On the Cohomology Comparison Theorem

ON THE COHOMOLOGY COMP ARISON THEOREM ALIN ST ANCU DEP AR TMENT OF MA THEMA TICS, COLUMBUS ST A TE UNIVERSITY, COLUMBUS, GA 3190 7 , USA Abstract. A relative derived ca tegory fo r the category of mod- ules ov er a presheaf of algebras is constructed to iden tify the rela- tive Y oneda a nd Ho chsc hild co homologies with its homomo rphism groups. The pr op erties of a functor b etw een this categor y and the relative deriv ed categor y of mo dules ov er the alg ebra a sso ciated to the preshea f are studied. W e obtain a genera lization of the S pecial C ohomolog y C omparison T heore m of M. Gerstenhab er and S. D. Schac k. 2000 Mathematics Subje ct Classific atio n. Primary : 18, Seco ndary: 16. Keywords: Ho chsc hild, Derived Category . 1 2 ALIN ST ANCU 1. Introduction Ho c hsc hild cohomology o f a k - algebra A , denoted here H • ( A, − ), pla ys a n imp ortant role in the study of asso ciativ e algebras, by serv- ing as a to ol in the deformation theory of this class of algebras where, broadly sp eaking, deformations of A are parameterizations A t , of a s- so ciativ e algebras, suc h that for t = 0 one obtains A . W e mention here only tw o of its man y other interes ting prop erties: firs t, se parable algebras A are c haracterized by H 1 ( A, − ) = 0 and second, as disco v- ered by Gerstenhab er, H • ( A, A ) has a rich a lgebraic structure (of G - algebra). In fact, one need not to restrict to a single algebra and, as M. G erstenhab er and S. D. Sc hac k did, ma y consider deformations of preshea v es of algebras, or mo r e general o f diagrams o f algebras, where the naturally defined Ho c hsc hild cohomology pla ys a similar role. The Ho c hsc hild cohomology of preshea v es is interes ting as a step to subsum- ing the deformation theory of complex manif o lds in the deformation theory o f asso ciativ e algebras. The authors men tioned ab ov e asso ci- ated to eac h presheaf of a lg ebras A a single alg ebra A ! and pro v ed the S pecial C ohomol og y C ompar ison T heor em whic h states that Y oneda and Ho c hsc hild cohomologies of the presheaf and the algebra asso ciated to the presheaf are isomorphic. ON THE COHOMOLO GY COMP ARISO N THEOREM 3 Note t ha t Y oneda and Ho chs c hild cohomologies are relative theories since k is a comm utativ e ring that is not necessarily a field. In this pap er we deve lop a relativ e deriv ed category , D − k ( A − bimo d), of the category of bimo dules ov er a presheaf A of k -alg ebras, one where the relative Y oneda cohomology , Ex t i A − A ,k ( M , N ), so in particu- lar Ho c hsc hild cohomology , can b e regarded as homomorphism groups, M or D − k ( A − bimod) ( M • , N • [ i ]). The r eader should b e aw are that t he term ‘presheaf of k -alg ebras’ is used to describ e functors A , defined on p o sets C , with images in the category of k - algebras. In this con text, we also sho w that the functor !, induced betw e en the relativ e derive d cate- gories of A -bimo d and A ! -bimo d, is full and faithful and w e obtain a generalization of the S pecial C ohomol og y C om par ison T heor em . This natural construction ma y be part of pro viding a more concep- tual in terpretation for the Ho c hsc hild cohomolog y of a presheaf of alge- bras together with its Gerstenhab er br a c k et, that of the Lie algebra