Gerstenhaber-Schack diagram cohomology from operadic point of view

Gerstenhab er-Sc hac k diagram cohomology from op eradic p oin t of view Martin Doub ek ∗ , Charl es Univ ersity , Pragu e mart indo ubek@ seznam.cz April 25, 202 2 Abstract W e sho w that the op eradic cohomology for an y t yp e of algebras o v er a non-symmetric op erad A can b e computed as Ex t in the category of op eradic A -mo dules. W e use this principle to pro ve that the Gerstenhab er-S c hac k d iagram cohomology is op eradic co- homology . 1 In tro duction The O p eradic Cohomology (OC) giv es a systematic wa y of constru cting cohomology theories for algebras A o ve r an op erad A . It reco v ers the classical cases: Ho c hsc hild, Chev alle y-Eilenberg, Harrison etc. It also applies to algebras o v er coloured op erads (e.g. morphism of algebras) and o v er PR OPs (e.g. bialgebras). T he OC ﬁrst app eared in pap ers [9], [8] by M. Markl. Abstractly , the OC is isomorp hic to the triple cohomolog y , at least for algebras ov er Koszul op erads [1]. It is also isomorph ic to the Andr´ e-Quillen Cohomolog y (AQC). In fact, the deﬁnition of OC is analogous to that of A QC: It co mpu tes the deriv ed functor of the functor Der of deriv atio ns lik e A QC, but does so in the cate gory of op erads. While AQC oﬀers a wider f r eedom f or the c hoice of a r esolution of the giv en algebra A , OC uses a particular universal resolution f or all A -algebras (this resolution is implicit, tec hnically O C r esolv es the op er ad A ). Th us there is, for example, a u n iv ersal construction of an L ∞ structure on the complex computing OC [12], whose generalized Maurer-Cartan equation describ es formal deformations of A . The success of O C is du e to the Koszul dualit y theory [6], whic h allo ws u s to construct resolutions of Koszul op erads explicitly . Koszul theory has r eceiv ed a lot of atten tion recen tly [17] an d n o w goes b ey ond op erads . Ho w ev er, it still has its limitations: On one hand , it is b ound to quadratic relations in a present ation of the op erad A . The problem with higher relations can b e remedied by u sing a diﬀerent pr esentati on, ∗ The author was supp or ted by GA ˇ CR 201/0 9/H012 . 1 but it comes at the cost of increasing the size of the r esolution (e.g. [18]). This is not a ma jor problem in applications, b ut the minimal resolutions hav e some n ice prop erties - namely they are unique up to an isomorph ism th us p ro viding a cohomology theory unique already at th e c hain lev el. So the construction of the minimal r esolutions is still of in terest. On the other h and, there are qu adratic op erads whic h are not Koszul and for those v ery little is kno wn [14]. In this pap er, we show that OC is isomorp hic to Ext in the category of op eradic A -mo dules. Th us ins tead o f resolving the op erad A , it suﬃces to ﬁ nd a pro jectiv e resolution of a sp eciﬁc A -mo dule MD A asso ciated to A . The id eas u sed h ere were already sk etc hed in th e pap er [12] by M. Markl. The resolution of op eradic mo du les are probably muc h easier to construct explicitly than resolutions of op erads, though this has to b e explored ye t. This sim p liﬁcation allo w s us to make a small step b eyond Koszul theory: An interesting examp le of a non-Koszul op erad is the coloured op erad describ ing a diagram of a ﬁxed shap e consisting of algebras ov er a ﬁxed op erad and m orphisms of those algebras. The case of a single morp h ism b et ween tw o algebras o v er a K oszul op erad is long wel l u ndersto o d . F or a m orp hism b et we en algebras o ve r a general op erad as well as for diagrams of a few simp le shap es, some partial results w ere ob tained in [11]. Th ese are h o w ev er not explicit enough to write do wn the O C. On the other hand, a satisfactory cohomology for diagrams w as inv ented by Ger- stenhab er and Sc hac k [4] in an ad-hoc manner. In [2], the authors pro v ed that the Gerstenhab er-Schac k cohomology of a single morphism of asso ciativ e or Lie algebras is op eradic cohomology . W e use our theory to extend this r esult to arb itrary diagrams. The metho d u sed can pr obably b e app lied in a more general con text to sho w that a gi ven co homology theory is isomorph ic to OC. The original example is [12] (and similar app roac h also a pp ears in [16]), wher e the author pro v es that Gerstenhab er- Sc hac k bialgebr a cohomol ogy is the op eradic cohomology . Also the metho d m ight give an insigh t in to th e stru cture of op eradic resolutions th ems elves, the problem we w on’t men tion in this p ap er. On th e wa y , w e obtain a mo d iﬁcation of the usual O C wh ic h includes the quotient b y inﬁ nitesimal automorph isms (Section 3.3). Also an explicit description of a free resolution of the op erad A with adjoine d derivation is giv en if a free r esolution of A is explicitly give n. This app eared already in [12] an d pro duces sev eral n ew examples of m in imal resolutions and as such migh t b e of an indep enden t interest. W e assume the reader is familiar with the language of op erads (e.g. [15],[7]). Finally , I w ould lik e to thank Martin Markl for many usefu l d iscussions. 1.1 C on v en tion. As our main ob ject of in terest is a diagram of asso ciativ e algebras, w e will get b y with non-symmetric op erads, that is op erads with no actio n of the p ermutatio n group s . The resu lts can p robably b e generalized in a straigh tforward w a y to symmetric op erads. 2 Con ten ts 1 In tro duction 1 2 Basics 4 2.1 Op eradic mo dules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 T ree comp osition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 F ree pro d uct of op erads . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Op eradic cohomology of algebras 13 3.1 Reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Algebras with deriv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Augmen ted cotangen t complex . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 In termediate resolution of D A . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 F rom op erads to op eradic m o dules . . . . . . . . . . . . . . . . . . . . . 26 4 Gerstenhab er-Sc hack diagram cohomology is op eradic cohomology 28 4.1 Op erad for diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Gerstenhab er-Schac k d iagram cohomolog y . . . . . . . . . . . . . . . . . 29 4.3 Resolution of MD A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 In Section 2 , we brieﬂy recall b asic notions of the op erad th eory with focus on c olour e d op erad s (see also [10],[11 ]). W e pa y sp ecial atte ntio n to op eradic mo dules. W e in tro du ce the notion of tree comp osition whic h is just a conv enient wa y to write do wn complicat ed op eradic comp ositions. In Sectio n 2.3, w e discuss free pro d uct of op erads and obtain a form of the K ¨ unneth f orm ula compu ting homology of the free pro du ct. In Section 3 , we dev elop the theory sk etc hed b y M. Markl in App endix B of [12]. W e give full details for coloured op erads. W e b egin b y recalling the op eradic cohomology . In Section 3.2 , we constru ct an explicit resolution of the operad D A describing algebras ov er A with adjoined deriv ation assuming we kno w an explicit resolution of the op erad A . In Section 3.3, we clarify the signiﬁcance of op eradic deriv ations on the resolution of D A with v alues in E nd A . Th is leads to an augmen tation of th e cotangen t complex whic h has n ice inte rp retation in terms of formal deformation theory . I n S ections 3.4 and 3.5, w e realize that all the information needed to construct augmen ted cohomology is con tained in a certain op eradic mo dule . This mo dule is in trinsically characte rized by b eing a resolution (in the category of op eradic mo d ules) of MD A , a certain mo d ule constru cted f r om A in a very simp le wa y . In Section 4 , we apply the theory to p ro v e that the Gerstenhab er-Sc hac k diagram cohomology is isomorphic to the operadic cohomolo gy . W e b egin by exp lainin g ho w a diagram of asso ciativ e algebras is describ ed b y an op erad A . W e also mak e the as- so ciated mo dule MD A explicit. Then w e recall the Gerstenhab er-Schac k cohomology and obtain a candidate for a resolution of MD A . In Section 4.3, w e v erify that the candidate is a v alid resolution. This computation is complicated, b ut still d emons trates the tec hnical adv an tage of passing to the mo dules. 3 2 Basics Fix the follo wing sym b ols: • C is a set of colours. • k is a ﬁeld of c haracteristics 0. • N 0 is the set of natural num b ers in clud ing 0. W e will also us e the follo wing n otations and con ve ntio ns: • V ector spaces ov er k are called k -mo dules , c hain complexes of v ector spaces o v er k with diﬀerential of degree − 1 are called dg- k -mo dules and morph isms of c hain complexes are cal led just maps . Chain complexes are assum ed non-negativ ely graded unless stated otherwise. • | x | is the degree of an elemen t x of a dg- k -mo d ule. • H ∗ ( A ) is homology of the ob ject A , whatev er A is. • k h S i is the k -linear sp an of the set S . • ar( v ) is arity of th e ob ject v , wh atev er v is. • Quism is a map f of d g- k -mo dules suc h that th e in duced map H ∗ ( f ) on homology is an isomorphism. 2.1 Deﬁnit ion. A dg- C - collection X is a set  X  c c 1 , . . . , c n  | n ∈ N 0 , c, c 1 , . . . , c n ∈ C  of dg- k -mo dules. W e call c the output colour of elemen ts of X  c c 1 ,...,c n  , c 1 , . . . , c n are the input colours, n is the arity . W e also adm it n = 0. When the ab o v e dg- k -mo du les ha ve ze ro diﬀerential s, w e talk just ab ou t graded C -collection. If moreo ver no g radin g is giv en, we talk just about C -collecti on. All notions that follo w ha v e similar analogues. If the context is clear, we might omit the preﬁxes dg- C completely . A dg- C -op erad A is a dg- C -collection A together with a set of of dg- k -mo du le maps ◦ i : A  c c 1 , . . . , c k  ⊗ A  c i d 1 , · · · , d l  → A  c c 1 , · · · , c i − 1 , d 1 , · · · , d l , c i +1 , · · · c k  , called op er adic c omp ositions , one for eac h c hoice of k , l ∈ N 0 , 1 ≤ i ≤ k and c, c 1 , . . . , c k , d 1 , . . . , d l ∈ C , and a set of un its e : k → A  c c  , one for eac h c ∈ C . These maps satisfy the usual asso ciativit y and unit axioms, e.g. [15]. The initial dg- C -op erad is denoted I . 4 Equiv alentl y , d g- C -op erad is a monoid in the monoidal catego ry of dg- C -collections with the comp osition pro duct ◦ : ( A ◦ B )  c c 1 , . . . , c n  := M k ≥ 0 , i 1 ,...,i k ≥ 0 , d 1 ,...,d k ∈ C A  c d 1 , . . . , d k  ⊗ B  d 1 c 1 , . . . , c i 1  ⊗ · · · ⊗ B  d k c i 1 + ··· i k − 1 +1 , . . . , c n  . In contrast to the uncoloured op erads, the comp osition is deﬁn ed only for the “correct” colours and there is one un it in eve ry colour, i.e. I  c c  = k for eve ry c ∈ C . Hence we usu ally talk ab out the un its . W e denote b y 1 c the image of 1 ∈ k = I  c c  , hence Im e = L c ∈ C k h 1 c i . The notation 1 c for units coincides with th e notation for iden tit y morphisms. The right meaning will alwa ys b e clear from the conte xt. F or a dg- C -op erad A , we can consid er its homology H ∗ ( A ). Th e op eradic comp o- sition d escends to H ∗ ( A ). Obviously , the un its 1 c are concent rated in degree 0 and by our conv ention on non-negativit y of the grading, 1 c deﬁnes a homology class [ 1 c ]. It is a unit in H ∗ ( A ). It can happ en th at [ 1 c ] = 0 in which case it is easily seen that H ∗ ( A )  c 0 c 1 ,...,c n  = 0 whenev er an y of c i ’s equals c . If all [ 1 c ]’s are nonzero, then H ∗ ( A ) is a graded C -operad. Let M 1 and M 2 b e t wo dg- C -col lections. Then dg- C -collection morphism f is a set of dg maps f  c c 1 , . . . , c n  : M 1  c c 1 , . . . , c n  → M 2  c c 1 , . . . , c n  , one for eac h n ∈ N 0 , c, c 1 , . . . , c n ∈ C . The dg- C -collection m orphisms are comp osed “colourwise” in the obvious wa y . A dg- C -op erad morphism is a d g- C -colle ction morphisms preserving the op - eradic comp ositions and un its. Recall th at give n a dg- k -mo d ule A , the endomorp hism op er ad E nd A is equipp ed with the diﬀerenti al ∂ E nd A f := ∂ A f − ( − 1) | f | f ∂ A ⊗ n for f ∈ E nd A ( n ) homogeneous. Let th ere b e a d ecomp osition ( A, ∂ A ) = M c ∈ C ( A c , ∂ A c ) . Then the endomorphism op erad is naturally a d g- C -op er ad via E nd A  c c 1 , . . . , c n  := Hom k ( A c 1 ⊕ · · · ⊕ A c n , A c ) . An algebra ov er a dg- C -operad A is a dg- C -op erad morphism ( A , ∂ A ) → ( E nd A , ∂ E nd A ) . 