♦ ♦ ♦ ♦ ✐ ✐ ✐ ✐ ✐ ✐ ✐ α + ϕ ϕ ϕ ❂ ❂ ϕ ϕ ✁ ✁ g ϕ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❖ ❖ ❖ ❖ α ♦ ♦ ♦ ♦ ❥ ❥ ❥ ❥ ❥ ❥ ❥ α    − P    ( − 1) | g | ϕ ❂ ❂ ϕ ϕ ✁ ✁ ϕ ❚ ❚ ❚ ❚ ❚ ❚ g ❖ ❖ ❖ ❖ α ϕ ♦ ♦ ♦ ♦ ϕ ✐ ✐ ✐ ✐ ✐ ✐ α + ϕ ❂ ❂ ϕ ϕ ✁ ✁ ϕ ❯ ❯ ❯ ❯ ❯ ❯ ϕ ❖ ❖ ❖ ❖ α g ♦ ♦ ♦ ♦ ϕ ❥ ❥ ❥ ❥ ❥ ❥ α    mo dels the equation α [ ϕ, α [ ϕ, g ]] = ( α ⋆ α )[ ϕ, g ] − α [ ϕ, ( ⋆ α ( ϕ ) , g )] (the sign ( − 1) | g | app ears when w e p ermute α and g ). Th us ∂ ϕ 2 ( g ) = ( ∂ ( α ) + α ⋆ α )[ ϕ, g ] − α [ ϕ, ( ∂ ( ϕ ) + ⋆ α ( ϕ ) , g )] . Since α is an op er a dic twisting morphism and ϕ is an algebra ic twisting morphism with resp ect to α , this c o ncludes the pro of.  2.4. The cotangen t complex of an algebra ov e r an op erad. F r o m now on, we trace through the isomor phisms of Theo rem 1 .4.2 in order to ma ke the diﬀerential on the cota ngent complex explicit. Finally , for appro pr iate diﬀerentials, we obtain the isomo rphism o f diﬀerent ial gr aded mo dules Der A ( P ( C ) , M ) ∼ = Hom M P A ( A ⊗ P C, M ) , where P ( C ) is a quasi-fr ee resolution of A . W e hav e in mind the resolutions obtained by means of the augmented bar cons tr uction on the level of ope r ad, applied to an algebra , or the Ko szul complex on an a lgebra or the co augmented cobar constr uction on the level of co o per ads, applied to a homo topy a lg ebra. The space Der A ( P ( C ) , M ) is endow ed with the following diﬀerential ∂ ( f ) = d M ◦ f − ( − 1) | f | f ◦ d α , where d α was deﬁned in Section 1.4.1. ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 17 2.4.1. Prop os ition. With the ab ove notations, we have the fol lowing isomorphism of dg mo dules (Der A ( P ◦ α C, M ) , ∂ ) ∼ = (Hom g M od K ( C, M ) , ∂ ϕ = ∂ + α [ ϕ, − ]) , wher e C = C ( A ) . Proof. First, the isomorphism of K -mo dules b etw een Der A ( P ( C ) , M ) and Hom g M od K ( C, M ) is given b y the restriction o n the generators C . W e verify that this isomorphism commutes with the res pective diﬀerentials. W e ﬁx the notations ¯ f := f | C and n := | ¯ f | = | f | . On the one hand, we hav e ∂ ( f ) | C = ( d M ◦ f ) | C − ( − 1 ) | f | ( f ◦ d α ) | C = d M ◦ ¯ f − ( − 1) n f ◦ ( d P ◦ id C + id P ◦ ′ d C − δ l α ) | C . Moreov er, ( d P ◦ id C ) | C = 0 since ( d P ) |P (1) = 0 and f ◦ ( id P ◦ ′ d C ) | C = ¯ f ◦ d C . Thus ∂ ( f ) | C = d M ◦ ¯ f − ( − 1) n ¯ f ◦ d C + ( − 1 ) n f ◦ δ l α | C . On the other hand, ∂ ϕ ( ¯ f ) = d M ◦ ¯ f − ( − 1) n ¯ f ◦ d C + α [ ϕ, ¯ f ] . With the signs α ⊗ ¯ f = ( − 1 ) | α || ¯ f | ( id ⊗ ¯ f ) ⊗ ( α ⊗ id ) and using the fact that f is a deriv ation, we verify that ( − 1) n f ◦ δ l α | C = α [ ϕ, ¯ f ].  Let us construct a twisted diﬀerential on the free A -mo dule A ⊗ P C a s follows. Since A ⊗ P C is a quotient of P ( A, C ), we deﬁne a map δ l 1 ( n ) : P ( A, C ) id P ( id A , ∆ C ( n )) − − − − − − − − − − → P ( A, ( C ( n ) ⊗ C ⊗ n ) S n ) id P ( id A , α ⊗ ϕ ⊗ n − 1 ⊗ id C ) − − − − − − − − − − − − − − − − → P ( A, P ( n ) ⊗ A ⊗ n − 1 ⊗ C ) → ( P ◦ P )( A, C ) γ ( id A , id C ) − − − − − − − → P ( A, C ) . This map sends the elements µ ⊗ γ A ( ν 1 ⊗ a 1 ⊗ · · · ⊗ a i 1 ) ⊗ · · · ⊗ c ⊗ · · · ⊗ γ A ( ν k ⊗ · · · ⊗ a n ) and γ P ( µ ⊗ ν 1 ⊗ · · · ⊗ ν k ) ⊗ a 1 ⊗ · · · ⊗ a i 1 ⊗ · · · ⊗ c ⊗ · · · ⊗ a n to the same image, for c ∈ C and a j ∈ A and µ, ν j ∈ P . So δ l 1 ( n ) induces a map on the quo tien t δ l α, ϕ ( n ) : A ⊗ P C → A ⊗ P C. W e write δ l 1 := P δ l 1 ( n ) and δ l α, ϕ := P δ l α, ϕ ( n ), or simply δ l ϕ . W e deﬁne the t wisted diﬀeren tial ∂ α, ϕ , or simply ∂ ϕ on Ho m M P A ( A ⊗ P C, M ) b y ∂ ϕ ( f ) := ∂ ( f ) + ( − 1) | f | f ◦ δ l ϕ = d M ◦ f − ( − 1) | f | f ◦ ( d A ⊗ P C − δ l ϕ ) , where the diﬀerential d A ⊗ P C is induced by the natur al diﬀer e n tial on P ( A, C ). So we consider the t wisted diﬀerent ial d ϕ := d A ⊗ P C − δ l ϕ on A ⊗ P C . O nce ag ain, the notation ∂ ϕ stands fo r several diﬀeren tials a nd the relev an t one is given witho ut ambiguit y by the c o ntext. 2.4.2. Theorem. W ith the ab ove notations, the fol lowing t hr e e dg mo dules ar e isomorphic (Der A ( P ◦ α C, M ) , ∂ ) ∼ = (Hom g M od K ( C, M ) , ∂ ϕ ) ∼ = (Hom M P A ( A ⊗ P C, M ) , ∂ ϕ ) . Proof. W e already know the isomorphism of K -mo dules given by the restriction (Hom M P A ( A ⊗ P C, M ) , ∂ ) ∼ = (Hom g M od K ( C, M ) , ∂ ) from the pr eliminaries. W e now verify that this isomo rphism co mmutes with the diﬀere ntials. With the notation ¯ f := f | C , we hav e ∂ ϕ ( ¯ f ) = d M ◦ ¯ f − ( − 1) | ¯ f | ¯ f ◦ d C + α [ ϕ, ¯ f ] and ∂ ϕ ( f ) | C = ( d M ◦ f − ( − 1) | f | f ◦ d A ⊗ P C + ( − 1 ) | f | f ◦ δ l ϕ ) | C . Since ( f ◦ d A ⊗ P C ) | C = ¯ f ◦ d C , w e just need to show the equa lit y α [ ϕ, ¯ f ] = ( − 1) | f | ( f ◦ δ l ϕ ) | C . This holds since M ∈ M P A and f is a morphism o f A -mo dules over P and the structure of A -mo dule on C into A ⊗ P C is just the pro jectio n P ( A, C ) ։ A ⊗ P C .  18 JOAN M ILL ` ES Finally , when P ◦ α C ∼ − → A is a q uasi-free reso lution of A , the c hain complex ( A ⊗ P C, d ϕ = d A ⊗ P C − δ l ϕ ) is a r epresentation of the cotangent complex. In our cases, we hav e C = C ( A ). Then a represen- tation of the cota ngent complex is given by ( A ⊗ P C ( A ) , d ϕ = d A ⊗ P C ( A ) − δ l ϕ + δ r ϕ ) , where δ l ϕ is induced by P ( A, C ( A )) id P ◦ ( id A , ∆ p ◦ id A ) − − − − − − − − − − − − → P ( A, ( C ◦ (1) C )( A )) id P ◦ ( id A , α ◦ (1) id C ◦ id A ) − − − − − − − − − − − − − − − → P ( A, ( P ◦ (1) C )( A )) ֌ ( P ◦ P )( A, C ( A )) γ ◦ ( id A , id C ( A ) ) − − − − − − − − − → P ( A, C ( A )) and δ r ϕ is induced b y P ( A, C ( A )) id P ◦ ( id A , ∆ p ◦ id A ) − − − − − − − − − − − − → P ( A, ( C ◦ (1) C )( A )) id P ◦ ( id A , id C ◦ (1) α ◦ id A ) − − − − − − − − − − − − − − − → P ( A, ( C ◦ (1) P )( A )) ֌ P ( A, C ( A, P ( A ))) id P ◦ ( id A , id C ◦ ( id A , γ A )) − − − − − − − − − − − − − − − − → P ( A, C ( A )) . Remark. Applying this description to the r e solutions of alg ebras obtained b y means of the a ug- men ted bar construction or b y means o f the K oszul complex, w e obtain t wo diﬀerent chain c o m- plexes whic h a llow us to compute the Andr´ e-Quillen coho mology . The one using the Ko szul resolution is smaller since P ¡ ֌ B ( P ). Howev er the diﬀerential on the one using the augmented bar cons truction is simpler as the diﬀerential stro ngly depe nds on the co pr o duct. The co o per ad P ¡ is o ften given up to isomorphism, therefor e it is diﬃcult to make it explicit. 3. Applica tion s and new examples of coho mology theories W e apply the pr e vious gener al deﬁnitions to nea rly all the o pe r ads we know. W e ex pla in which re solution can b e used each time. Sometimes, it c orresp onds to known chain co mplex es. W e also s how that the cotriple coho mology co rresp onds to Andr´ e -Quillen c ohomology . Among the new examples, we make the Andr´ e-Quillen cohomolo gy for a lg ebras over the op er ad P er m explicit. W e do the s a me for ho mo topy P -a lgebras . F r om now o n, we assume that the algebr as are no n-negatively gra ded. 3.1. Applications. F or some op era ds, a n explicit c hain complex computing the cohomolog y the- ory fo r the as so ciated algebr a s ha s alrea dy b een prop o sed by v arious authors. • When P = A ss is the op erad of ass o ciative algebr as, A ⊗ P P ¡ ( A ) ∼ = ( K ⊕ A ) ⊗ B ( A ) ⊗ ( K ⊕ A ) (b y 0.6 .1) is the nor malized Ho chsc hild co mplex (see Section 1 . 