of an algebraic group ( i.e a group v alued functor). In the case of a single algebra ov er a field B. Keller, in [6], iden tifies H • ( A, A ) with the Lie al- gebra of an alg ebraic g roup b y regarding H i ( A, A ) as a homomor phism group M or D ( A − bimod) ( A • , A • [ i ]) in the deriv ed category D ( A − bimod) and then establishing a bijection b et w een the latter g roups and cer- tain infinitesimal de formatio ns of A whic h ha v e a natural Lie brac k et. 4 ALIN ST ANCU Since the Gerstenhab er brac k et exists on the Ho c hsc hild cohomology of preshea v es of algebras presumably a similar in terpretation exists fo r this situation too . T o adapt Keller’s tec hnique to this c ase one needs to find the “correct” deriv ed category that allo ws the in terpretation of the relativ e Ho c hsc hild cohomology as Hom gr oups. Note : This pap er w as inspired by [7] and it w ould hav e not b een p ossible without the supp ort o f Sam uel D. Sc hac k. 2. Res olutions, adjoint functors and the functor ! Let k b e a commu tative ring and C a p oset view ed as a category in the usual w a y: for eac h i ≤ j there is a unique map ϕ ij : i − → j . When A is a k -algebra and M an y A bimo dule w e assume M to b e symmetric ov e r k . ( i.e. ax = xa for all x ∈ M and a ∈ k .) A presheaf of k -algebras o v er C is a functor A : C op − → k - alg . W e will denote A ( i ) b y A i . A preshe af as a b o v e is a s p ecial case o f functor defined from a small cat ego ry to the category of k algebras. In [1] these functors are called a “diagrams”. The category A -bimo d is the category whose ob jects are A -bimo dules and the maps are maps of bimo dules. An A -bimo dule M is a presheaf of ab elian groups suc h tha t M i is an A i -bimo dule ( ∀ ) i ∈ C and f o r ON THE COHOMOLO GY COMP ARISO N THEOREM 5 all i ≤ j the map T ij M : M j − → M i is an A j -bimo dule map. An A - bimo dule map η : M − → N is a natural tr ansformation in whic h η i is an A i -bimo dule map ( ∀ ) i ∈ C . In defining Y oneda cohomolo gy of the cat ego ry A - bimo d ‘allo w able’ maps pla y a vital role. A map η : M − → N is allo w able if ( ∀ ) i ∈ C the map η i : M i − → N i admits a k - bimo dule splitting map k i : N i − → M i satisfying η i k i η i = η i . W e do not require the splitting maps k i to b e na tural. An A - bimo dule P is a r elativ e pro ject iv e if for ev ery allo w able epimorphism M − → N the induced map H om A − A ( P , M ) − → H om A − A ( P , N ) is an epimorphism of sets. A relativ e pro jective allow ab le resolu tion of an A -bimo dule M is an exact sequence · · · − → P n · · · − → P 1 − → P 0 − → M − → 0 in whic h all P n are relativ e pro jectiv e A - bimo dules and all maps are al- lo w able. The category A -bimo d has enough relativ e pro jectiv e bimo d- ules and eac h bimo dule has a relativ e pro jectiv e allow able resolution. Moreo v er, there is a functorial w a y of getting this t yp e of resolutions. The construction of suc h a resolution is due to M. G erstenhaber and S. D. Sc hac k (see [1]) and is based on tw o facts: First, the ‘forg etf ul’ functor A - bimo d − → K -bimo d has a left adjoint A ⊗ K − ⊗ K A , where K is the constan t presheaf K i = k , ( ∀ ) i ∈ C . F or eac h N ∈ A - bimo d w e 6 ALIN ST ANCU set ( A ⊗ K N ⊗ K A ) i = A i ⊗ k N i ⊗ k A i and the map A j ⊗ k N j ⊗ k A