5 2.1 Op eradic mo dules 2.2 Deﬁnit ion. Let A = ( A , ∂ A ) b e a dg- C -op erad. An (op eradic) dg- A -mo dule M is a dg- C -c ollection  M  c c 1 , . . . , c n  | n ∈ N 0 , c, c 1 , . . . , c n ∈ C  with structure maps ◦ L i : A  c c 1 , · · · , c k  ⊗ M  c i d 1 , · · · , d l  → M  c c 1 , · · · , c i − 1 , d 1 , · · · , d l , c i +1 , · · · c k  , ◦ R i : M  c c 1 , · · · , c k  ⊗ A  c i d 1 , · · · , d l  → M  c c 1 , · · · , c i − 1 , d 1 , · · · , d l , c i +1 , · · · c k  , one for eac h c hoice of c, c 1 , · · · , d 1 , · · · ∈ C and 1 ≤ i ≤ k . T hese structure maps are required to satisfy the exp ected axioms: ( α 1 ◦ j α 2 ) ◦ L i m =      ( − 1) | α 2 || m | ( α 1 ◦ L i m ) ◦ R j +ar( m ) − 1 α 2 . . . i < j α 1 ◦ L j ( α 2 ◦ L i − j +1 m ) . . . j ≤ i ≤ j + ar( α 2 ) − 1 ( − 1) | α 2 || m | ( α 1 ◦ L i − ar( α 2 )+1 ) ◦ R j α 2 . . . i ≥ j + ar( α 2 ) , m ◦ R i ( α 1 ◦ j α 2 ) = ( m ◦ R i α 1 ) ◦ R j + i − 1 α 2 , ( α 1 ◦ L i m ) ◦ R j α 2 =      ( − 1) | α 2 || m | ( α 1 ◦ j α 2 ) ◦ L i +ar( α 2 ) − 1 m · · · j < i α 1 ◦ L i ( m ◦ R j − i +1 α 2 ) · · · i ≤ j ≤ i + ar( m ) − 1 ( − 1) | α 2 || m | ( α 1 ◦ j − ar( m )+1 α 2 ) ◦ L i m · · · j ≥ i + ar( m ) and 1 c ◦ 1 m = m = m ◦ i 1 c i . . . 1 ≤ i ≤ ar( m ) for α 1 , α 2 ∈ A and m ∈ M  c c 1 , ··· ,c ar( m )  in the correct colours. W e usually omit the upp er indices L, R , w riting on ly ◦ i for all the op erations. A morphism of dg- A -modules M 1 , M 2 is a d g- C -colle ction m orp hism M 1 f − → M 2 satisfying f ( a ◦ L i m ) = a ◦ L i f ( m ) , f ( m ◦ R i a ) = f ( m ) ◦ R i a. W e expand the d eﬁnition of dg- A -mo dule. Recall that eac h M  c c 1 ,...,c n  is a a dg- k -mo dule. The diﬀeren tials of these dg- k -mo du les deﬁne a dg- C -collection morp hism ∂ M : M → M of degree − 1 sati sfying ∂ 2 M = 0. The structure m ap s ◦ L i and ◦ R i comm ute w ith the d iﬀeren tials on the tensor pro du cts. Hence ∂ M : M → M is a deriv ation in the f ollo w in g sense: ∂ M ( a ◦ i m ) = ∂ A a ◦ i m + ( − 1) | a | a ◦ i ∂ M m, ∂ M ( m ◦ i a ) = ∂ M m ◦ i a + ( − 1) | m | m ◦ i ∂ A a. As in the case of mo du les o v er a ring, A -mo dules form an ab elian category . W e hav e “colourwise” ke rn els, coke rn els, sub m o dules etc. There is a fr ee A -mo du le generated b y a C -col lection M , denoted A h M i 6 and satisfying the u sual universal p rop erty . As an example of an explicit description of A h M i , let A := F ( M 1 ) b e a free C -o p erad generated b y a C -collection M 1 . Then A h M 2 i is spanned b y all planar trees, wh ose exactl y o ne verte x is d ecorated by an elemen t of M 2 and all the other v ertices are decorated by elemen ts of M 1 suc h that the colours are resp ected in the obvio us sense. W e wa rn the reader that the notion of op eradic mo du le v aries in th e literature. F or example the monograph [3] uses a diﬀerent deﬁ nition. While dealing with A -mo dules, it is useful to in tro du ce the follo wing inﬁnitesimal comp osition pro duct A ◦ ′ ( B , C ) of C -collec tions A , B , C : ( A ◦ ′ ( B , C ))  c c 1 , . . . , c n  := (1) M k ≥ 0 ,l> 0 , 0 ≤ i 1 ≤ ... ≤ i k − 1 ≤ n, d 1 ,...,d k ∈ C A  c d 1 ,..., d k  ⊗ B  d 1 c 1 ,..., c i 1  ⊗ · · · ⊗ C  d l c i l − 1 +1 ,..., c i l  | {z } l th − posi tion ⊗ · · · ⊗B  d k c i k − 1 +1 ,..., c n  . See also [7]. W e denote by A ◦ ′ l ( B , C ) the pro jection of A ◦ ′ ( B , C ) onto the comp onent w ith ﬁ xed l . F or the free mo dule, we hav e the follo w ing d escription using the inﬁ nitesimal com- p osition pr o duct: A h M i ∼ = A ◦ ′ ( I , M ◦ A ) . 2.2 T ree comp osition An (un orien ted) gr aph (without lo ops) is a set V of vertic es, a set H v of half edges for ev ery v ∈ V and a set E of (distinct) unordered pairs (called e dges ) of distinct elemen ts of V . If e := ( v , w ) ∈ E , w e sa y that the vertic es v , w are adjacen t to the edge e and the edge e is adjacen t to th e vertic es v , w . Denote E v the set of all edges adjacen t to v . Similarly , f or h ∈ H v , w e sa y that th e v ertex v is adjacen t to the half edge h and vice v ersa. A p ath connecting vertice s v , w is a sequence ( v , v 1 ) , ( v 1 , v 2 ) , . . . , ( v n , w ) of distinct edges. A tr e e is a graph suc h that for ev ery t wo vertic es v , w there is a path connecting them iﬀ v 6 = w . A r o ote d tr ee is a tree with a c hosen half edge, called r o ot . The r o ot vertex is the un ique v ertex adjacent to the ro ot. The half edges other than the ro ot are called le aves . F or eve ry vertex v except for the ro ot v ertex, there is a unique edge e v ∈ E v con tained in the unique p ath connecting v to the ro ot ve rtex. T h e edge e v is called output and the other edges and h alf edges adj acen t to v are called le gs or inputs of v . Th e ro ot is, b y deﬁnition, the outpu t of the root v ertex. The n umber of legs of v is called arity of v and is denoted ar( v ) . Notice we also admit v ertices w ith no legs, i.e. vertic es of arit y 0. A planar tree is a tree with a given ordering of th e set H v ⊔ E v − { e v } for eac h v ∈ V (the notatio n ⊔ stands for the disjoint union). The planarit y induces an ordering on the s et of all leav es, e.g. by num b erin g them 1 , 2 , . . . . F or example, ro ot 1 2 1 2 1 2 7 is a planar tree w ith 3 v ertices, 3 half edges and 2 edges. W e use th e con v ent ion that the topm ost half edge is alw a ys the ro ot. Th en there are 2 lea ve s. The planar ordering of legs of all v ertices is denoted b y small n umb ers an d the induced ordering of lea ve s is denoted by big num b ers. Let T 1 , T 2 b e tw o planar ro oted tr ees, let T 1 ha v e n lea v es and for j = 1 , 2 let V j resp. H j,v resp. E j denote the set of vertices resp. half edges resp. edges of T j . F or 1 ≤ i ≤ n we hav e the grafting op eration ◦ i pro du cing a planar r o oted tree T 1 ◦ i T 2 deﬁned as follo ws: ﬁrst denote l the i th leg of T 1 and denote v 1 the vertex adjacent to l , den ote r the ro ot of T 2 and denote v 2 the ro ot v ertex of T 2 . Then the set of v ertices of T 1 ◦ i T 2 is V 1 ⊔ V 2 , th e set H v of h alf edges is H v =        H 1 ,v . . . v ∈ V 1 − { v 1 } H 1 ,v 1 − { l } . . . v = v 1 H 2 ,v . . . v ∈ V 2 − { v 2 } H 2 ,v 2 − { r } . . . v = v 2 , and ﬁnally th e set of edges is E 1 ⊔ E 2 ⊔ { ( v 1 , v 2 ) } . The planar s tr ucture is inh erited in the ob vious wa y . F or example, 1 2 1 2 1 2 ◦ 2 1 2 1 2 = 1 2 1 2 1 1 2 3 2 . F rom th is p oin t on, tr e e will alw a ys mean a planar rooted tree. Suc h trees can b e used to enco de comp ositio ns of elemen ts of an op erad includ ing th ose of arit y 0. Let T b e a tree with n vertice s v 1 , . . . , v n . Su pp ose moreo v er that the vertices of T are ordered, i.e. there is a bijection b : { v 1 , . . . , v n } → { 1 , . . . , n } . W e denote su c h a tree with ordered v ertices by T b . No w we exp lain how the bijection b induces a structure of tree with leve ls on T b suc h that eac h vertex is on a diﬀeren t lev el. Intuitiv ely , b en co d es in wh at or der are elemen ts of an op erad comp osed. W e form alize this as follo ws: Let p 1 , . . . , p n ∈ P be elemen ts of a d g- C -op erad P such that if t w o v ertices v i , v j are adjacen t to a common edge e , which is simultaneously the l th leg of v i and the output of v j , then the l th input colour of p i equals the output colour of p j . W e sa y that v i is de c or ate d by p i . Deﬁne inductiv ely: Let i b e su c h that v i is th e ro ot ve rtex. Deﬁne T 1 := v i , T 1 b ( p 1 , . . . , p n ) := p i . Here w e are identifying v i with the corresp ond ing corolla. Assume a subtree T k − 1 of T and T k − 1 b ( p 1 , . . . , p n ) ∈ P are already deﬁned . Consider th e set J of all j ’s such that v j 6∈ T k − 1 and there is an edge e b et ween v j and some v ertex v in T k − 1 . Let i ∈ J b e suc h that b ( v i ) = min { b ( v j ) : j ∈ J } . Let l b e th e num b er of the leg e of v ertex v in the planar ordering of T and deﬁne T k := T k − 1 ◦ l v i , T k b ( p 1 , . . . , p n ) := T k − 1 b ( p 1 , . . . , p n ) ◦ l p i . 8 In the upp er equation, w e are u sing the op eration ◦ l of grafting of trees. Fi nally T b ( p 1 , . . . , p n ) := T n b ( p 1 , . . . , p n ) . T b ( p 1 , . . . , p n ) is called tree comp osition of p 1 , . . . , p n along T b . If T and p i ’s are ﬁxed, c hanging b ma y change the sign of T b ( p 1 , . . . , p n ). Observe that if P is concent rated in ev en d egrees (in particular 0) then the sign d o esn’t c hange. If b is under s to o d and ﬁxed, we usu ally omit it. F or example, let T := v 1 v 2 v 3 F or i = 1 , 2 , 3, let p i b e an elemen t of degree 1 and arity 2. Let b ( v 1 ) = 1, b ( v 2 ) = 2, b ( v 3 ) = 3 an d b ′ ( v 1 ) = 1, b ′ ( v 2 ) = 3, b ( v 3 ) = 2. Then T b ( p 1 , p 2 , p 3 ) = ( p 1 ◦ 1 p 2 ) ◦ 3 p 3 and T b ′ ( p 1 , p 2 , p 3 ) = ( p 1 ◦ 2 p 3 ) ◦ 1 p 2 and b y the asso ciativit y axiom T b ( p 1 , p 2 , p 3 ) = − T b ′ ( p 1 , p 2 , p 3 ) . A u seful observ ation is that we can alwa ys reindex p i ’s s o that T b ( p 1 , . . . , p n ) = ( · · · (( p 1 ◦ i 1 p 2 ) ◦ i 2 p 3 ) · · · ◦ i n − 1 p n ) (2) for some i 1 , i 2 , . . . , i n − 1 . T ree comp ositions are a conv enient n otation for dealing with op eradic d er iv ations. 2.3 F ree pro duct of op erads 2.3 Deﬁnit ion. F ree product A ∗ B of dg- C -op erads A , B is the co pr o duct A ` B in the category of dg- C -operads. Let A, B b e d g- k -mo dules. The usu al K ¨ unneth formula states th at th e map H ∗ ( A ) ⊗ H ∗ ( B ) ι − → H ∗ ( A ⊗ B ) (3) [ a ] ⊗ [ b ] 7→ [ a ⊗ b ] is a natur al isomorphism of dg- k -mo dules, wh er e [ ] denotes a homology class. Our aim here is to pro ve an analogue of the K ¨ unneth formula for th e free pro duct of op erads, that is H ∗ ( A ) ∗ H ∗ ( B ) ∼ = H ∗ ( A ∗ B ) (4) naturally as C -op er ad s . First w e describ e A ∗ B m ore exp licitly . In tuitiv ely , A ∗ B is spanned by trees whose v ertices are decorated by elemen ts of A or B s u c h that no t wo v ertices adjacent to a common edge are b oth decorated by A or b oth by B . Unfortunately , this is not quite true - there are problems with un its of the op erads. 9 Recall a dg- C -operad P is called augmente d iﬀ th ere is a dg- C -operad morph ism P a − → I in v erting the unit of P on the left, i.e. the comp osition I e − → P a − → I is 1 I . The k ernel of a is d enoted by P and usually called augmenta tion ideal. If A , B are augmen ted, we let the v ertices b e decorated by the augmen tation ideals A , B instead of A , B and the ab o v e description of A ∗ B works wel l. I n fact, this has b een already treated in [10]. Ho w ev er we will w ork without the augment ation assump tion. Ch o ose a sub - C - collect ion A of A such that A ⊕ 1 A = A , (5) Im ∂ A ⊂ A , (6) where 1 A := M c ∈ C k h 1 c i . This is p ossible iﬀ [ 1 c ] 6 = 0 for all c ∈ C, that is if H ∗ ( A ) is a graded C -o p erad. This m igh t not b e the case generally as we h a v e already seen at th e b eginning of Section 2 so let’s assume it. Choose B for B similarly . F or giv en A , B , a free pro duct tree is a tree T together with c ( v ) , c 1 ( v ) , c 2 ( v ) , . . . , c ar( v ) ( v ) ∈ C for eac h vertex v and a map P : vertic es of T → {A , B } (7) suc h that if vertices v 1 , v 2 are adjacen t to a common edge, whic h is simultaneo usly th e l th leg of v 1 and the output of v 2 , then c l ( v 1 ) = c ( v 2 ) and P ( v 1 ) 6 = P ( v 2 ) . Finally , the description of the free pro d uct is as follo ws: A ∗ B := M c ∈ C k h 1 c i ⊕ M T O v P ( v )  c ( v ) c 1 ( v ) , . . . , c ar( v ) ( v )  , (8) where T runs o v er all isomorphism classes of free pro duct trees and v run s ov er a ll v ertices of T and 1 c ’s are of degree 0. If the v ertices of T are v 1 , . . . , v n then every elemen t of N v P ( v )  c ( v ) c 1 ( v ) ,... ,c ar( v ) ( v )  can b e wr itten as a tree comp osition T ( x 1 , . . . , x n ) where x i ∈ P ( v i ). W e say that v i is d ecorated by x i . The op eradic comp osition T ( x 1 , . . . , x n ) ◦ i T ′ ( x ′ 1 , . . . , x ′ m ) in A ∗ B is deﬁned in the ob vious wa y by grafting T an d T ′ (the result T ◦ i T ′ of the grafting ma y not b e a free pr o duct tree) and th en (rep eatedly) applying the follo wing r e duci ng op er ations : 10 1. Su pp ose w 1 , w 2 are v ertices of T ◦ i T ′ adjacen t to a common edge e which is sim ultaneously the l th leg of w 1 and the output of w 2 . Sup p ose moreo v er that w 1 is decorated by p 1 and w 2 b y p 2 . If b oth p 1 , p 2 are elemen ts of A or b oth of B , then con tract e and d ecorate th e resu lting v ertex by the comp osition of p 1 ◦ l p 2 . 2. If a v ertex is decorated by a unit fr om 1 A or 1 B (this ma y happ en s ince neither A nor B is generally closed under the comp osition!), omit it u n less it is the only remaining v ertex of the tree. After sev eral app lications of the ab o v e reducing op erations, we obtain a f ree pro d u ct tree or a tree with a single v ertex decorate d b y a unit. Ob viously , 1 c ’s are units f or th is comp osition. The diﬀeren tial ∂ on A ∗ B is determined by (8) and the requirement that ∂ ( 1 c ) = 0 for ev ery c ∈ C . It has the deriv ation prop ert y and equals the diﬀeren tial on A resp. B up on the r estriction o n the corresp onding sub- C -op er ad of A ∗ B . Exp licitly , for T ( x 1 , . . . , x n ) ∈ A ∗ B with x i ∈ A or B , assuming (2), we h a v e ∂ ( T ( x 1 , . . . , x n )) = n X i =1 ǫ i T ( x 1 , . . . , ∂ ( x i ) , . . . , x n ) , where ǫ i := ( − 1) P i − 1 j =1 | x j | . It is ea sily seen th at the dg- C -op erad ( A ∗ B , ∂ ) j ust describ ed has the required unive rsal p rop erty of th e copro duct. No w we are prepared to prov e a v ersion of (4) in a certain sp ecial case: 2.4 L e mma. Let ( A , ∂ A ) α − → ( A ′ , ∂ A ′ ) and ( B , ∂ B ) β − → ( B ′ , ∂ B ′ ) b e quism s of dg- C -op erads, that is we assume homology of A , A ′ , B , B ′ are graded C -op er ad s and H ∗ ( α ) , H ∗ ( β ) are g raded C -op erad isomorph isms. Then there are graded C -o p erad isomorphisms ι, ι ′ suc h that the follo wing diagram commutes: H ∗ ( A ) ∗ H ∗ ( B ) ι ✲ H ∗ ( A ∗ B ) H ∗ ( A ′ ) ∗ H ∗ ( B ′ ) H ∗ ( α ) ∗ H ∗ ( β ) ❄ ι ′ ✲ H ∗ ( A ′ ∗ B ′ ) H ∗ ( α ∗ β ) ❄ Pr o of. Cho ose A , B so that (5) and (6) hold. No w we wan t to c ho ose A ′ ⊂ A ′ so that A ′ ⊕ 1 A ′ = A ′ , α ( A ) ⊂ A ′ (9) and c ho ose B ′ ⊂ B ′ similarly . T o see th at th is is p ossible, we obser ve α ( A ) ∩ 1 A ′ = 0: If α ( a ) ∈ 1 A ′ for some a ∈ A , there is u ∈ 1 A suc h that α ( u ) = α ( a ), hen ce α ( a − u ) = 0 and ∂ A ( a − u ) = 0 since b oth a and u are of degree 0. Since α is a quism, a − u = ∂ A a for some a ∈ A and by the prop ert y (6) of A w e ha ve a − u ∈ A . But this im p lies u ∈ A , a con tradiction. 11 No w use the explicit description (8) of the free pro duct A ∗ B and the u sual K ¨ unneth form ula (3) to ob tain an isomorp hism H ∗ ( A ) ∗ H ∗ ( B ) = M T O v H ∗ ( P ( v )) ι − → H ∗ ( M T O v P ( v )) = H ∗ ( A ∗ B ) and similarly for A ′ , B ′ . Assume w e are giv en a f r ee pro duct tree T and its v ertex v . The tree T comes equipp ed with P as in (7). Let P ′ ( v ) := ( P ( v )) ′ = ( A ′ for P ( v ) = A B ′ for P ( v ) = B and deﬁne a map π ( v ) : P ( v ) → P ′ ( v ) , π ( v ) =  α for P ( v ) = A β for P ( v ) = B . This is justiﬁed by (9). Then the f ollo w ing diagram M T O v H ∗ ( P ( v )) ι ✲ H ∗ ( M T O v P ( v )) M T O v H ∗ ( P ′ ( v )) L T N v H ∗ ( π ( v )) ❄ ι ′ ✲ H ∗ ( M T O v P ′ ( v )) H ∗ ( L T N v π ( v )) ❄ comm utes by the naturality of the usu al K ¨ un n eth form ula. The horizon tal C -collection isomorphism ι is giv en in terms of tree comp ositio ns b y the formula ι ( T ([ x 1 ] , [ x 2 ] , . . . )) = [ T ( x 1 , x 2 , . . . )] , where x 1 , x 2 , . . . ∈ A or B . No w we verify th at ι preserves the op eradic comp osition: ι ( T x ([ x 1 ] , . . . )) ◦ i ι ( T y ([ y 1 ] , . . . )) = ι ( T x ([ x 1 ] , . . . ) ◦ i T y ([ y 1 ] , . . . )) . The left-hand side equals [ T x ( x 1 , . . . ) ◦ i T y ( y 1 , . . . )], so w e chec k [ T x ( x 1 , . . . ) ◦ i T y ( y 1 , . . . )] = ι ( T x ([ x 1 ] , . . . ) ◦ i T y ([ y 1 ] , . . . )) . W e wo uld lik e to p erform the same reducing op erations on T x ( x 1 , . . . ) ◦ i T y ( y 1 , . . . ) and T x ([ x 1 ] , . . . ) ◦ i T y ([ y 1 ] , . . . ) p arallely . F or the ﬁ rst r educing op eration, this is OK. F or the second one, if, s a y , [ x 1 ] ∈ 1 H ∗ ( A ) , then x 1 = u + ∂ A a for some u ∈ 1 A and a ∈ A . Hence T x ( x 1 , . . . ) = T x ( u, . . . ) + T x ( ∂ A a, . . . ). So we ca n go on with T x ( u, . . . ) ◦ i T y ( y 1 , . . . ) and T x ([ x 1 ] , . . . ) ◦ i T y ([ y 1 ] , . . . ), ommiting th e verte x v 1 decorated by u resp. [ x 1 ], but w e also hav e to apply the r educing op erations to T x ( ∂ A a, . . . ) ◦ i T y ( y 1 , . . . ). As it turn s out, this tree comp osition is a b oundary in A ∗ B . W e lea v e the d etails to the reader.  