1 . 14 of [Lo d98]). The Andr´ e-Quillen cohomolo gy of as so ciative alg ebras is the Ho chsc hild coho mology (see also Chapter IX, Section 6 of [CE99]). • When P = L ie is the o per ad of Lie algebras, A ⊗ P P ¡ ( A ) ∼ = U e ( A ) ⊗ Λ( A ) since L ie ¡ ( A ) ∼ = Λ( A ). The Andr ´ e-Quillen co homology of Lie algebras is Chev alley-Eilenberg co homology (see Chapter XII I of [CE9 9]). • When P = C om is the op erad of comm utative alg ebras, the complex A ⊗ P P ¡ ( A ) ∼ = ( K ⊕ A ) ⊗ C om ¡ ( A ), only v alid in characteristic 0, gives the coho mo logy theory of commutativ e algebras de ﬁned by Quillen in [Qui70]. It c o rresp onds to Harriso n cohomology deﬁned in [Har62]. W e r efer to [Lod9 8] for the relationship b etw een the diﬀeren t deﬁnitions. • When P = D ias is the o per ad o f diasso ciative algebr as and with the Koszul resolutio n, we ge t the chain complex and the a sso ciated cohomo lo gy deﬁned b y F rab etti in [F ra 01]. • When P = L eib is the op erad o f Leibniz alg ebras and with the Ko szul resolutio n, the Andr´ e-Quillen coho mology o f Leibniz alg ebras is the cohomolog y deﬁned by Lo day and Pirashvili in [LP93]. • F or the op erad P = P oiss enco ding Poisso n algebra s, F resse follow ed, as in this pap er, the ideas of Quillen to make a coho mology o f Poisson alg ebras explicit with the Koszul resolution [F re0 6]. ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 19 • When P = P r el ie and with the Koszul resolution, the Andr´ e-Quillen cohomolo gy of pre- Lie algebr as is the one deﬁned b y Dzh umadil’daev in [Dzh99]. • When P = Z inb , or equiv alently L eib ! , and with the K oszul resolution, the Andr´ e-Q uillen cohomolog y of Zinbiel a lgebras is the one g iven in [Bal98]. More genera lly , • Balvoine introduces a chain co mplex in [Ba l98]. When the op erad P is a binar y K oszul op erad, the chain complex computing the Andr´ e-Q uillen cohomolo g y obtained with the Koszul resolution corres po nds to the o ne deﬁned by Ba lav oine. Thus, the cohomolog y theories are the same in this case. 3.2. The case of P erm algebras. W e denote by P er m the o per ad co rresp onding to Perm a lge- bras deﬁned in [Cha01]. Let us r ecall that a bas is for P er m ( n ) is g iven b y cor ollas in s pace with n leav es lab elled by 1 to n with o ne lea f under line d. So P er m ( n ) is of dimension n . The comp osition pro duct in P erm is given b y the path traced through the upper underlined leaf from the r o ot. F or example, γ    1 2 3 ❏ ❏ t t 1 2 3 ❚ ❚ ❚ ❚ t t    = 1 2 3 4 5 ❚ ❚ ❚ ❚ ❏ ❏ t t ❥ ❥ ❥ ❥ and γ    1 2 3 ❏ ❏ t t 1 2 3 ❚ ❚ ❚ ❚ ❥ ❥ ❥ ❥    = 1 2 3 4 5 ❚ ❚ ❚ ❚ ❏ ❏ t t ❥ ❥ ❥ ❥ . In [CL01], the authors show that the Ko szul dua l op era d of the op erad P er m is the op er ad P rel i e and that the op erad P rel i e is Ko szul. It follows that the op erad P erm is Ko szul (see [GK94] fo r general facts ab out Koszul dualit y of oper a ds). Since P er m ¡ ∼ = P rel i e ∨ , it is p ossible to understand the copro duct on P erm ¡ if we k now the pro duct on P r el ie . Cha po ton a nd Livernet gav e an explicit bas is for P r el ie and made explicit the pro duct. This bas is of P r el ie is given by the ro o ted tree s of degr ee n , that is with n v ertices, denoted RT ( n ). Then w e need to understand the co pr o duct on P r el ie ∗ which is given b y ∆ : P r el ie ∗ t γ − → ( P r el ie ◦ P r el ie ) ∗ ≃ − → P r el ie ∗ ◦ P r el ie ∗ , where P rel ie ∗ ( n ) := P r el ie ( n ) ∗ and t γ ( f ) := f ◦ γ . A ro oted tree is re pr esented as in [CL 0 1], with its ro ot a t the b ottom. W e make ex plicit the copro duct on a particular element ∆  '&%$ !"# 1 ✾ ✾ '&%$ !"# 3 ✆ ✆ '&%$ !"# 2  = '&%$ !"# 1 ◦ 1 '&%$ !"# 1 ✾ ✾ '&%$ !"# 3 ✆ ✆ '&%$ !"# 2 + '&%$ !"# 1 '&%$ !"# 2 ◦ 2 '&%$ !"# 2 '&%$ !"# 1 + '&%$ !"# 2 '&%$ !"# 1 ◦ 1 '&%$ !"# 1 '&%$ !"# 2 + '&%$ !"# 1 ✾ ✾ '&%$ !"# 3 ✆ ✆ '&%$ !"# 2 ◦ 1 '&%$ !"# 1 + '&%$ !"# 1 ✾ ✾ '&%$ !"# 3 ✆ ✆ '&%$ !"# 2 ◦ 2 '&%$ !"# 1 + '&%$ !"# 1 ✾ ✾ '&%$ !"# 3 ✆ ✆ '&%$ !"# 2 ◦ 3 '&%$ !"# 1 . Let A be a P er m -alg ebra. The cotangent complex has the following for m A ⊗ P P ¡ ( A ) = A ⊗ P RT ( A ) ∼ = RT ( A ) ⊕ ❏ ❏ ❏ t t t A RT ( A ) ⊕ ❏ ❏ ❏ t t t RT ( A ) A ∼ = RT ( A ) ⊕ A ⊗ RT ( A ) ⊕ RT ( A ) ⊗ A, where RT ( A ) = ⊕ n RT ( n ) ⊗ S n A ⊗ n . 3.2.1. When the algebra is trivi al. W e assume ﬁrst that A is a trivial algebra , that is γ A ≡ 0. T o make the diﬀerential on the cotangent co mplex ex plicit, we just need to describ e the r estriction RT ( A ) → RT ( A ) ⊗ A ⊕ A ⊗ RT ( A ) since the diﬀerential is zero o n A ⊗ RT ( A ) ⊕ RT ( A ) ⊗ A . Let T b e in RT ( n ). The r e are several p os sibilities: i) the ro oted tree T has the for m ?>=< 89:; T 1 ?>=< 89:; 1 , where T 1 is in RT ( n − 1 ). In that ca se, the term '&%$ !"# 2 '&%$ !"# 1 ◦ 2 T 1 app ears in ∆ RT ( T ), so the image o f T ⊗ a 1 ⊗ · · · ⊗ a n under d ϕ in A ⊗ RT ( A ) contains − a 1 ⊗ ( T 1 ⊗ a 2 ⊗ · · · ⊗ a n ); 20 JOAN M ILL ` ES ii) there exis ts T 2 in RT ( n − 1) such that the ro oted tree T can b e written ?>=< 89:; 1 ?>=< 89:; T 2 . In that case, the term '&%$ !"# 1 '&%$ !"# 2 ◦ 2 T 2 app ears in ∆ RT ( T ), so the imag e of T ⊗ a 1 ⊗ · · · ⊗ a n under d ϕ in RT ( A ) ⊗ A con tains − ( T 2 ⊗ a 2 ⊗ · · · ⊗ a n ) ⊗ a 1 ; iii) the ro o ted tre e ha s the for m ?>=< 89:; T 3 7654 0123 n , where T 3 is in RT ( n − 1). In that case, the term '&%$ !"# 1 '&%$ !"# 2 ◦ 1 T 3 app ears in ∆ RT ( T ), so the image of T ⊗ a 1 ⊗ · · · ⊗ a n under d ϕ in A ⊗ RT ( A ) contains − a n ⊗ ( T 3 ⊗ a 1 ⊗ · · · ⊗ a n − 1 ); iv) there exists T 4 in RT ( n − 1) such that the ro oted tree ca n b e w r itten 7654 0123 n ?>=< 89:; T 4 . In that ca se, the ter m '&%$ !"# 2 '&%$ !"# 1 ◦ 1 T 4 app ears in ∆ RT ( T ), so the image o f T ⊗ a 1 ⊗ · · · ⊗ a n under d ϕ in RT ( A ) ⊗ A con tains − ( T 4 ⊗ a 1 ⊗ · · · ⊗ a n − 1 ) ⊗ a n ; A ro oted tree T has the shap e i) and iv), or ii) and iii), o r ii) and iv), or i) only , or ii) o nly , or iii) only , or iv) only , or ﬁnally a shap e not describ ed in i) to iv). In this last c ase, the diﬀerential is 0. Otherwise, the imag e under the diﬀeren tial of an element T ⊗ a 1 ⊗ · · · ⊗ a n in RT ( A ) is g iven by the sum of the corre sp o nding terms in i) to iv). F or example, if T c a n be written ?>=< 89:; T 1 ?>=< 89:; 1 and 7654 0123 n ?>=< 89:; T 4 , we ge t d ϕ ( T ⊗ a 1 ⊗ · · · ⊗ a n ) = − a 1 ⊗ ( T 1 ⊗ a 2 ⊗ · · · ⊗ a n ) − ( T 4 ⊗ a 1 ⊗ · · · ⊗ a n − 1 ) ⊗ a n . 3.2.2. F or an y P erm algebra. F or a g eneral P er m -a lgebra A , we no longer assume a prior i that the res triction of the diﬀere n tial d A ⊗ P P ¡ ( A ) to P ¡ ( A ), that is d α , is zero . F or a ro o ted tree T in RT ( n ), we deﬁne the function f by f ( T , i, j ) = 1 if T = ?>=< 89:; T 2 ?>=< 89:; T 3 7654 0123 i ?>=< 89:; j ?>=< 89:; T 1 for some ro o ted tree T 1 and some families of ro oted trees T 2 and T 3 , and f ( T , i, j ) = 0 other wise. There exists a ro oted tree T i in RT ( n − 1 ) such that T appea rs in the pro duct T i ◦ i '&%$ !"# 2 '&%$ !"