j − → A i ⊗ k N i ⊗ k A i corresp onding to i ≤ j in C is just ϕ ij ⊗ T ij N ⊗ ϕ ij . The corresp onding categorical bar resolution, of [2], of an A -bimo dule N , denoted B • ( N ), is allow able and since B q ( N ) = A ⊗ K B q − 1 ( N ) ⊗ K A w e ha v e that B q ( N ) i is a relativ e pro jectiv e A i -bimo dule ( ∀ ) i ∈ C . In addition, the resolution has a functorial contracting homotopy x q : B q ( N ) − → B q +1 ( N ), x q ( a ) = 1 ⊗ a ⊗ 1 . Second, observ e that ( ∀ ) i ∈ C the functor ( i ) ∗ : A -bimo d − → A i - bimo d defined by ( i ) ∗ M = M i admits a left adjoin t ( i ) ! : A i -bimo d − → A -bimo d, where ( i ! M ) h = A h ⊗ A i M ⊗ A i A h if h ≤ i and ( i ! M ) h = 0 otherwise. If h ≤ j ≤ i the map ( i ! M ) j − → ( i ! M ) h is ϕ hj ⊗ I d M ⊗ ϕ hj and it is zero otherwise. Com bining the functors ( i ) ∗ w e obtain a single exact functor R : A - bimo d − → Q i ∈C ( A i -bimo d), defined on ob jects b y R M = Q i ∈C M i and whose left adjoint L is defin ed on ob jects b y L M i = ` i ∈C ( i ) ! M i . Ap- plying again the categorical bar resolution of [2] we o btain an allow able resolution with a functorial con tracting homot op y . W e denote this res- olution b y S • . Thus S p = ( LR ) p +1 = LRS p − 1 and the b oundary ma ps d p : S p +1 − → S p are defined inductiv ely by d p = ε S p − LR d p − 1 , where d − 1 = ε is the counit of the adjunction. The con tracting ho mo t o p y is the unit η RS p : RS p − → RS p +1 . ON THE COHOMOLO GY COMP ARISO N THEOREM 7 Here is a more direct description o f S • . Let [p] b e the linearly ordered set { 0 < 1 < · · · < p } . A co v a r ian t functor σ : [ p ] → C is called a p - simplex. Thu s p - simplices a r e ob jects of the functor categor y C [ p ] . The domain of σ is defined as σ (0 ) and is denoted by dσ . Similarly , the co domain of σ is defined as σ ( p ) a nd is denoted b y cσ . F or eac h p - simplex σ w e write σ = ( σ 01 , . . . , σ p − 1 ,p ) and define σ r =                    ( σ 12 , . . . , σ p − 1 ,p ) if r=0 ( σ 01 , . . . , σ r − 1 ,r + 1 , . . . , σ p − 1 ,p ) if 0 < r < p ( σ 01 , . . . , σ p − 2 ,p − 1 ) if r=p Note that dσ r = dσ = σ (0 ) if r 6 = 0 and dσ 0 = σ (1). Similarly , cσ r = cσ = σ ( p ) if r 6 = p and cσ p = σ ( p − 1 ) . Also, note that dσ ≤ dσ r and cσ r ≤ cσ and recall t hat the structure maps defining preshea v es and bimo dules are con trav ariant. F or N ∈ A - bimo d and p ≥ 0 w e ha v e S p N = ` σ ∈C [ p ] S σ p N , where S σ p N = ( dσ ) ! ( A dσ ⊗ A cσ N cσ ⊗ A cσ A dσ ) and A dσ is an A cσ -bimo dule via the map ϕ dσ ,cσ : A cσ − → A dσ . F or p ≥ 0, the b oundary ∂ : S p N − → S p − 1 N is a sum ∂ = P p r =0 ( − 1) r ∂ r where the restriction of ∂ r to S σ p is denoted ∂ σ r : S σ p N = ( dσ ) ! ( A dσ ⊗ A cσ N cσ ⊗ A cσ A dσ ) − → ( dσ r ) ! ( A dσ r ⊗ A cσ r N cσ r ⊗ A cσ r A dσ r ) = S σ r p − 1 N . 