12 3 Op e radic coho mology of alg ebras 3.1 Remi nder Let ( R , ∂ R ) ρ − → ( A , ∂ A ) b e d g- C -op erad o v er ( A , ∂ A ), i.e. ρ is a dg- C -op erad morphism. Let ( M , ∂ M ) b e a d g- A -mo dule. Deﬁne a k -mo d ule Der n A ( R , M ) consisting of all C -collec tion morph ism s θ : R → M of degree | θ | = n in all colours satisfying θ ( r 1 ◦ i r 2 ) = θ ( r 1 ) ◦ R i ρ ( r 2 ) + ( − 1) | θ || r 1 | ρ ( r 1 ) ◦ L i θ ( r 2 ) for an y r 1 , r 2 ∈ R and any 1 ≤ i ≤ ar( r 1 ). Denote Der A ( R , M ) := M n ∈ Z Der n A ( R , M ) . F or θ ∈ Der A ( R , M ) homogeneous, let δ θ := θ ∂ R − ( − 1) | θ | ∂ M θ . (10) Extending b y linearit y , the abov e form ula deﬁnes a map δ from the k -mo dule Der A ( R , M ). 3.1 L e mma. δ maps deriv ations to d er iv ations and δ 2 = 0. Pr o of. The degree of δ obvio usly equals − 1 and δ 2 θ = ( θ ∂ R − ( − 1) | θ | ∂ M θ ) ∂ R − ( − 1) | δ θ | ∂ M ( θ ∂ R − ( − 1) | θ | ∂ M θ ) = = θ ∂ 2 R − ( − 1) | θ | ∂ M θ ∂ R − ( − 1) | θ | +1 ∂ M θ ∂ R − ( − 1) | θ | +1+ | θ | +1 ∂ 2 M θ = = 0 . The follo wing computation sh o ws that δ maps deriv ations to d eriv ations: ( δ θ )( r 1 ◦ i r 2 ) = θ  ∂ r 1 ◦ i r 2 + ( − 1) | r 1 | r 1 ◦ i ∂ r 2  + − ( − 1) | θ | ∂  θ r 1 ◦ i ρr 2 + ( − 1) | θ || r 1 | ρr 1 ◦ i θ r 2  = = θ ∂ r 1 ◦ i ρr 2 + ( − 1) | θ | ( | r 1 | +1) ρ∂ r 1 ◦ i θ r 2 + + ( − 1) | r 1 | θ r 1 ◦ i ρ∂ r 2 + ( − 1) ( | θ | +1) | r 1 | ρr 1 ◦ i θ ∂ r 2 + − ( − 1) | θ | ∂ θ r 1 ◦ i ρr 2 − ( − 1) | r 1 | θ r 1 ◦ i ∂ ρr 2 + − ( − 1) | θ | ( | r 1 | +1) ∂ ρr 1 ◦ i θ r 2 − ( − 1) ( | θ | +1) | r 1 | + | θ | ρr 1 ◦ i ∂ θ r 2 = = ( δ θ ) r 1 ◦ i ρr 2 + ( − 1) | δ θ | ·| r 1 | ρr 1 ◦ i ( δ θ ) r 2 where w e hav e ommited the s ubscripts of ∂ .  A p articular example of this construction is ( R , ∂ R ) ∼ − → ρ ( A , ∂ A ) , 13 a coﬁbrant [10] resolution of a dg- C -oper ad A , and M := ( E nd A , ∂ E nd A ) , whic h is a d g- A -mo dule via a d g- C -op erad morp hism ( A , ∂ A ) α − → ( E nd A , ∂ E nd A ) determining an A -algebra structur e on a dg- k -mo d ule ( A, ∂ A ) = L c ∈ C ( A c , ∂ A c ). Let ↑ C denote the susp ension of a graded ob ject C , that is ( ↑ C ) n := C n − 1 . Anal- ogously ↓ denote the desusp ension . 3.2 Deﬁnit ion. ( C ∗ ( A, A ) , δ ) := ↑  Der −∗ (( R , ∂ R ) , E nd A ) , δ  (11) is called op eradic cotangen t complex of the A -algebra A and H ∗ ( A, A ) := H ∗ ( C ∗ ( A, A ) , δ ) is called op eradic cohomology of A -algebra A . The c h ange of grading ∗ 7→ 1 − ∗ is pu rely conv entional . F or example, if A is the op erad for asso ciativ e algebras and R is its minimal r esolution, u nder ou r con v en tion w e reco v er the grading of the Hochsc h ild complex for whic h the bilinear co c hains are of degree 1. 3.2 Algebras with d eriv ation Let A b e a dg- C -op erad. Consider a C -collec tion Φ := k h φ c | c ∈ C i , suc h that φ c is of arit y 1, degree 0 and the input and output colours are b oth c . Let D b e the ideal in A ∗ F (Φ) generated b y all elemen ts φ c ◦ 1 α − n X i =1 α ◦ i φ c i (12) for n ∈ N 0 , c, c 1 , . . . , c n ∈ C and α ∈ A  c c 1 ,...,c n  . Denote D A :=  A ∗ F (Φ) D , ∂ D A  , where ∂ D A is th e deriv ation give n b y the formulas ∂ D A ( a ) := ∂ A ( a ) , ∂ D A ( φ c ) := 0 for a ∈ A and c ∈ C . An alge br a o ver D A is a pair ( A, φ ), where A = L c ∈ C A c is an algebra o v er A and φ is a deriv ation of A in the follo wing sense: φ is a collectio n of d egree 0 dg-maps φ c : A c → A c suc h th at φ c ( α ( a 1 , . . . , a n )) = n X i =1 α ( a 1 , . . . , φ c i ( a i ) , . . . , a n ) 14 for α ∈ A  c c 1 ,...,c n  and a j ∈ A c j , 1 ≤ j ≤ n . Giv en a fr e e resolution R := ( F ( X ) , ∂ R ) ρ R − − → ( A , ∂ A ) , (13) where X is a dg- C -collection, it is surp risingly easy to explicitly construct a free r eso- lution of ( D A , ∂ D A ). Consid er the fr ee graded C -op erad D R := F ( X ⊕ Φ ⊕ X ) , where X := ↑ X . W e denote by x the element ↑ x ∈ X corresp ondin g to x ∈ X . T o describ e the diﬀerent ial, let s : F ( X ) → D R b e a degree +1 deriv ation d etermin ed by s ( x ) := x for x ∈ X . Then deﬁne a degree − 1 deriv ation ∂ D R : D R → D R by ∂ D R ( x ) := ∂ R ( x ) , ∂ D R ( φ c ) := 0 , (14) ∂ D R ( x ) := φ c ◦ 1 x − n X i =0 x ◦ i φ c i − s ( ∂ R x ) . 3.3 C on v en tion. F rom no w on w e w ill assume n ∈ N 0 , c, c 1 , . . . , c n ∈ C , x ∈ X  c c 1 , . . . , c n  whenev er an y of these sym b ols app ears. W e w ill usu ally omit th e lo wer in d ices c and c i ’s f or φ . 3.4 L e mma. ∂ D R 2 = 0. Pr o of. Using the tree comp ositions of Section 2.2, let ∂ R ( x ) = P i T i ( x i 1 , · · · , x in i ). ∂ D R 2 ( x ) = ∂ D R   φ ◦ 1 x − n X j =1 x ◦ j φ − s ( ∂ R ( x ))   = = φ ◦ 1 ∂ D R ( x ) − n X j =1 ∂ D R ( x ) ◦ j φ + − ∂ D R   X i n i X j =1 ǫ ij T i ( x i 1 , . . . , x ij , . . . , x in i )   If we assume (2), then ǫ ij = ( − 1) P j − 1 l =1 | x il | . The last application of ∂ D R on the double 15 sum can b e rewritten as X i n i X j =1 X 1 ≤ k ≤ n i , k 6 = j ˜ ǫ ij k T i ( x i 1 , . . . , ∂ R ( x ik ) , . . . , x ij , . . . , x in i ) + + X i n i X j =1 T i ( x i 1 , . . . , φ ◦ 1 x ij , . . . , x in i ) + − X i n i X j =1 ar( x ij ) X k =1 T i ( x i 1 , . . . , x ij ◦ k φ, . . . , x in i ) + − X i n i X j =1 T i ( x i 1 , . . . , s ( ∂ R ( x ij )) , . . . , x in i ) , where ˜ ǫ ij k = ǫ ij ǫ ik if k < j and ˜ ǫ ij k = − ǫ ij ǫ ik if k > j . The s econd an d third lines sum to X i φ ◦ 1 T i ( x i 1 , . . . , x in i ) − X i ar( T i ) X j =1 T i ( x i 1 , . . . , x in i ) ◦ j φ = = φ ◦ 1 ∂ D R ( x ) − n X j =1 ∂ D R ( x ) ◦ j φ, while the ﬁrst and last rows sum to − s∂ D R X i T i ( x i 1 , · · · , x in i ) ! = − s∂ D R 2 = 0 and this concludes the computation.  F rom n o w on, w e will r efer by D R also to the dg - C -op erad ( D R , ∂ D R ). Deﬁn e a C -op erad morp hism ρ D R : D R → D A by ρ D R ( x ) := ρ R ( x ) , ρ D R ( φ c ) := φ c , ρ D R ( x ) := 0 . 3.5 T heorem. ρ D R is a free r esolution of D A . 3.6 E xample. Let’s see what w e get for A := A ss = F ( µ ) / ( µ ◦ 1 µ − µ ◦ 2 µ ) and its minimal resolution (see e.g. [11]) R := A ss ∞ = ( F ( X ) , ∂ R ) ρ R − − → ( A ss , 0), wh ere X = k  x 2 , x 3 , . . .  