# 1 if and only if f ( T , i, i + 1) = 1 (take T i = 7654 0123 T ′ 2 7654 0123 T ′ 3    7654 0123 i 7654 0123 T ′ 1 where T ′ j is the family of trees T j with vertices k > i replaced by k + 1 ). Similar ly there exists a ro oted tree T i in RT ( n − 1) such that T app ears in the pro duct T i ◦ i '&%$ !"# 1 '&%$ !"# 2 if a nd only if f ( T , i + 1 , i ) = 1. W e deﬁne E 1 ( T ) := { i | f ( T , i, i + 1 ) = 1 } and E 2 ( T ) := { i | f ( T , i + 1 , i ) = 1 } . W e obtain d α ( T ⊗ a 1 ⊗ · · · ⊗ a n ) = P i ∈ E 1 ( T ) T i ⊗ a 1 ⊗ · · · ⊗ γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a i ⊗ a i +1 ) ⊗ · · · ⊗ a n + P i ∈ E 2 ( T ) T i ⊗ a 1 ⊗ · · · ⊗ γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a i ⊗ a i +1 ) ⊗ · · · ⊗ a n , where T i is the r o oted tree s uch that T app ear s in the pr o duct T i ◦ i '&%$ !"# 2 '&%$ !"# 1 or T i ◦ i '&%$ !"# 1 '&%$ !"# 2 . Finally , on RT ( A ), the diﬀeren tial on the c otangent complex is given by d ϕ = d α − δ l ϕ . W e describ e now the diﬀerential δ l ϕ on A ⊗ RT ( A ) thanks to the description i) - iv) o f the previous section. ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 21 i)-ii) The term γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a 0 ⊗ a 1 ) ⊗ ( T i ⊗ a 2 ⊗ · · · ⊗ a n ) app ears in δ l ϕ ( a 0 ⊗ ( T ⊗ a 1 ⊗ · · · ⊗ a n )) (with i = 1 o r 2); iii)-iv) the term γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a 0 ⊗ a n ) ⊗ ( T i ⊗ a 1 ⊗ · · · ⊗ a n − 1 ) app ear s in δ l ϕ ( a 0 ⊗ ( T ⊗ a 1 ⊗ · · · ⊗ a n )) (with i = 3 o r 4). Similarly , we describ e the diﬀerential δ l ϕ on RT ( A ) ⊗ A . i) The term γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a 1 ⊗ a n +1 ) ⊗ ( T 1 ⊗ a 2 ⊗ · · · ⊗ a n ) a pp ea r s in δ l ϕ (( T ⊗ a 1 ⊗ · · · ⊗ a n ) ⊗ a n +1 ); ii) the term ( T 2 ⊗ a 2 ⊗ · · · ⊗ a n ) ⊗ γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a 1 ⊗ a n +1 ) app ears in δ l ϕ (( T ⊗ a 1 ⊗ · · · ⊗ a n ) ⊗ a n +1 ); iii) the term γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a n ⊗ a n +1 ) ⊗ ( T 3 ⊗ a 1 ⊗ · · · ⊗ a n − 1 ) a pp ea r s in δ l ϕ (( T ⊗ a 1 ⊗ · · · ⊗ a n ) ⊗ a n +1 ); iv) the term ( T 4 ⊗ a 1 ⊗ · · ·⊗ a n − 1 ) ⊗ γ A ( ❄ ❄ ⑧ ⑧ − ⊗ a n ⊗ a n +1 ) a pp ea r s in δ l ϕ (( T ⊗ a 1 ⊗ · · · ⊗ a n ) ⊗ a n +1 ). Finally , the diﬀerential o n the cotangent complex RT ( A ) ⊕ A ⊗ RT ( A ) ⊕ RT ( A ) ⊗ A is given by d α + id A ⊗ d α + d α ⊗ id A − δ l ϕ . 3.3. The case of A ∞ -algebras. Mark l g av e in [Ma r 92] a deﬁnition for a cohomolog y theory for homotopy asso cia tive a lgebras. In this section, we make explicit the Andr´ e - Quillen cohomo logy for homo to p y a sso ciative algebr as and we recover the complex deﬁned b y Mark l. The o p e rad A ∞ = Ω( A ss ¡ ) = F ( ❄ ❄ ⑧ ⑧ , ❄ ❄ ⑧ ⑧ , ❖ ❖ ❄ ❄ ⑧ ⑧ ♦ ♦ , . . . ) is the fre e o pe rad o n one gener ator in each degree g reater than 1. W e have the resolution R := A ∞ ◦ A ss ¡ ( A ) ∼ − → A and we get L R | A = M l,l 1 ,l 2 ≥ 0 k ≥ 0 M i 1 + ··· + i k = l 1 j 1 + ··· + j k = l 2 A ⊗ i 1 | · · · | A ⊗ i k | A ⊗ l | A ⊗ j k | · · · | A ⊗ j 1 . Actually , an element in L R | A should b e seen as a planar tr ee ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ✴ ✴ ✴ ✴ ✴ ✴ ✎ ✎ ✎ ✎ ✎ ✎ ② ② ② ② ② ② ② ② ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ✴ ✴ ✴ ✴ ✴ ✴ ✎ ✎ ✎ ✎ ✎ ✎ ② ② ② ② ② ② ② ② ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ✴ ✴ ✴ ✴ ✴ ✴ ✎ ✎ ✎ ✎ ✎ ✎ ② ② ② ② ② ② ② ② ✴ ✴ ✴ ✴ ✴ ✴ ✎ ✎ ✎ ✎ ✎ ✎ a 1 1 ··· a 1 i 1 c 1 1 ··· c 1 j 1 a k 1 ··· a k i k c k 1 ··· c k j k b 1 ··· b l where so me i t or j t may b e 0. An element in L R | A is wr itten a 1 1 · · · a 1 i 1 | · · · | a k 1 · · · a k i k [ b 1 · · · b l ] c k 1 · · · c k j k | · · · | c 1 1 · · · c 1 j 1 . A structur e o f A ∞ -algebra on A is given by maps µ n : A ⊗ n → A satisfying compatibility relations and a structure of A -mo dule ov er the op era d A ∞ on M is given by maps µ n,i : A ⊗ i − 1 ⊗ M ⊗ A ⊗ n − i → M for n ≥ 2 and 1 ≤ i ≤ n s atisfying so me compatibility relatio ns. In this ca se, the twisting morphism α is the injection A ss ¡ ֌ Ω( A ss ¡ ) and the twisting mor- phism o n the level of (co )algebras ϕ is the pro jection A ss ¡ ( A ) ։ A . When d A = 0, the diﬀerential on the cotangent co mplex is the sum of three ter ms that we will make explicit. O therwise, we hav e to add a term induced b y d A . The ﬁrs t part o f the diﬀerential is d A ⊗ A ∞ A ss ¡ ( A ) given b y d α and d A ∞ . W e use the fact that ∆ p : A ss ¡ → A ss ¡ ◦ (1) A ss ¡ is given by the for mu la ∆ p ( µ c n ) = X λ, k ( − 1) λ + k ( l − λ + k ) µ c l +1 − k ⊗ ( id ⊗ · · · ⊗ id | {z } λ ⊗ µ c k ⊗ id ⊗ · · · ⊗ id | {z } l − λ − k ) 22 JOAN M ILL ` ES to give on A ss ¡ ( A ) the diﬀeren tial d α ([ b 1 · · · b l ]) = X λ, k ( − 1) λ + k ( l − λ − k )+( | b 1 | + ··· + | b λ | )( k − 1) [ b 1 · · · b λ µ k ( b λ +1 · · · b λ + k ) b λ + k +1 · · · b l ] . Contrary to A ss and A ss ¡ , A ∞ has a non-zero diﬀerential which induce s a no n-zero diﬀerential on L R | A (also deno ted d A ∞ by abuse of notations). W e get d A ∞ ( a 1 1 · · · a 1 i 1 | · · · | a k 1 · · · a k i k [ b 1 · · · b l ] c k 1 · · · c k j k | · · · | c 1 1 · · · c 1 j 1 ) = − P ε λ, k, t a 1 1 · · · | a t 1 · · · µ k ( a t λ +1 · · · a t λ + k ) · · · a t i t | · · · [ · · · ] c k 1 · · · | · · · | · · · c 1 j 1 − P ε λ, k, t a 1 1 · · · | a t 1 · · · a t λ | a t λ +1 · · · a t i t | · · · [ · · · ] · · · | c t 1 · · · c t k − i t + λ − 1 | c t k − i t + λ · · · c t j t | · · · c 1 j 1 − P ε λ, k, t a 1 1 · · · | · · · | · · · a k i k [ · · · ] c k 1 · · · | c t 1 · · · µ k ( c t λ +1 · · · c t λ + k ) · · · c t j t | · · · c 1 j 1 , where ε λ, k, t = ( − 1) i 1 + j 1 + ··· + i t − 1 + j t − 1 + λ + k ( i t + j t +1 − λ + k ) . The second part of the diﬀerential is the twisted one induced b y δ l ϕ . W e get δ l ϕ ( a 1 1 · · · a 1 i 1 | · · · | a k 1 · · · a k i k [ b 1 · · · b l ] c k 1 · · · c k j k | · · · | c 1 1 · · · c 1 j 1 ) = X λ, k ǫ · a 1 1 · · · | b 1 · · · b λ [ b λ +1 · · · b λ + k ] b λ + k +1 · · · b l | · · · c 1 j 1 , where ǫ := ( − 1) i 1 + j 1 + ··· + i k + j k +( | a 1 1 | + ··· + | a k i k | )( l − k +1)+( | b 1 | + ··· + | b λ | )( k − 1)+ λ + k ( l − λ + k ) . 3.4. The case of L ∞ -algebras. The case of L ∞ -algebra s can be made explicit in the same wa y , with trees in space instead of planar trees. W e recover then the deﬁnitions given b y Hinich and Schec h tman in [HS93]. 3.5. The case of P ∞ -algebras. The g eneral cas e of homotopy P -alg ebras can b e trea ted similarly as follows. Let P b e a Kos zul op erad and let P ∞ := Ω( P ¡ ) b e its Ko szul r esolution. Any P ∞ - algebra A admits a res olution P ∞ ◦ ι P ¡ ◦ ι A ∼ − → A , wher e ι : P ¡ → P ∞ = Ω( P ¡ ) is the universal t wisting mo rphism. The c o tangent c o mplex has the same form as in the pre v ious cas es. 4. The cot angent compl ex and the modul e of K ¨ ahler differential forms In this sec tion, we show that the Andr´ e-Q uille n coho mology of a P -algebra A is an Ext-functor ov er the env eloping algebra of A if a nd only if the c o tangent co mplex o f A is a resolution of the module o f K¨ ahler diﬀeren tial forms. More ov er, we prove that the Andr´ e-Q uillen cohomolo gy theory of an o p e r ad is an E xt-functor ov er its env eloping algebr a. W e recall tha t we consider o nly non-negatively graded P -alg ebras in or der to ha ve coﬁbr ant resolutio ns . 