8 ALIN ST ANCU W e obtain that for h ≤ dσ and a ⊗ n ⊗ a ′ ∈ ( S σ p N ) h = A h ⊗ A cσ N cσ ⊗ A cσ A h , ∂ σ r ( a ⊗ n ⊗ a ′ ) = a ⊗ T cσ r ,cσ N ( n ) ⊗ a ′ ∈ ( S σ r p − 1 N ) h . Here T cσ r ,cσ N is the structure map of the bimo dule N corresp onding to cσ r ≤ cσ . In particular, when r = 0 w e get ∂ σ 0 ( a ⊗ n ⊗ a ′ ) = a ⊗ n ⊗ a ′ , and when r = p w e get ∂ σ p ( a ⊗ n ⊗ a ′ ) = a ⊗ T cσ p ,cσ N ( n ) ⊗ a ′ . The augmen tation map ε : S 0 N = ` i ∈C S i 0 = ` i ∈C ( i ) ! ( A i ⊗ A i N i ⊗ A i A i ) − → N is defined on the comp onents ( i ) ! ( A i ⊗ A i N i ⊗ A i A i ). F or h ≤ i , ( i ) ! ( A i ⊗ A i N i ⊗ A i A i ) h = A h ⊗ A i N i ⊗ A i A h − → N h is g iv en by 1 ⊗ n ⊗ 1 − → T hi N ( n ). F or i ∈ C the con tracting homotopy κ i p : ( S p N ) i − → ( S p +1 N ) i is giv en comp onen t wise by ( S σ p N ) i − → ( S ( i,σ ) p +1 N ) i = identity , w here ( S ( i,σ ) p +1 N ) = 0 if i  dσ . If i ≤ dσ , then ( i, σ ) is the simplex ( i, dσ = σ (0) , . . . , σ ( p ) ) . In general the ab o v e resolution is not a relativ e pro jectiv e resolution, but it is when each N i is a relativ e pro jectiv e A i -bimo dule. Th us, to construct a relativ e pro jectiv e allow able resolution of an A -bimo dule N we tak e the resolution B • ( N ) − → N , determined b y the forg etful functor and its left adjoint, and then apply S • to it to o btain a double complex S • B • ( N ). T ake now the total complex of this double complex to get the desired resolution. The Ho chs c hild cohomology of a presheaf A is defined to b e the relativ e Y oneda cohomology of A . ON THE COHOMOLO GY COMP ARISO N THEOREM 9 That is, H • ( A , − ) = Ext • A − A ( A , − ) . It plays a crucial role in the study o f deformatio ns o f diagrams of alge- bras and it has the same ric h structure as the Ho c hsc hild cohomolog y of a single algebra. If P • → A is a relative pro jectiv e a llo w able resolution of A then H • ( A , − ) is the homology of the complex Hom A − A ( P • , − ). T o eac h presheaf o f algebras A o v er C w e can asso ciate a single al- gebra A ! = ro w-finite C × C matrices ( a ij ) with a ij ∈ A i if i ≤ j and a ij = 0 otherw ise. The addition is comp onent wis e and the m ultiplica- tion ( a ij )( b ij ) = ( c ij ) is induc ed b y the matrix m ultiplication w ith the understanding that, fo r h ≤ i ≤ j , the summand a hi b ij of c hj is regarded as a hi b ij = a hi ϕ hi ( b ij ). F or our purp ose it is con v enien t to use the equiv- alen t represen tation A ! = Q i ∈C ` i ≤ j A i ϕ ij , as k -bimo dule. Here ϕ ij serv e to distinguish distinct copies of A i from one ano ther. The general elemen t of A i ϕ ij will b e denoted a i ϕ ij . T he multiplication is defined comp onen t wise and sub ject to the rule: ( a h ϕ hi )( a j ϕ j l ) = a h ϕ hi ( a j ) ϕ hl if i = j a nd 0 otherwise. Let 1 i the unit elemen t of A i . S ince ( a h ϕ hi )(1 i ϕ ij ) = a h ϕ hj and (1 i ϕ hi )( a i ϕ ij ) = ϕ hi ( a i ) ϕ hj w e may a bbreviate 1 i ϕ ij to ϕ ij . The maps ϕ ij are then elemen ts of A ! and ϕ hi ϕ ij = ϕ hj ; ϕ hi ϕ j l = 0 if i 6 = j . 10 ALIN ST ANCU W e define the functor ! : A -bimo d − → A !-bimo d, suc h that A − → A !, by setting for any A -bimo dule M , M ! = Q i ∈C ` i ≤ j M i ϕ ij as a k - bimo dule. The actions of A ! are defined by : ( a h ϕ hi )( m i ϕ ij ) = a h T hi M ( m i ) ϕ hj ( m h ϕ hi )( a i ϕ ij ) = m h ϕ hi ( a i ) ϕ hj ( a h ϕ hi )( m j ϕ j l ) = 0 = ( m h ϕ hi )( a j ϕ j l ) , if i 6 = j. F or η ∈ H om A − A ( N , M ) define η ! ∈ H om A ! − A ! ( N ! , M !) b y η !