is the collectio n spann ed by x n in arit y n and degree | x n | = n − 2 and ∂ R is a deriv ation diﬀeren tial give n b y ∂ R ( x n ) := X i + j = n +1 i X k =1 ( − 1) i +( k + 1)( j +1) x i ◦ k x j 16 and the quism ρ R : R = A ss ∞ → A ss = A is giv en b y ρ R ( x 2 ) := µ, ρ R ( x n ) := 0 for n ≥ 3 . Then the asso ciated op erad with d eriv ation is D A := A ss ∗ F (Φ) ( φ ◦ µ − µ ◦ 1 φ − µ ◦ 2 φ ) , where Φ := k h φ i with φ a generator of arit y 1. Its free resolution is D R := ( F ( X ⊕ Φ ⊕ X ) , ∂ D R ) ρ D R − − − → ( D A , 0) , where the diﬀerentia l ∂ D R is given b y ∂ D R ( x ) := ∂ R ( x ) , ∂ D R ( φ ) := 0 , ∂ D R ( x n ) := φ ◦ 1 x n − n X i =1 x n ◦ i φ − X i + j = n +1 i X k =1 ( − 1) i +( k + 1)( j +1) ( x i ◦ k x j + ( − 1) i x i ◦ k x j ) and the quism ρ D R b y ρ D R ( x ) := ρ R ( x ) , ρ D R ( φ ) := φ, ρ D R ( x ) = 0 . Pr o of (of The or em 3.5). Ob viously ρ D R has degree 0 and comm utes w ith diﬀerential s b ecause of the r elations in DA . Let’s abbreviate ∂ D R =: ∂ . Firs t we wan t to use a sp ectral s equ ence to s plit ∂ suc h that ∂ 0 , the 0 th page part of ∂ , is non trivial only on the generators from X . Let’s put an additional grad in g gr on the C -collec tion X ⊕ Φ ⊕ X of generators: gr( x ) := | x | , gr( φ ) := 1 , gr( x ) := | x | . This in duces a grad in g on D R determined b y th e requirement that the comp osition is of gr degree 0. Let F p := p M i =0 { z ∈ D R | gr( z ) = i } . Ob viously ∂ D R F p ⊂ F p . Consider the sp ectral sequence E ∗ asso ciated to the ﬁltration 0 ֒ → F 0 ֒ → F 1 ֒ → · · · of D R . O n DA we hav e the trivial ﬁltration 0 ֒ → D A and the asso ciated sp ectral sequence E ′∗ . W e will sho w that ρ D R induces quism ( E 1 , ∂ 1 ) ∼ − → ( E ′ 1 , ∂ ′ 1 ). Th en w e can use the comparison theorem since b oth ﬁ ltrations are ob viously b ounded b elo w and exhaustiv e (e.g. [19 ], p age 126, T h eorem 5.2.12, and page 135, Theorem 5.5.1). 17 Then the 0 th page satisﬁes E 0 ∼ = F ( X ⊕ Φ ⊕ X ) and is equip p ed w ith the deriv ation diﬀeren tial ∂ 0 : ∂ 0 ( x ) = 0 = ∂ 0 ( φ c ) , ∂ 0 ( x ) = φ c ◦ 1 x − ar( x ) X i =1 x ◦ i φ c i . Denote b y D the ideal in F ( X ⊕ Φ) generated b y φ c ◦ 1 x − ar( x ) X i =1 x ◦ i φ c i (15) for all x ∈ X  c c 1 ,...,c ar( x )  of arb itrary colours. 3.7 Sublemma. H ∗ ( E 0 , ∂ 0 ) ∼ = F ( X ⊕ Φ) D Once this sublemma is prov ed, ∂ 1 on E 1 ∼ = H ∗ ( E 0 , ∂ 0 ) ∼ = F ( X ⊕ Φ) D will b e given by ∂ 1 ( x ) = ∂ R ( x ) , ∂ 1 ( φ c ) = 0 . (16) W e immed iately see that E ′ 1 ∼ = D A and it is equipp ed with the diﬀerenti al ∂ ′ 1 = ∂ D A . T o see that ρ D R 1 : E 1 → E ′ 1 induced b y ρ D R is a qu ism, observe that we can use the r elations (12) in D A to ”mo v e all the φ ’s to the b ottom of the tree comp ositions”, hence, denoting Φ ′ := F (Φ) , w e hav e D A ∼ = A ◦ Φ ′ . The comp osition and the diﬀerent ial on A ◦ Φ ′ are trans f erred along this isomorphism from D A . S imilarly , F ( X ⊕ Φ) D ∼ = F ( X ) ◦ Φ ′ . (17) Under these quisms ρ D R 1 b ecomes ρ R ◦ 1 Φ ′ . (18) It remains to use the usu al K ¨ unneth formula (3) to ﬁ nish the p ro of. Pr o of (of Sublemma 3.7). Denote φ m c := φ c ◦ 1 · · · ◦ 1 φ c the m -fold comp osition of φ c . Let D R 0 := F ( X ⊕ Φ) and, for n ≥ 0, let D R n +1 ⊂ D R b e spann ed by element s φ m c ◦ 1 x ◦ ( x 1 , . . . , x ar( x ) ) and φ m c ◦ 1 x ◦ ( x 1 , . . . , x ar( x ) ) 18 for all x ∈ X  c c 1 ,...,c ar( x )  , m ≥ 0 , x i ∈ D R n  c i ···  , 1 ≤ i ≤ ar( x ). In other wo rds , D R n +1 = Φ ′ ◦ ( X ⊕ X ) ◦ DR n . D R n is ob viously closed u nder ∂ 0 and D R 0 ֒ → D R 1 ֒ → · · · → colim n D R n ∼ = D R , where the colimit is tak en in the category of dg- C -colle ctions. Before we go furth er, w e m ust make a short notatio nal digression. Let T ( g 1 , . . . , g m ) b e a tree comp osition with g i ∈ X ⊔ Φ ⊔ X for 1 ≤ i ≤ m . Recall the tree T has vertices v 1 , . . . , v m decorated b y g 1 , . . . , g m (in that order). W e say that g j is in depth d in T ( g 1 , . . . , g j , . . . , g m ) iﬀ th e sh ortest p ath from v j to the root ve rtex passes through exactly d v ertices (includ ing v j and the ro ot v ertex) d ecorated by elemen ts of X ⊔ X . As an example, consider g 1 g 2 g 3 g 4 If g 1 , g 3 , g 4 ∈ X and g 2 ∈ Φ, then g 1 , g 2 are in d epth 1 and g 3 , g 4 are in depth 2. Using the notion of depth, the d eﬁ nition of D R n can b e rephrased as follo ws: D R n is sp anned by T ( g 1 , . . . , g m ) with g i ∈ X ⊔ Φ ⊔ X , 1 ≤ i ≤ m , suc h that if g j ∈ X for some j , then g j is in d epth ≤ n in T ( g 1 , . . . , g m ). Consider the quotien t Q n of F ( X ⊕ Φ) by the id eal generated by elemen ts T ( g 1 , . . . , g j − 1 , φ c ◦ 1 x j − ar( x j ) X i =0 x j ◦ i φ c i , g j +1 , . . . , g m ) for any tree T , an y g 1 , . . . , g j − 1 , g j +1 , . . . , g m ∈ X ⊔ Φ and any x j ∈ X in depth ≤ n in T ( g 1 , . . . , g j − 1 , x j , g j +1 , . . . , g m ). Th ere are obvio us pro jections F ( X ⊕ Φ) = Q 0 ։ Q 1 ։ · · · → colim n Q n ∼ = F ( X ⊕ Φ) D . T o see the la st isomorphism, observ e that w e ca n us e the relatio ns d eﬁning Q n to ”mo v e” the φ c ’s in tree comp ositions so that th ey are all in depth ≥ n or in p ositions suc h that their in puts are lea ves, then use (17). F or example, consider the follo wing computation in Q 2 , where the b lac k v ertices are decorated by X and white vertice s b y Φ: = + = = + 2 + + + . Notice that we can’t get the white v ertices any deep er in Q 2 . 