4.1. Andr´ e-Quil len cohomolo gy as an E xt-functor. L e t R b e a co ﬁbrant reso lution of a P -algebra A . Then there is a map L R/ A = A ⊗ P R Ω P ( R ) → A ⊗ P A Ω P ( A ) ∼ = Ω P ( A ) . If the functor A ⊗ P − Ω P ( − ) sends coﬁbrant reso lutions to co ﬁbrant resolutions, then the Andr´ e- Quillen cohomology is the following E xt-functor H • P ( A, M ) ∼ = Ext • A ⊗ P K (Ω P ( A ) , M ) . Moreov er, we will see in this s ubsection that the reverse implication is true . Let X • ∼ − → Ω P ( A ) be a coﬁbr ant re solution in M P A and consider a quas i- free r esolution R = P ◦ C ( A ) of A . The cotangent complex L R/ A ∼ = A ⊗ P C ( A ) is a q uasi-free A -mo dule over P since R is quasi- free, s o this realization of the cotang en t c o mplex is a coﬁbrant A -mo dule over P . The mo del catego ry structure o n M P A and the comm utative diagram 0 / /     X • ∼     A ⊗ P C ( A ) / / : : ✉ ✉ ✉ ✉ ✉ ✉ Ω P ( A ) ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 23 give a map A ⊗ P C ( A ) → X • . This las t map induces a map H • P ( A, M ) ← H • P (Hom A ⊗ P K - mod ( X • , M )) ∼ = Ext • A ⊗ P K (Ω P ( A ) , M ) . When this map is an is o morphism, the Andr´ e-Quil len c ohomo lo gy is an Ext-functor over the P - enveloping algebr a . W e prov e the follo wing homolog ical lemmas. 4.1.1. Lemma. L et ϕ : V → W b e a map of dg ve ctor sp ac es. If ϕ ∗ : V ∗ ← W ∗ is an isomorphi sm then ϕ : V → W is an isomorphi sm, wher e V ∗ := Hom K ( V , K ) . Proof. Let x ∈ V non zero a nd H be a supplementary o f K x in V = K x ⊕ H . Since ϕ ∗ is surjective, there ex is ts g ∈ W ∗ such that x ∗ = ϕ ∗ ( g ) = g ◦ ϕ , where x ∗ is the map in V ∗ which is 1 on x and 0 on H . Th us 1 = x ∗ ( x ) = g ◦ ϕ ( x ), so ϕ ( x ) 6 = 0 a nd ϕ is injective. Dually we show that ϕ is surjective.  4.1.2. Lemma. L et S b e a dg un it ary asso ciative algebr a over K and let ϕ : M → N b e a map of dg left S -mo dules. If ϕ ∗ : Hom S - mod ( M , M ′ ) ∼ ← − Hom S - mod ( N , M ′ ) is a qu asi-isomorph ism for al l dg left S -mo dule M ′ , then ϕ : M ∼ − → N . Proof. W e endow Hom K ( S, K ) with a structure o f dg left S -mo dule b y s · f ( x ) := f ( s − 1 · x ) for s ∈ S and f ∈ Hom K ( S, K ) and x ∈ S . W e ha ve the adjunction Hom S - mod ( M , Hom K ( S, K )) ∼ = Hom K ( M ⊗ S S, K ) ∼ = Hom K ( M , K ) , which is an is omorphism of dg left S -mo dules (where K is endowed with a trivia l structure). Thus ϕ ∗ induces a qua si-isomor phism Hom K ( M , K ) ∼ ← − Hom K ( N , K ). Since the diﬀerential on K is 0, we ge t H • (Hom K ( M , K )) ∼ = Hom K (H • ( M ) , K ). W e conclude using Lemma 4.1.1.  4.1.3. Lemma. L et P b e a dg op er ad and let A b e a P -algebr a. L et C b e a c o o p er a d and let α : C → P b e an op er adic twisting morphism such that P ◦ C ( A ) is a quasi-fr e e r esolution of A . Ther e exists a sp e ctr a l se quenc e which c onver ges to the c oh omolo gy of A with c o eﬃcie nts in M , such that E p, q 2 ∼ = Ext p A ⊗ P K (H q ( A ⊗ P C ( A )) , M ) ⇒ H p + q P ( A, M ) . Proof. The arguments of Sec tio n 5 . 3 . 1 of [Bal98] are still v alid here and give the co nv ergence of the sp ectra l sequence.  4.1.4. Theorem. L et P b e a dg op er ad and let A b e a P -algebr a. Le t R b e a c oﬁbr ant r esolution of A . The fol lowing pr op erties ar e e qu ivalent: ( P 0 ) the Andr ´ e-Quil len c oho molo gy of A is an Ext-fun ctor over the enveloping algebr a A ⊗ P K , that is H • P ( A, M ) ∼ = Ext • A ⊗ P K (Ω P ( A ) , M ) ; ( P 1 ) the c otangent c omplex is quasi-isomorp hic to t he mo dule of K ¨ ahler diﬀer ential forms, that is L R/ A ∼ − → Ω P ( A ) . Proof. A repr esentation o f the cotangent complex is given by A ⊗ P C ( A ), where C is a co op erad and α : C → P is a Koszul morphism, e.g. C = B ( P ) and α = π . When A ⊗ P C ( A ) ∼ − → Ω P ( A ), a s A ⊗ P C ( A ) is a qua si-free A ⊗ P K -mo dule, the Andr´ e-Quillen coho mology is by deﬁnition an Ext- functor and the pr op erty ( P 1 ) implies the prop er t y ( P 0 ). Conv ersely , we assume that H • P ( − , A ) is a n Ext-functor. W e apply L e mma 4 .1.2 to S = A ⊗ P K , to M = A ⊗ P C ( A ) and to N = X • a coﬁbrant re solution of Ω P ( A ). This g ives that the prop er t y ( P 0 ) implies the pro p e rty ( P 1 ).  4.2. Andr´ e-Quil len cohomol ogy of op erads as an Ext-functor. Rezk deﬁned a cohomolo gy theory for op erads following the ideas of Quillen in [Rez96]. Ba ues, Jibladz e and T onks prop o sed in [B J T97] a cohomolo g y theo r y for monoids in par ticular monoidal ca tegories, which includes the case of op erads. Later Merkulov and V a llette gav e in [MV09] the cohomolog y theor y “ ` a la Quillen” for prop era ds, a nd so for o pe r ads. Merkulov a nd V alle tte deﬁne the cotangent co mplex asso ciated to the res olution of an op erad. Let Ω( C ) ∼ − → P b e a co ﬁbrant resolution of the op era d P . W e get L Ω( C ) / P ∼ = P ◦ (1) ( s − 1 C ◦ P ) → Ω op. ( P ) ∼ = P ◦ (1) ( P ◦ P ) / ∼ ∼ = P ◦ (1) P , 24 JOAN M ILL ` ES where Ω op. ( P ) is the left P -mo dule of K¨ ahler diﬀerential forms (we c a n se e Ω op. ( P ) as Ω S ( P ) with S the coloure d op erad whose alg ebras a re oper ads). The diﬀer ent ial on L Ω( C ) / P is made explicit as a trunca tion of the functorial co tangent complex deﬁned in Section 5.1. This enables to deﬁne the Andr ´ e-Quil len c oho molo gy of an op er ad with c o eﬃcients in an inﬁnitesimal P -bimo dule . 4.2.1. Inﬁnitesim al bi mo dul e. An inﬁnitesimal P -bimo dule is an S -mo dule M endow ed with t wo deg ree 0 maps P ◦ ( P , M ) → M and M ◦ P → M sa tisfying the commut ativity of certain diagrams. W e refer to Sectio n 3 of [MV09] for an ex plicit deﬁnition. The no tion of o per ad is a generaliza tion of the notion of ass o ciative a lgebra. Thus, the following lemma c a n b e seen as a generalizatio n o f the one in the ca se of asso ciative alg e bra. 4.2.2. Lemma. L et P b e an augmente d dg op er ad and Ω( C ) ∼ − → P b e a c oﬁbr a nt r esolution. The map P ◦ (1) ( s − 1 C ◦ P ) → Ω op. ( P ) ∼ = P ◦ (1) P is a quasi-isomorphism. Proof. Since the r esult do es not depend o n the co ﬁbrant res olution, we s how it in the under lying case C = B ( P ). W e ﬁlter the complex P ◦ (1) ( s − 1 B ( P ) ◦ P ) by the total num ber of elements of P in B ( P ) ◦ P F p P ◦ (1) ( s − 1 B ( P ) ◦ P ) := M w + k ≤ p P ◦ (1) ( s − 1 B ( w ) ( P ) ◦ ( I ⊕ P |{z} k times )) . The diﬀerential in P ◦ (1) ( s − 1 B ( P ) ◦ P ) is g iven by d P ◦ (1) ( s − 1 B ( P ) ◦P ) − δ l + δ r . The term − δ l decreases w and po ssibly k . The par t of d P ◦ (1) ( s − 1 B ( P ) ◦P ) induced b y d P keeps w + k co nstant and the part induced by d 2 of B ( P ) keeps w + k consta nt when the applicatio n of γ is given by P ◦ I ∼ = P ∼ = I ◦ P and decreases w + k by one otherwis e. T he term δ r behaves as the par t o f the diﬀerential induced by d 2 . Then, the diﬀerent ial r esp ects the ﬁltration. The ﬁltra tion is b o unded below a nd exha ustive so we can apply the classica l theorem of conv ergence of sp ectr al se quence (cf. Theorem 5 . 5 . 