( n i ϕ ij ) = η i ( n i ) ϕ ij . W e will use the fo llowing prop osition due to M. G erstenhab er and S. D. Sc hac k. Prop osition 2.1. The functor ! : A -bim o d − → A ! -bimo d is exact, pr e- serves al lowability and is ful l and faithful. Pr o of. see[2]  In fact, M. Gerstenhab er and S. D. Sc hac k pro v ed in [2] the “Sp ecial Cohomology Comparison Theorem” (SCCT). Theorem ( SCCT ) . L et C b e an arbi tr ary p oset and A a pr eshe af over C . The functor ! induc es a n isomorphism of r elative Y on e da c o- homolo gies E xt • A − A (( − ) , ( − )) ∼ = E xt • A ! − A ! (( − )! , ( − )!) . In p articular, we ON THE COHOMOLO GY COMP ARISO N THEOREM 11 have an isomorphism of r elative Ho chschild c o homolo gies H • ( A , ( − )) ∼ = H • ( A ! , ( − )!) . An imp orta nt consequence of t his theorem is that the deformation theories of A a nd of A ! are equiv alen t, if the p oset C has a terminator. Another is that H • ( A ! , A !) has a G -a lgebra structure. T hese results can b e found in their full generalization to diagrams in [2], but w e will not deal with them here. W e will how ev er generalize the SCCT to deriv ed categories and prov e theorems 3 .9 and 4.1 . The SC CT fo llows as a corollary fr o m these theorems. T o do this w e need to in tro duce a sub category of t he category of A !-bimo d. The image of ! lies in a f ull sub category of A !-bimod. This is the category of aligned bimo dules, A !-albimo d. The main reason to consider it here is tha t the functor ! has a left adjoint when restricted to ! : A -bimo d − → A !-albimo d. Th us, for ev ery A ! bimo dule X w e set X al = Q i ∈C ` i ≤ j ϕ ii X ϕ j j with the ob vious A ! bimo dule structure. Definition 2.2. An A ! bimo dule X is said to b e aligned if the k linear map X − → Q i ∈C Q j ∈C ϕ ii X ϕ j j , x − → < ϕ ii xϕ j j > induces an A ! bimo dule isomorphism α X : X − → X al = Q i ∈C ` i ≤ j ϕ ii X ϕ j j . F or each A !-bimo dule map f : X − → Y , the restriction o f f to ϕ ii X ϕ j j is a k linear, ev en a A i - A j -bimo dule map f ij : ϕ ii X ϕ j j − → 12 ALIN ST ANCU ϕ ii Y ϕ j j since f ( ϕ ii xϕ j j ) = ϕ ii f ( x ) ϕ j j lies in ϕ ii Y ϕ j j . Thus , f g iv es rise to a family of k linear maps f ij : ϕ ii X ϕ j j − → ϕ ii Y ϕ j j suc h that f hj ( a h ϕ hi · x ) = a h ϕ hi · f ij ( x ) and f iq ( x · a j ϕ j q ) = f ij ( x ) · a j ϕ j q ∀ x ∈ ϕ ii X ϕ j j , a h ∈ A h , a j ∈ A j and h ≤ i ≤ j ≤ q in C . In fact these are exactly the conditions necessary on suc h a collection of maps f or f al = Q i ∈C ` i ≤ j f ij to b e an A !- bimo dule map X al − → Y al . One can easily see that A !-albimo d is ab elian, and that both the inclusion f unctor inc : A !-a lbimo d − → A !-bimo d and the alignmen t functor ( − ) al : A !-bimo d − → A !-a lbimo d, X − → X al are exact and preserv e allow abilit y and that α : I d A ! − albimod − → ( − ) al ◦ inc is a natural isomorphism. No w, w e describ e a metho d of pro ducing relativ e pro jectiv e allow - able resolutions o f aligned bimo dules of the form N ! that w e will use to replace complexes of aligned bimo dules with relative pro jectiv e ones in a suitable deriv ed category . W e b egin with a result du e to M. Ger- stenhab er and S. D. Sc hac k. Prop osition 2.3. 1. F or e ach i ≤ j in C the r estriction functor ( − ) ij : A ! -albimo d − → A i -mo d- A j , X − → ϕ ii X ϕ j j is exac t and pr e- serves al lowability. 2. The functor ( − ) ij has a left adjoint L ij that pr eserves r elative pr o- je ctivity. ON THE COHOMOLO GY COMP ARISO N THEOREM 13 Pr o of. P art 1 is ob vious. F or 2, define L ij : A i -mo d- A j − → A !