19 In p articular, Q n +1 ∼ = X ◦ Q n . (19) Ob viously H ∗ ( D R 0 , ∂ 0 ) ∼ = Q 0 and w e claim that H ∗ ( D R n , ∂ 0 ) ∼ = Q n for n ≥ 1. Supp ose th e claim holds for n and we pr o v e it for n + 1. The idea is to use a sp ectral sequence to get rid of the last su m in the formula ∂ 0 ( φ m c ◦ 1 x ◦ ( x 1 , . . . , x ar( x ) )) = φ m +1 c ◦ 1 x ◦ ( x 1 , . . . , x ar( x ) ) + − ar( x ) X i =1 φ m ◦ 1 x ◦ i φ ◦ ( x 1 , . . . , x ar( x ) ) + +( − 1) | x | ar( x ) X i =1 ( − 1) P i − 1 j =1 | x j | φ m ◦ 1 x ◦ ( x 1 , . . . , ∂ 0 ( x i ) , . . . , x ar( x ) ) . Consider the sp ectral sequence E 0 ∗ on D R n +1 asso ciated to the ﬁltration 0 ֒ → G 0 ֒ → G 1 ֒ → · · · ֒ → D R n , where G k is s panned b y φ m ◦ 1 g ◦ ( x 1 , . . . , x ar( x ) ) for all m ≥ 0, g ∈ X ⊕ X , x i ∈ D R n and P ar( x ) i =1 | x i | ≤ k . Obvio usly ∂ 0 : G k → G k . W e will use the comparison theorem for the obvious pr o jection D R n +1 pr − → Q n +1 . W e consider the zero d iﬀerential on Q n +1 . It is easily seen that pr ∂ 0 = 0, hen ce pr is dg- C -collection morphism. W e equ ip Q n +1 with th e trivial ﬁ ltration 0 ֒ → Q n +1 and consider the asso ciated sp ectral sequ ence E ′ 0 ∗ . Again, b oth ﬁltrations are b ound ed b elo w and exh austiv e. On th e 0 th page E 00 ∼ = D R n +1 , the diﬀeren tial ∂ 00 has the desired form: ∂ 00 ( φ m ◦ 1 x ◦ ( x 1 , . . . , x ar( x ) )) = φ m +1 c ◦ 1 x ◦ ( x 1 , . . . , x ar( x ) ) − ar( x ) X i =1 φ m ◦ 1 x ◦ i φ ◦ ( x 1 , . . . , x ar( x ) ) and ∂ 00 is zero on other elemen ts. F or this diﬀerential ∂ 00 , it is (at last!) clear h o w its k ernel lo oks (compare to ∂ 0 ), namely Ker ∂ 00 = F ( X ⊕ Φ) ◦ D R n . Hence H ∗ ( E 00 , ∂ 00 ) ∼ = X ◦ D R n . This is E 01 and the diﬀeren tial ∂ 01 is equal to the restriction of ∂ 0 on to X ◦ D R n . 20 F or E ′ 0 ∗ ev erything is trivial, E ′ 01 ∼ = Q n +1 and ∂ ′ 01 = 0. Then pr 1 : E 01 → E ′ 01 induced by pr is quism, b ecause H ∗ ( E 01 , ∂ 01 ) ∼ = X ◦ H ∗ ( D R n , ∂ 0 ) ∼ = X ◦ Q n ∼ = Q n +1 , where the ﬁrst isomorphism follo ws fr om the usual K ¨ u n neth formula (3), the second one follo ws from th e in duction hypothesis and the last on e wa s already observ ed in (19). This conclud es the pro of of the claim H ∗ ( D R n , ∂ 0 ) ∼ = Q n . Finally H ∗ ( D R , ∂ 0 ) ∼ = H ∗ (colim n D R n ) ∼ = colim n H ∗ ( D R n ) ∼ = colim n Q n ∼ = F ( X ⊕ Φ) D pro ve s S ublemma 3.7.  No w that the sublemma is pro v ed, we easily go through all the isomorphisms to c hec k (16) and (18) .  3.3 Augmen ted cotangen t c omplex Let ( A , ∂ A ) α − → ( E nd A , ∂ E nd A ) b e an A -algebra stru cture on A . W e b egin by extracting the op eradic cohomolog y fr om D R . Let F ( X ⊕ Φ ⊕ X ) → A b e the d g- C -op erad morp hism whic h equals ρ R on X and v anishes on th e other generators. Hence D R = F ( X ⊕ Φ ⊕ X ) is a dg- C -op erad over A . F or M one of the su bsets X , X ⊕ Φ, X ⊕ X of D R deﬁ ne Der M A ( D R , E nd A ) := { θ ∈ Der A ( D R , E nd A ) | ∀ m ∈ M θ ( m ) = 0 } . (20) W e will abb reviate th is by Der M . Let δ b e the diﬀerenti al on Der M deﬁned b y δ θ := θ ∂ D R − ( − 1) | θ | ∂ E nd A θ . A c hec k similar to that for (10) v eriﬁes this is wel l d eﬁned. Obviously Der X = Der X ⊕ X ⊕ Der X ⊕ Φ . Recall w e assume the dg- k -mo dule A is graded by the colours, that is A = L c ∈ C A c . Hence we h a v e Der X ⊕ X ∼ = Hom C − coll . (Φ , E nd A ) ∼ = M c ∈ C Hom k ( A c , A c ) . Imp ortantly , Der X ⊕ Φ is closed und er δ . 3.8 L e mma. (Der X ⊕ Φ , δ ) ∼ = ↓ (Der A ( R , E nd A ) , δ ) as dg- k -mo dules. 21 Pr o of. Recall ↓ δ = − δ . Deﬁne a d egree + 1 map Der X ⊕ Φ f 1 − → Der A ( R , E nd A ) b y the formula ( f 1 θ ′ )( x ) := θ ′ ( x ) for θ ′ ∈ Der X ⊕ Φ . Its inv erse, f 2 of degree − 1, is deﬁned for θ ∈ Der A ( R , E nd A ) by the f orm ulas ( f 2 θ )( a ) = 0 = ( f 2 θ )( φ ) , ( f 2 θ )( x ) = θ ( x ) . Ob viously f 2 f 1 = 1 and f 1 f 2 = 1 and it r emains to chec k f 1 δ = − δ f 1 . ( f 1 ( δ θ ′ ))( x ) = ( δ θ ′ )( x ) = θ ′ ( ∂ D R x ) − ( − 1) | θ ′ | ∂ E nd A ( θ ′ ( x )) , ( − δ ( f 1 θ ′ ))( x ) = − ( f 1 θ ′ )( ∂ R x ) + ( − 1) | f 1 θ ′ | ∂ E nd A (( f 1 θ ′ )( x )) . No w we c hec k θ ′ ( ∂ D R x ) = − ( f 1 θ ′ )( ∂ R x ). Let ∂ R x = P i T i ( x i 1 , . . . , x in i ). θ ′ ( ∂ D R x ) = θ ′ ( φ ◦ x − X j x ◦ j φ − s ( ∂ R x )) = = − θ ′ ( s X i T i ( x i 1 , . . . , x in i )) = = − θ ′ ( X i n i X j =1 ǫ ij T i ( x i 1 , . . . , x ij , . . . , x in i )) = = − X i X j ǫ 1+ | θ ′ | ij T i ( ρ R ( x i 1 ) , . . . , θ ′ ( x ij ) , . . . , ρ R ( x in i )) , − ( f 1 θ ′ )( ∂ R x ) = . . . = − X i X j ǫ | f 1 θ ′ | ij T i ( ρ R ( x i 1 ) , . . . , ( f 1 θ ′ )( x ij ) , . . . , ρ R ( x in i )) , where w e hav e denoted ǫ ij := ( − 1) P j − 1 l =1 | x il | .  3.9 Deﬁnit ion. W e call C ∗ aug ( A, A ) := ((Der X ) −∗ , δ ) augmen ted op eradic cotangen t complex of A and its cohomology H ∗ aug ( A, A ) := H ∗ ( C ∗ aug ( A, A ) , δ ) augmen ted op eradic cohomology of A . The inte rp r etation of the augmenta tion (Der X ⊕ X ) −∗ δ − → (Der X ⊕ Φ ) −∗ ∼ = C ∗ ( A, A ) of the usual cotangen t complex C ∗ ( A, A ) is via inﬁnitesimal automorphisms of th e A - algebra structure on A . T h is su ggests a relation b et w een H ∗ aug ( A, A ) and H ∗ ( A, A ). It is b est seen in an example: 22 3.10 Exa mple. Con tin uing Example 3.6, let A b e k -mo dule with a structure of an asso ciativ e algebra, that is A ss α − → E nd A . W e hav e C 0 aug ( A, A ) = Der X ⊕ X ∼ = Hom k ( A, A ) , C n aug ( A, A ) = (Der X ⊕ Φ ) − n ∼ = Hom C − coll . ( X , E nd A ) − n ∼ = ∼ = Hom C − coll . ( X n , E nd A ) ∼ = E nd A ( n + 1) = Hom k ( A ⊗ n +1 , A ) and, for f ∈ C n aug ( A, A ), δ f = ( − 1) n +1 µ ◦ 2 f + n X k =1 ( − 1) n +1 − k f ◦ k µ + µ ◦ 1 f . (21) So the augmented cotangen t complex is the Ho c h sc hild complex w ithout the term C − 1 ( A, A ) = Hom k ( k , A ) ∼ = A , while the ordin ary cotangen t complex w ould b e addi- tionally missing C 0 ( A, A ): C 0 ( A, A ) δ − → C 1 ( A, A ) δ − → C 2 ( A, A ) δ − → · · · | {z } C ∗ ( A,A ) | {z } C ∗ aug ( A,A ) T o generalize the conclusion of the examp le, recall fr om [13] that T J -grading on a free r esolution R = ( F ( X ) , ∂ ) ρ − → ( A , 0) is induced by a grading X = L i ≥ 0 X i on the C -collection of generators, den oted b y upp er indices, R i , and satisfying 1. ∂ maps X i to F  L j

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