1 of [W ei94]) to obtain that the s pectr al s equence asso cia ted to the ﬁltratio n conv erges to the homology of P ◦ (1) ( s − 1 B ( P ) ◦ P ). The diﬀerential d 0 on the E 0 p, • page is given by d P ◦ (1) id s − 1 B ( P triv ) ◦P triv + id P ◦ (1) d s − 1 B ( P triv ) ◦P triv , where P tr iv is the underlying dg S - mo dule of P endow ed with a trivial comp osition structure, that is γ P triv ≡ 0 . By Maschk e’s theorem, since K is a ﬁeld of c haracteris tic 0, every K [ S n ]-mo dule is pr o jective. Then, by the K¨ unneth formula, we ge t H • ( E 0 p, • ) = H • ( P ◦ (1) ( s − 1 B ( P tr iv ) ◦ P tr iv )) = H • ( P ) ◦ (1) H • ( s − 1 B ( P tr iv ) ◦ P tr iv ) . Similarly to the pro of of I ∼ − → B ( P ) ◦P (see Theorem 2 . 19 in [GJ9 4]), we see that P ∼ − → s − 1 B ( P ) ◦P . Then, for P tr iv , we hav e H • ( E 0 p, • ) = H • ( P ) ◦ (1) H • ( s − 1 B ( P tr iv ) ◦ P tr iv ) = H • ( P ) ◦ (1) H • ( P ) = H • ( P ◦ (1) P ) . Finally , the sp ectral sequence collapse s a t rank 1 and the Lemma is true.  As a co rollar y of the previous Lemma, w e get 4.2.3. Theorem . Th e Andr ´ e-Quil len c ohomolo gy of op er ads with c o eﬃ cients in an inﬁnitesimal P -bimo dule is t he Ext-functor H • ( P , M ) ∼ = Ext • P ◦ (1) ( I ◦P ) (Ω op. ( P ) , M ) . Proof. W e combine Theo rem 4.1 .4 and Lemma 4.2.2.  5. The functorial cot angent complex In this sectio n, w e introduce a functorial c o tangent c omplex and a functorial m o dule of K¨ ahler diﬀer entia l forms , dep e nding only on the op er ad. W e prove that when the Andr´ e-Quillen co ho- mology is an Ext- functor then the map b etw een these tw o complexes is a quas i-isomorphism. W e deﬁne the mo dule o f obstructions a nd we show that it is acyclic when the Andr´ e-Quillen cohomol- ogy is an Ext-functor , giving a wa y , by contrap osition, to show that the Andr´ e-Q uillen cohomolo gy isn’t a n Ext-functor. ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 25 Under s ome PBW c onditions , w e prov e that the mo dules o f obstructions is acyclic if and only if the Andr ´ e-Quillen cohomolo gy is an Ext-functor. 5.1. Deﬁnitio n of the functorial cotangen t comple x. As we expla in in Section 1.2, the resolutions of algebras we us e in this pa pe r co me from o pe r adic resolutions. They all hav e the form P ◦ α C ∼ − → I , wher e α : C → P is an op eradic twisting mo rphism. W e call such a twisting morphism a Koszul morphism . W e deﬁne (a r e presentation o f ) the fun ct orial c otangent c omplex based on suc h t ype of res olutions as fo llows. W e consider the dg inﬁnitesima l P -bimo dule L P := P ( I , C ◦ P ) = P ◦ (1) ( C ◦ P ) endowed with the diﬀerential d L P := d P ( I , C ◦P ) − δ l L P + δ r L P , wher e δ l L P is deﬁned b y the compo site P ◦ (1) ( C ◦ P ) id P ◦ (1) (∆ p ◦ id P ) − − − − − − − − − − − → P ◦ (1) (( C ◦ (1) C ) ◦ P ) id P ◦ (1) ( α ◦ id C ◦ id P ) − − − − − − − − − − − − − → P ◦ (1) (( P ◦ (1) C ) ◦ P ) ֌ ( P ◦ P ◦ P ) ◦ (1) ( C ◦ P ) γ ◦ γ ◦ (1) id C◦P − − − − − − − − → P ◦ (1) ( C ◦ P ) and δ r L P is deﬁned b y the comp osite P ◦ (1) ( C ◦ P ) id P ◦ (1) (∆ p ◦ id P ) − − − − − − − − − − − → P ◦ (1) (( C ◦ (1) C ) ◦ P ) id P ◦ (1) ( id C ◦ α ◦ id P ) − − − − − − − − − − − − − → P ◦ (1) (( C ◦ (1) P ) ◦ P ) ֌ P ◦ (1) ( C ◦ P ◦ P ) id P ◦ (1) ( id C ◦ γ ) − − − − − − − − − → P ◦ (1) ( C ◦ P ) . The right action is given b y P ◦ (1) ( C ◦ P ) ◦ P ֌ ( P ◦ P ) ◦ (1) ( C ◦ P ◦ P ) γ ◦ (1) id C ◦ γ − − − − − − − → P ◦ (1) ( C ◦ P ) . 5.1.1. Prop os ition. L et A b e a P -algebr a. With t he ab o ve not ations, t her e is an isomorphism of chain c omplexes L P ◦ P A ∼ = A ⊗ P C ( A ) . Proof. W e write L P ◦ A ∼ = P ( A, C ◦ P ( A )). W e use the description of the relative co mpo sition pro duct ◦ P and of the desc r iption A ⊗ P N to get L P ◦ P A ∼ = A ⊗ P C ( A ). The equality o f the diﬀerentials comes from the same des criptions.  5.1.2. Co rollary . L et V b e a dg trivial P -algebr a, that is γ V ≡ 0 . Ther e is an isomorphism of chain c omplexes ( L P ◦ P I ) ◦ V ∼ = V ⊗ P C ( V ) . Proof. When the P -algebra V is trivia l, we get the isomo rphism of underlying dg mo dules ( L P ◦ P I ) ◦ V ∼ = L P ◦ P V , wher e I can b e seen as a left P -mo dule with a trivia l structure. The equality of the diﬀere n tials follows from their deﬁnitions.  W e denote L P := L P ◦ P I . 5.2. Deﬁnitio n of the functorial mo dul e of K¨ ahler di ﬀeren tial forms. Let P b e a dg op erad. W e deﬁne the functorial mo dule of K¨ ahler diﬀer ential forms as the following c o equal- izer diagr am in the catego ry of inﬁnitesimal P -bimo dules (see 10 . 3 of [F re09] for an equiv alen t deﬁnition) P ◦ (1) ( P ◦ P ) id P ◦ (1) γ / / c 2 / / P ◦ (1) P / / / / Ω P , where c 2 is given by the comp osite P ◦ (1) ( P ◦ P ) id P ◦ (1) ( id P ◦ ′ id P ) − − − − − − − − − − − − → P ◦ (1) ( P ◦ ( P , P )) ֌ ( P ◦ P ◦ P ) ◦ (1) P γ ◦ γ ◦ (1) id P − − − − − − − → P ◦ (1) P . The r ight P -mo dule action on Ω P is induced b y the right P -mo dule action o n P ◦ (1) P given b y ( P ◦ (1) P ) ◦ P ֌ ( P ◦ P ) ◦ (1) ( P ◦ P ) γ ◦ (1) γ − − − − → P ◦ (1) P . 5.2.1. Prop os ition. L et A b e a P -algebr a. Ther e is an isomorphism of chain c omplexes Ω P ◦ P A ∼ = Ω P ( A ) . Proof. W e write A ⊗ P A ∼ = ( P ◦ (1) P ) ◦ P A and A ⊗ P P ( A ) ∼ = ( P ◦ (1) P ◦ P ) ◦ P A . Thank s to the descr iption of Ω P ( A ) given at the end of Lemma 1.1.5, w e g et the result.  26 JOAN M ILL ` ES 5.2.2. Coroll ary . L et V b e a dg trivial P -algebr a. Ther e is an isomorphism of chain c omplexes (Ω P ◦ P I ) ◦ V ∼ = Ω P ( V ) . Proof. When the P -alg ebra V is trivial, we get (Ω P ◦ P I ) ◦ V ∼ = Ω P ◦ P ◦ V .  W e denote Ω P := Ω P ◦ P I . 5.3. Homo to p y category. Let P b e an augmented dg op er ad a nd C b e a co augmented dg co op erad such that P ◦ α C ∼ − → I . W e deﬁne the following s urjective map o f inﬁnitesimal P - bimo dules L P = P ( I , C ◦ P ) ։ P ( I , I ◦ P ) / ∼ ∼ = P ◦ (1) P / ∼ ∼ = Ω P . This ma p induces a map A ⊗ P C ( A ) ∼ = L P ◦ P A ։ Ω P ◦ P A ∼ = Ω P ( A ) which coincides with the map g iven in Section 4.1. The diﬀerential o n L Ω( C ) / P and the a ugmentation L Ω( C ) / P → P ◦ (1) ( I ◦ P ) induce a diﬀerential on the cone s L Ω( C ) / P ⊕ P ◦ (1) ( I ◦ P ). With this diﬀerential, we hav e L P ∼ = s L Ω( C ) / P ⊕ P ◦ (1) ( I ◦ P ). Then, L P is well-deﬁned in the homotopy ca tegory of inﬁnitesimal P -bimo dules. The same is true for L P = L P ◦ P I and we call its imag e in the homotopy categ ory of inﬁnitesima l left P -mo dules the functorial c otangent c omplex , that we denote by L P . W e denote by Ω P the image of Ω P := Ω P ◦ P I in the homotopy categ ory . 5.4. Filtration on the cotangen t complex. Let α : C → P b e a K oszul morphism b etw een a connected weigh t graded dg co op era d a nd a w eight gr aded dg o pe r ad. L e t A be a dg P -a lgebra. W e ﬁlter L P ◦ A ∼ = P ( A, C ◦ P ( A )) by the weight in the ﬁr s t P and the weigh t in C : F p ( L P ◦ A ) := M m + n ≤ p P ( n ) ◦ (1) ( C ( m ) ◦ P ) ◦ A. With the pro jection L P ◦ A ։ L P ◦ P A ∼ = A ⊗ P C ( A ), it induces a ﬁltration on A ⊗ P C ( A ) by F p ( A ⊗ P C ( A )) := im ( F p ( L P ◦ A ) ։ A ⊗ P C ( A )). The diﬀer e n tial on A ⊗ P C ( A ) is given by id L P ◦ ′ P d A + d P ( I , C ◦P ) ◦ P id A − δ l L P ◦ P id A + δ r L P ◦ P id A . The term depending o n d C keeps the sum n + m constant (since it pres erves the weight ), the part id L P ◦ ′ P d A − δ l L P ◦ P id A and the term dep ending on d P may decre a se the sum n + m and the pa r t δ r L P ◦ P id A decreases the sum n + m (at least by 1 since C is co nnected w eight gr aded). It follows that the diﬀerential o n the c otangent complex resp ects this ﬁltratio n. 5.4.1. Lemma. F or any P -algebr a A , the sp e ctr al se quenc e asso ci ate d to the ﬁ lt r ation F p c onver ges to the homolo g y of the c otangent c o mplex E 1 p, q = H p + q ( F p ( A ⊗ P C ( A )) /F p − 1 ( A ⊗ P C ( A ))) ⇒ H p + q ( A ⊗ P C ( A )) . Proof. This ﬁltra tion is exhaustive and b ounded be low so w e can apply the classica l theor em of conv ergence of s p ectr al sequences (cf. Theorem 5 . 