- a lbimo d as follo ws: L ij ( N ) hl =          A h ⊗ A i | N | j l if h ≤ i ≤ j ≤ l 0 otherwise Here, | N | j l is N view ed as a left A i -mo dule a nd a right A l -mo dule via the map ϕ j l . The actions of A ! are given b y a r ϕ r h ( a h ⊗ n ) = a r ϕ r h ( a h ) ⊗ n ∈ L ij ( N ) r l ( a h ⊗ n ) a l ϕ lm = a h ⊗ nϕ j l ( a l ) ∈ L ij ( N ) hm , for a h ⊗ n ∈ L ij ( N ) hl and a r ϕ r h , a l ϕ lm ∈ A !. One can c hec k now t ha t w e ha v e a natural is omorphism H om A ! − albimo d ( L ij ( N ) , X ) ⇆ H om A i − A j ( N , X ij ) for all X ∈ A !-albimo d and N ∈ A i -mo d- A j . If P ∈ A i -mo d- A j is relativ e pro jectiv e then t he natural isomorphism H om A ! − albimo d ( L ij ( P ) , − ) ∼ = H om A i A j ( P , ( − ) ij ) = H om A i A j ( P , − ) ◦ ( − ) ij is a comp osite of functors whic h preserv e allow able epimorphisms, so L ij ( P ) is relative pro jectiv e. ( for more details see [2])  Mo deled on the M. Gerstenhab er - S. D. Sc hac k resolution S • , C. B. Kullmann obtained in [3] an allow a ble resolution T • N − → N ! in A !-albimo d as follows. F or p ≥ 0 let T p N = ` σ ∈C [ p ] T σ p N , where the 14 ALIN ST ANCU copro duct is take n in A !-albimo d ( constructed b y applying ( − ) al to that in A -bimo d ), where T σ p N = L dσ ,cσ ( A dσ ⊗ A cσ N cσ ). F or h ≤ dσ ≤ cσ ≤ l we ha v e a natural isomorphism ( T σ p N ) hl = A h ⊗ A dσ | A dσ ⊗ A cσ N cσ | cσ,l ∼ = A h ⊗ A cσ | N cσ | cσ.l and w e use this iden tification to define the differen tials. If p ≥ 1 we define d : T p N − → T p − 1 N as a sum d = P p r =0 ( − 1) r d r , where eac h d r is determined by its restriction to T σ p N and for h ≤ dσ ≤ cσ ≤ l and a ⊗ n ∈ ( T σ p N ) hl = A h ⊗ A cσ | N cσ | cσ,l , w e ha v e d σ r ( a ⊗ n ) = a ⊗ T cσ r ,cσ N ( n ) ∈ A h ⊗ A cσ r | N cσ r | cσ r ,l = ( T σ p − 1 N ) hl . If p = 0 the map ε T : T 0 N = ` i T ( i ) 0 N − → N ! is determined b y ( T ( i ) 0 N ) hl = A h ⊗ A i | N i | il − → N ! hl = N h ϕ hl , a ⊗ n − → aT hi N ( n ) ϕ hl , for h ≤ i ≤ l . It is easy t o c hec k that T • N − → N ! is a chain complex and it is in fact an allow able resolution since it has a contracting homotopy induced b y κ p : ( T σ p N ) hl − → ( T ( h,σ ) p +1 N ) hl , κ p = identity , where ( h, σ ) is t he simplex ( h, σ (0) , . . . , σ ( p )) if h ≤ σ (0) and T ( h,σ ) p +1 N = 0 if h  dσ . In general T p N is not a relativ e pro jectiv e aligned A !-bimo dule, but it is when eac h N i is relativ e pro jectiv e A i -bimo dule. T o obtain a relative pro jectiv e aligned resolution, for eac h A - bimo dule N !, tak e the relativ e pro jectiv e resolution B • ( N ), apply ! and then T • to obtain a double complex. Now , take the total complex to obtain the desired resolution. ON THE COHOMOLO GY COMP ARISO N THEOREM 15 W e conclude this section with a result whic h connects T • and S • via a left adjoint of !. Because the only source for the follow ing theorem is [3] the pro of is included in the App endix A. Theorem 2.4. 