5 . 1 of [W ei94]) to obta in the r esult.  W e denote by d 0 the diﬀerential on E 0 p, • , which dep ends on i d L P ◦ ′ P d A , on d P ( I , C ◦P ) ◦ P id A and on δ l L P ◦ P id A . W e denote by d r the diﬀerential o n E r p, • . 5.5. Filtration of the m o dule of K¨ ahler diﬀerential forms. By its deﬁnition as a co equa lizer, Ω P ( A ) is a quotient o f A ⊗ P A . W e ﬁlter A ⊗ P A by the weight in P and this induces a ﬁltration F ′ p on Ω P ( A ). The diﬀerential on Ω P ( A ) resp ects the ﬁltr ation. W e denote by d ′ 0 the diﬀerential on E ′ 0 p, • , which is induced by the diﬀerential on Ω P ( A ). Then, for a ny P -algebra A , the ﬁltration F ′ p is exhaustive and bounded b elow, s o the sp ectra l sequence a s so ciated to F ′ p conv erges to the homology o f the mo dule of K¨ ahler diﬀerential forms E ′ 1 p, q = H p + q ( F ′ p (Ω P ( A )) /F ′ p − 1 (Ω P ( A ))) ⇒ H p + q (Ω P ( A )) . The map A ⊗ P C ( A ) → Ω P ( A ) c o nsidered ab ov e is compatible with the ﬁltrations F • and F ′ • . ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 27 5.6. The cota ngent complex and the mo dule of K¨ ahler di ﬀeren tial fo rm s. As in the previous sections, w e endo w the enveloping algebra A ⊗ P K with a ﬁltratio n given by the weigh t in P . F o llowing the notatio ns given by Pirashvili in the rev iew of [F r a01], we say that P is an op erad satisfying the PBW prop er t y if for any P -algebr a A , ther e is a n iso morphism g r ( A ⊗ P K ) ∼ = A tr ⊗ P K , where A tr is the under lying space of A endowed with the tr ivial P -alg e bra structur e γ A tr ≡ 0. The study o f the diﬀerential on the cotangent complex A ⊗ P C ( A ) shows that P is an op erad satisfying the PBW pro per ty if and o nly if for an y P -algebra A , w e hav e the isomorphism g r ( A ⊗ P C ( A )) ∼ = A tr ⊗ P C ( A tr ). W e similarly say that the P -K¨ ahler diﬀere ntials sa tisfy the PBW prop erty when for any P - algebra A we have the isomorphis m g r (Ω P ( A )) ∼ = Ω P ( A tr ). These notio ns are diﬀerent from the notion of PBW-op erad deﬁned in [Hof10]. T o shor ten the follo wing theorem, w e s ay that P is a PBW-op er ad when • it is a connected weight gra ded op era d (in this case, we can consider the co op era d C = B P which is connected weight gr a ded), • it s a tisﬁes the PBW prop erty , and • P -K¨ a hler diﬀeren tials s atisfy the PBW pro p e r ty . W e complete Theo rem 4.1 .4 as follows. 5.6.1. Theorem. The assertion ( P 0 ) the Andr´ e-Quil len c ohomo lo gy is an Ex t-functor over t he enveloping algebr a A ⊗ P K for any P -algebr a A ; implies the fol lowing pr op erties e quivalent assertions: ( P ′ 1 ) the c otangent c omplex is qu asi-isomorphi c to the mo dule of K¨ ahler diﬀer ential forms for any dg ve ctor sp ac e V , se en as an algebr a with t rivial structur e, t hat is L R/V ∼ − → Ω P ( V ) ; ( P 2 ) the functorial c otangent c omplex L P is quasi-isomorphi c to t he functorial mo dule of K¨ ahler diﬀer entia l forms Ω P , that is L P ∼ − → Ω P . Under the assumption that P is a PBW-op er ad, we have the e quivalenc es ( P 0 ) ⇔ ( P ′ 1 ) ⇔ ( P 2 ) . Proof. The implication ( P 0 ) ⇒ ( P ′ 1 ) is clea r by Theo rem 4.1.4. The equiv alence ( P ′ 1 ) ⇔ ( P 2 ) follows from t he equalities ( V ⊗ P C ( V ) , d ϕ ) = ( L P ◦ V , d L P ◦ V ) = ( L P ◦ V , d L P ◦ id V ) a nd (Ω P ( V ) , d Ω P ( V ) ) = ( Ω P ◦ V , d Ω P ◦ id V ). W e a ssume now that P be a connec ted weight gr aded op er a d satisfying the PBW prop erty such that the P -K ¨ ahler diﬀerent ials also satisfy the PBW prop erty . W e prov e the implica tion ( P 0 ) ⇐ ( P ′ 1 ). Supp ose that the cotang e n t complex to b e quasi-iso mo rphic to the mo dule o f K¨ ahler diﬀerential forms for any dg vector space. Le t A b e a P -algebra and denote by V the underlying dg vector space of A considered as a triv ial algebr a. W e use the ﬁltrations and the s pec tr al sequences of the tw o pr evious se c tio ns. W e hav e E 0 p, q =  g r ( A ⊗ P C ( A )) , d 0 A  ∼ =  A tr ⊗ P C ( A tr ) , d 0 A tr  by the fact that P satisﬁes the PBW pro p e rty . Similarly , E ′ 0 p, q =  g r (Ω P ( A )) , d ′ 0 A  ∼ =  Ω P ( A tr ) , d ′ 0 A tr  bec ause the P -K ¨ ahler diﬀerentials also satisfy the PB W pro per ty . By ( P ′ 1 ), w e have a quasi- isomorphism ( E 0 , d 0 ) ∼ − → ( E ′ 0 , d ′ 0 ). This map is induced by the counit C → I and it follows that E 0 is a sub quo tient of A ⊗ P I ( A ) ( C is coaugmented). Then the higher diﬀerentials d r A and d ′ r A coincide thro ug h the (quasi-)isomo rphism b eca us e they only depend on d A and d P . So we obtain ( P 1 ), and therefore ( P 0 ) by Theorem 4.1.4.  Remark. When P is an o p e r ad concentrated in homolo gical deg ree 0, Ω P is an S -mo dule c o n- centrated in deg ree 0. In this case, we s ay that L P is acyclic when its homo lo gy is concentrated in deg r ee 0 and equal to Ω P . 28 JOAN M ILL ` ES 5.6.2. First applications. The op erads enco ding ass o ciative a lgebras and Lie algebra s a re PBW- op erads. Indeed they a re co nnected weight gr aded since they admit a homogeneo us quadratic presentation. Then the o pe r ad A ss sa tisfy the PB W prop erty by means of the computation made in the Examples after Prop ositio n 0.6.1. W e hav e g r ( A ⊗ A ss K ) ∼ = g r 0 ⊕ g r 1 ⊕ g r 2 = K ⊕ ( A ⊗ K ⊕ K ⊗ A ) ⊕ ( A ⊗ A ⊗ A ) ∼ = ( K ⊕ A tr ) ⊗ A tr ⊗ ( K ⊕ A tr ) ∼ = A tr ⊗ P K . F or the op erad L ie , the corre sp o nding sta temen t is the meaning of the Poincar´ e-Birkho ﬀ-Witt theorem. It remains to show that the A ss (resp. L ie )-K¨ a hler diﬀerentials satisfy the PBW prop erty . In the case of the op erad A ss , a computation similar as the computation for the env eloping algebra g ives g r (Ω P ( A )) ∼ = g r 0 ⊕ g r 1 ∼ = dA ⊕ dA ⊗ A ∼ = dA tr ⊕ dA tr ⊗ A tr ∼ = Ω P ( A tr ) , where we hav e noted dA the “linea r” co pie o f A in Ω P ( A ) ∼ = A ⊗ P dA/ ∼ and where we hav e made used o f the relatio n d ( a · b ) = da ⊗ b ± a ⊗ db and a ⊗ db ⊗ c = ± ( d ( a · b ) ⊗ c − a ⊗ d ( b · c )). The case o f the o per ad L ie can b e pr ov en analogous ly (we ﬁnd g r (Ω L ie ( A )) ∼ = S ( A ) ∼ = Ω L ie ( A tr )). W e now prov e the acyclic it y of the functorial cotangent complex in these ca s es. This gives a conceptual pr o of o f the fact that for these o p e rads the Andr´ e-Q uillen c o homology is an Ext-functor ov er the env eloping alg ebra A ⊗ P K . • W e hav e L A ss = A ss ¡ ⊕ ❄ ❄ ⑧ ⑧ A ss ¡ ⊕ ❄ ❄ ⑧ ⑧ A ss ¡ ⊕ A ss ¡ ❖ ❖ ❖ ❖ ♣ ♣ ♣ ♣ . Then L A ss ( n ) is genera ted b y the elements u n := 1 · · · n ◗ ◗ ◗ ◗ ● ● ● ✇ ✇ ✇ ♠ ♠ ♠ ♠ , r n := 1 · · · n ◗ ◗ ◗ ◗ ● ● ● ✇ ✇ ✇ ♠ ♠ ♠ ♠ ■ ■ ■ ♠ ♠ ♠ ♠ , l n := 1 · · · n ◗ ◗ ◗ ◗ ● ● ● ✇ ✇ ✇ ♠ ♠ ♠ ♠ ◗ ◗ ◗ ◗ t t t and v n := 1 · · · n ◗ ◗ ◗ ◗ ● ● ● ✇ ✇ ✇ ♠ ♠ ♠ ♠ ❱ ❱ ❱ ❱ ❱ ❱ ❣ ❣ ❣ ❣ ❣ ❣ . Since d ( u n ) = − l n − 1 − ( − 1) n − 1 r n − 1 , d ( r n ) = − v n − 1 = ( − 1) n − 1 d ( l n ) and d ( v n ) = 0, we deﬁne a homotopy h for d by h ( u n ) := 0, h ( l n ) = h ( r n )( − 1) n := − 1 2 u n +1 and h ( v n ) := − 1 2 (( − 1) n l n +1 + r n +1 ). • W e have L L ie = L ie ¡ 1 ⊕ ❄ ❄ ⑧ ⑧ L ie ¡ 1 2 ⊕ ❄ ❄ ❄ ⑧ ⑧ ⑧ ❄ ❄ L ie ¡ 1 2 3 ⊕ · · · ⊕ ✾ ✾ ✾ ✾ ⑧ ⑧ ⑧ ⑧ ⑧ ✴ ✴ L ie ¡ 1 · · · n -1 n ⊕ · · · . Then we can deﬁne the s a me homotopy as in [CE99], Theorem 7 . 1, Chap. XI I I. • F ollowing F r a b e tti in [F r a01], we show the acyclicity of L D ias . Remark. W e r ecall the following results. • Lo day a nd P irashvili show ed in [LP93] that the co homology of Leibniz a lgebras can b e written a s an Ext-functor. • Dzhum adil’daev showed in [Dzh99] that the coho mo logy of pr e-Lie algebra s can b e written as an Ext-functor. 5.6.3. The mo dule of obstructions. Let P b e an aug men ted dg op era d and let C b e a co aug- men ted dg co op erad and let α : C → P b e a twisting morphism. The map L P ։ Ω P is s urjective and we deﬁned O P := ker( L P ։ Ω P ) to get the following short ex act sequence of dg S -modules O P ֌ L P ։ Ω P . Since L P and Ω P are well-deﬁned in the homotopy c a tegory of inﬁnitesima l left P -mo dules, the same is tr ue for O P . Thus w e deﬁne the mo dule of obstructions O P by its imag e in the homotopy category o f inﬁnitesimal P - mo dules. W e get the following short exa ct sequence O P ֌ L P ։ Ω P . W e compute O P = Rel ⊕ ( P ( I , C ◦ P )) ◦ P I , ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 29 where R el is the image of the set of relations { ❖ ❖ ❄ ❄ ⑧ ⑧ ♦ ♦ + ❖ ❖ ❄ ❄ ⑧ ⑧ ♦ ♦ + ❖ ❖ ❄ ❄ ⑧ ⑧ ♦ ♦ , where ❖ ❖ ❄ ❄ ⑧ ⑧ ♦ ♦ = γ  ❄ ❄ ⑧ ⑧ ❖ ❖ ♦ ♦  and ❖ ❖ ♦ ♦ , ❄ ❄ ⑧ ⑧ ∈ P } in ( P ◦ (1) I ) ◦ P I . W e deduce the following theorem. 5.6.4. Theorem. Le t P b e an augmente d dg op er ad. The pr op ert y ( P 0 ) The Andr ´ e-Quil len c ohomol o gy is an Ext-fun ctor over the enveloping algebr a A ⊗ P K for any P -algebr a A ; implies the fol lowing pr op erty ( P 3 ) the homolo gy of the mo dule of obstructions O P is acyclic. When P b e a PBW-op er ad, we obtain the e quivalenc e ( P 0 ) ⇔ ( P 3 ) . Proof. The sho rt exact s equence O P ֌ L P ։ Ω P induces a long e x act sequence in homo logy which gives the equiv alence ( P 3 ) ⇔ ( P 2 ). Then the theorem fo llows fro m Theorem 5.6.1.  5.7. Another approac h. In the para llel work [F re09], F r esse studied the ho motopy prop erties of mo dules over op erads. His metho d applied to the present question provides the following s uﬃcient condition for the Andr´ e - Quillen c o homology to b e an E xt-functor. In this s ection, we show the relationship betw een the t w o a pproaches. Let P [1] be the S -mo dule deﬁned in [F re0 9] given by P [1]( n ) := P (1 + n ). The S n -action is given b y the action o f S n on { 2 , . . . , n + 1 } ⊂ { 1 , . . . , n + 1 } . Similarly to this de ﬁnitio n, we deﬁne the S - mo dule P [1] j by P [1] j ( n ) := P ( n + 1) where the S n -action is given by the actio n S n on { 1 , . . . , ˆ j , . . . , n + 1 } ⊂ { 1 , . . . , n + 1 } . Th us P [1 ] = P [1] 1 . W e hav e ( P ◦ (1) I )( n ) ∼ = P ( n ) ⊕ · · · ⊕ P ( n ) | {z } n times . As a right P -mo dule, we have ( P ◦ (1) I )( n ) ∼ = P [1] 1 ( n − 1) ⊕ · · · ⊕ P [1 ] n ( n − 1) | {z } n tim es . When P [1] is a qua si-free r ight P -mo dule, that is P [1] ∼ = ( M ◦ P , d ), w e get that P [1] j is a quasi-free rig ht P -mo dule ( M ◦ P , d ) tha nk s to the isomo rphism P [1] → P [1] j , µ 7→ µ · (1 · · · j ) . W e deﬁne M ′ ( n ) := ⊕ k ≥ 1 M ( n ) ⊕ · · · ⊕ M ( n ) | {z } k times . Then P ◦ (1) I is a retr a ct o f ( M ′ ◦ P , d ′ ), which is quasi- free. When P [1 ] is only a r etract of a quasi- free right P -mo dule, we get by the same arg umen t that P ◦ (1) I is a retra ct o f a quasi-free rig h t P -mo dule. Then, P [1] coﬁbr ant as a right P -mo dule implies that P ◦ (1) I coﬁbrant as a right P - mo dule. Finaly , when P [1] is a coﬁbrant rig ht P -mo dule, L P ∼ = ( P ◦ (1) I ) ⊗ ( C ◦ P ) is als o coﬁbrant. Thu s, when we assume mor eov er that Ω P is a coﬁbrant rig h t P - mo dule, the quas i-isomorphis m L P ∼ − → Ω P betw een co ﬁbrant right P - mo dules g ives a qua si-isomor phism A ⊗ P C ( A ) ∼ = L P ◦ P A ∼ − → Ω P ◦ P A ∼ = Ω P ( A ) (since A is coﬁbrant). Therefore, we hav e the following suﬃcient condition for the Andr´ e-Q uillen cohomolog y to be an E xt-functor. 5.7.1. Theorem (Theorem 1 7 . 3 . 4 in [F r e09]) . If P [1] and Ω P form c oﬁbr a nt right P -mo dules, t hen we have H • P ( A, M ) ∼ = Ext • A ⊗ P K (Ω P ( A ) , M ) . 30 JOAN M ILL ` ES 6. Is Andr ´ e-Quillen coho m o logy a n Ex t-functor ? In the prev ious section, w e show ed that when the Andr´ e-Q uillen co homology is a n Ext-functor, the module of obstructions O P is acyclic. It follo ws, b y con trap ositio n, that when the mo dule of obstruction O P is not acyclic, then the Andr´ e-Q uillen cohomo lo gy is no t an Ext-functor . Mor eov er, when P is a P BW-op erad, the mo dule o f obstr uctions O P is acyclic if and only if the Andr´ e- Quillen cohomolog y is an Ex t-functor. In this s ection, we apply these criteria to the o pe r ads C om , P e rm and to the minimal models of Koszul oper ads. In the case of the o p er ads C om a nd P e rm , we provide universal obstructions for the Andr´ e- Quillen cohomo logy to b e a n Ext- functor . In the case of an o per ad which is the cobar construction on a coo p er ad, we show that the obs tructions alwa ys v anish. W e apply this to the ca s e of homo topy a lgebras. 6.1. The case of com m utativ e alge bras. W e exhibit a non- trivial elemen t in the homolog y of the mo dule of obstructio ns . This gives a universal obstruction for the Andr´ e-Q uillen cohomolog y of co mm utative alg ebras to be an Ex t-functor ov er the enveloping alg ebra A ⊗ C om K . 6.1.1. Prop os ition. The mo dule of obstructions O C om is not acyclic. Mor e pr e cisely, we have H 1 ( O C om ) 6 = 0 . Proof. Co nsider the element ν := 1 2 ✼ ✼ ✞ ✞ in C om ¡ ֌ B ( C om ) a nd µ := 1 2 ✼ ✼ ✞ ✞ in C om . The element µ ⊗ ( ν ⊗ id ) = 1 2 ❀ ❀ ✄ ✄ 3 ❈ ❈ t t lives in O C om . W e compute d O C om ( µ ⊗ ( ν ⊗ id )) = 1 2 3 ❏ ❏ t t + 2 1 3 ❏ ❏ t t = 0 . Then µ ⊗ ( ν ⊗ id ) is a cycle in O C om . How ever, d O C om      1 2 3 ✼ ✼ ✼ ✞ ✞ ✞ ✞ ✞ − 1 2 3 ✼ ✼ ✼ ✼ ✼ ✞ ✞ ✞     = 1 2 ❀ ❀ ✄ ✄ 3 ❈ ❈ t t − 2 3 1 ❀ ❀ ✄ ✄ ❏ ❏ ④ ④ and it is imp ossible to obtain µ ⊗ ( ν ⊗ id ) a s a b oundary of an e lemen t in O C om . There fore, this shows that H 1 ( O C om ) 6 = 0.  Remark. The short exact sequence O C om ֌ L C om ։ Ω C om gives a long exact sequence in homology a nd, since H n (Ω C om ) = 0 for all n ≥ 1, w e get a lso H 1 ( L C om ) 6 = 0. It follows that there exists a comm utative algebra such that the cotangent complex is not acy c lic . Thanks to Theorem 5.6 .4, by co nt rap os ition, this gives a c o nceptual explanation to the fac t that the Andr´ e- Quillen cohomo lo gy of co mm utative algebra s cannot alwa ys be wr itten as an Ext- functor o ver the en veloping alg ebra A ⊗ C om K . 6.2. The case of P erm-algebras. The s ame a rgument applied to Perm algebr a s gives a co ncep- tual explanatio n to the fact that the Andr´ e- Q uillen co homology of Perm a lgebras cannot alwa ys be written as an E xt-functor over the en veloping algebra A ⊗ P er m K . 6.2.1. Prop os ition. We have H 1 ( O P er m ) 6 = 0 . Proof. The proo f is similar to the pro of o f Pr op osition 6.1.1.  ANDR ´ E-QUILLEN COHOMOLOGY OF ALGEBRAS O VER AN OPERAD 31 6.3. The case of alge bras up to h o motop y. W e show a new homoto p y prop erty for a lgebras ov er certa in coﬁbrant op erads. W e apply this in the case of P - algebras up to homotopy to prov e that the Andr´ e- Quillen coho mology is always a n E xt-functor ov er the env eloping a lgebra A ⊗ P ∞ K . 6.3.1. Theorem. L et C b e a c o augmente d weight gr ade d dg c o op er ad and P = Ω( C ) the c ob ar c onstruction on it. The Andr ´ e-Q u il len c ohomolo gy of Ω( C ) -algebr as is an Ext-functor over t he enveloping algebr a A ⊗ Ω( C ) K . Explicitly, for any Ω( C ) -algebr a A and any A -mo dule M , we have H • Ω( C ) ( A, M ) ∼ = Ext • A ⊗ Ω( C ) K (Ω Ω( C ) ( A ) , M ) . Proof. As in this case of A ∞ -algebra s, the twisting mor phis m α is the ma p C → Ω( C ) given in the examples after Theo r em 1.3 .2 a nd the twisting morphism on the le vel of (co)algebra s ϕ is the pro jection C ( A ) ։ A . As a dg S - mo dule, the module of obs tructions has the following form O Ω( C ) ∼ = Rel ⊕ M n ≥ 0 ( s − 1 C ) ◦ (1)  ( s − 1 C ) ◦ (1) · · ·  ( s − 1 C ) | {z } n tim es ◦ (1) C  · · ·  , where R el ⊂ L n ≥ 0 ( s − 1 C ) ◦ (1)  ( s − 1 C ) ◦ (1) · · ·  ( s − 1 C ) | {z } n times ◦ (1) I  · · ·  is deﬁned in Section 5 .6.3. F or any 1 ≤ j ≤ n , let µ c j, i j ∈ C ◦ (1) I , wher e i j is the emphasized entry and µ c j ∈ C ( m j ). L e t ν c ∈ C ( m ). F or σ ∈ S m 1 + ··· + m n + m − n , we deﬁne the map h b y h ( s − 1 µ c 1 , i 1 ⊗ s − 1 µ c 2 , i 2 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ ν c ⊗ σ ) = 0 a nd on Rel , h  P m n i n =1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ 1 c ⊗ σ  = ε n − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ µ c n ⊗ σ, where ε n − 1 = ( − 1) n − 1+ | µ c 1 , i 1 | + ··· + | µ c n − 1 , i n − 1 | . W e compute ( dh + hd )( s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ ν c ⊗ σ ) = = 0 + h  P · · · ⊗ ν ′ c ⊗ σ + ε n P m i =1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ s − 1 ν c i ⊗ 1 c ⊗ σ  = s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ ν c ⊗ σ, where ν ′ c ∈ C and i is the emphasized entry of ν c i , a nd ( dh + hd )  P m n i n =1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ 1 c ⊗ σ  = = d ( ε n − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ µ c n ⊗ σ ) + h  P m n i n =1 P n j =1 P ε j − 1 ( − 1) | µ ′ c j, i ′ j | s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ ′ c j, i ′ j ⊗ s − 1 µ ′′ c j, i ′′ j ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ 1 c ⊗ σ  + h  P m n i n =1 P n j =1 ε j − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ ( − s − 1 d C ( µ j, i j )) ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ 1 c ⊗ σ  = ε n − 1 P n − 1 j =1 ε j − 1 ( − 1) | µ ′ c j, i ′ j | s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ ′ c j, i ′ j ⊗ s − 1 µ ′′ c j, i ′′ j ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ µ c n ⊗ σ + ε n − 1 P ε n − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ s − 1 µ ′ c n, i ′ n ⊗ µ ′′ c n ⊗ σ + ε n − 1 ε n − 1 P m n i n =1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ 1 c ⊗ σ − ε n − 1 P n − 1 j =1 ε j − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 d C ( µ j, i j ) ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ µ c n ⊗ σ + ε n − 1 ε n − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ d C ( µ c n ) ⊗ σ − ε n − 1 P n − 1 j =1 ε j − 1 ( − 1) | µ ′ c j, i ′ j | s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ ′ c j, i ′ j ⊗ s − 1 µ ′′ c j, i ′′ j ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ µ c n ⊗ σ − ε n − 1 ( − 1) | µ ′ c n, i ′ n | P ε n − 1 ( − 1) | µ ′ c n, i ′ n | s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ s − 1 µ ′ c n, i ′ n ⊗ µ ′′ c n ⊗ σ + ε n − 1 P n − 1 j =1 ε j − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 d C ( µ j, i j ) ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ µ c n ⊗ σ − ε n − 1 ε n − 1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n − 1 , i n − 1 ⊗ d C ( µ c n ) ⊗ σ = P m n i n =1 s − 1 µ c 1 , i 1 ⊗ · · · ⊗ s − 1 µ c n, i n ⊗ 1 c ⊗ σ. Thu s dh + hd = id a nd h is a homotopy . Finally , O Ω( C ) is acyclic and Theorem 5.6.4 gives the theorem since a ny quasi- fr ee o p er ad P = Ω( C ) is a PBW- o p e rad (bec a use a fre e op erad has no relations).  32 JOAN M ILL ` ES W e conjecture that this theorem is true for a n y co ﬁbrant op era d. When P is a Kos z ul op erad, the prev ious theorem applied to C = P ¡ shows that the Andr´ e- Quillen cohomology of a homotopy alg ebra is a lwa ys an Ext-functor ov er its env eloping alg ebra. Let A b e a P -algebr a. The algebr a A is a P ∞ -algebra since there is a map o f op era ds P ∞ = Ω( P ¡ ) → P . Simila rly , an A -mo dule ov er the op er ad P is a lso an A -mo dule over the op era d P ∞ . This lea ds to the following result. 6.3.2. Prop ositio n . L et P b e a Koszul op er ad and let A b e a P -algebr a. The Andr´ e-Quil len c oho molo gy of t he P - algebr a A is e qual to the Andr ´ e-Quil len c ohomolo gy of the P ∞ -algebr a A . That is, H • P ( A, M ) = H • P ∞ ( A, M ) , fo r any A -mo dule M over t he op e r ad P . Proof. A res olution of A as a P -algebr a is given by P ◦ P ¡ ( A ) a nd a res o lution of A as a P ∞ - algebra is given by P ∞ ◦ P ¡ ( A ). Thus, by Theorem 2.4 .2, we have Hom M P ∞ A ( A ⊗ P ∞ P ¡ ( A ) , M ) = Hom g M od K ( P ¡ ( A ) , M ) = Hom M P A ( A ⊗ P P ¡ ( A ) , M ) . Moreov er, the diﬀerential on Hom g M od K ( P ¡ ( A ) , M ) is the same in b oth ca ses since the higher pro ducts P ¡ ( k ) ⊗ S k A ⊗ k → A for k ≥ 3 are 0 .  W e show ed that, for co mm utative algebr as and Perm a lgebras, the Andr´ e-Quillen coho mology of a P -a lgebra cannot always b e written as an Ext- functor ov er the env eloping algebr a A ⊗ P K . How ev er, by the following theorem, it ca n a lways b e written as a n Ex t-functor ov er the env eloping algebra A ⊗ P ∞ K . 6.3.3. Theorem. L et P b e a Koszul op er ad, let A b e a P -algebr a and let M b e an A -mo dule over the op er ad P . We have H • P ( A, M ) ∼ = Ext • A ⊗ P ∞ K (Ω P ∞ ( A ) , M ) . Proof. W e make us e of Theorem 6.3.1 and Prop ositio n 6.3.2.  Ackno wledgments I would like to thank my advisor Br uno V a llette for his useful a nd constant help. I am very grateful to Benoit F resse for ideas and useful and clarifying discussions and to Martin Ma rkl for discussions a bo ut his cohomolog y theo ry for homotopy as s o ciative algebr as and for many relev an t remarks on the ﬁrst version of the pap er. I am grateful to Henrik Stro hmay er for his car eful reading of my pap er a nd for all his corr ections and to Vladimir Dotsenko for p ointing a mistake in a previous version of Theorem 5.6 .1. I a lso wish to thank the referee for his n umerous comment s. References [And74] M. Andr´ e. Homolo gi e des alg` e b r es c ommu tatives . Springer-V erl ag, Berlin, 1974. [Bal98] D. Balav oine. H omol ogy and cohomology with co eﬃcien ts, of an algebra ov er a quadratic operad. J. Pur e Appl. 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