1. The functor ! : A -bim o d − → A ! -albim o d adm its a left adjo int ¡ : A ! -albimo d − → A -bimo d. 2. Ther e ar e natur al isomorphism s T p N ¡ − → S p N which ind uc e a nat- ur al isomorphism of c omple xes ( T • N − → N !) ¡ and ( S • N − → N ) . Pr o of. see App endix A.  3. Derive d ca t egories and Hochschild cohomology Let K om − ( A − bimo d) the category of b ounded to the rig ht com- plexes of A -bimo dules M • := · · · M n / / · · · · · · / / M 1 / / M 0 / / 0 A map b et w een t w o complexes M • and N • is a collection of maps f = ( f i ) : M i → N i , one fo r each p ositiv e in teger i , which comm ute with the differen tials of M • and N • . W e do not require the maps defining the complexes or the maps b et w een comple xes to b e k -split. Similarly , w e define K om − ( A ! − a lbimo d) and K om − ( A ! − bimo d). 16 ALIN ST ANCU Definition 3.1. 1 ) A map f : M • / / N • in K om − ( A ! − bimo d) (or K om − ( A ! − albimo d)) is a r elativ e quasi - isomorphism if its cone C ( f ) • is con tractible when considered a s a complex of k -bimo dules. 2) A map f : M • / / N • in K om − ( A − bimo d) is a relative quasi - isomorphism if the maps of complexes f i : M i • / / N i • ha v e con tractible cones, when considered as complexes of k - bimo dules, for all i ∈ C . The word “ relative ” in the ab ov e definition is used a s a reminder to the reader that Y oneda and Ho c hsc hild cohomologies are relativ e theories, since k is a comm utativ e ring that is not necessarily a field. It is the relativ e Y oneda groups that we wan t to view a s ho momorphism groups in a suitable category . Prop osition 3.2. L et A b e any k -a lgebr a and f : M • − → N • a map of c omplexes of A bimo dules in K om − ( A − bimod ) . T hen, f is a r elative quasi-isomorphism if and only if ther e exists γ : N • − → M • a map of c omplexes of k -bimo dules s uch that f γ ∼ id N • and γ f ∼ id M • in K om − ( k − bimod ) , wher e ‘ ∼ ’ stand s for homotopy e quivalenc e. Pr o of. ′ ⇒ ′ ON THE COHOMOLO GY COMP ARISO N THEOREM 17 Assume that f is a relativ e quasi-isomorphism. Th us C ( f ) • is con- tractible when regarded as a complex of k -bimo dules, so there ex- ist s = ( s n ) : C ( f ) n − 1 • − → C ( f ) n • maps of k - bimo dules suc h tha t sd C ( f ) • + d C ( f ) • s = id . W e ma y assume that s =     α γ β δ     and d C ( f ) • =     − d M • 0 f d N •     , where α : M •− 1 − → M • , β : M •− 1 − → N • +1 , γ : N • − → M • and δ : N • − → N • +1 are k linear maps. Since sd C ( f ) • + d C ( f ) • s = id , w e obtain − α d M • + γ f − d M • α = id M • , − β d M • + δ f + f α + d N • β = 0 , γ d N • − d M • γ = 0 and δ d N • + f γ + d N • δ = id N • . Th us, γ is a map of complexes of k -bimo dules and since δ d N • + d N • δ = id N • − f γ and αd M • + d M • α = γ f − id M • , w e hav e f γ ∼ id N • and γ f ∼ id M • in K om − ( k − bimod ). ′ ⇐ ′ Assume f γ ∼ id N • and γ f ∼ id M • in K om − ( k − bimod ), so there are ma ps s N • and s M • suc h that f γ − id N • = s N • d N • + d N • s N • and γ f − id M • = s M • d M • + d M • s M • . 18 ALIN ST ANCU The map s C ( f ) • =     s M • + γ ( s N • f − f s M • ) γ s N • ( f s M • − s N • f ) − s N •     is a homotop y . Indeed, s C ( f ) • d C ( f ) • + d C ( f ) • s C ( f ) • = =     id M • − γ s N • f d M • + γ f s M • d M • − d M • γ s N • f + d M • γ f s M • 0 s N • s N • f d M • + f γ s N • f − d N • s N • s N • f − s N • f γ f id N •     =     id M • 0 0 id N •     = id C ( f ) • . Th us C ( f ) • is contractible in K om − ( k − bimod ) .  Prop osition 3.2. allows us to conclude that if any tw o of f , g or f g are relativ e quasi-isomorphisms then so is the third. W e prov e no w t he follo wing Prop osition 3.3. The class of r elative quasi-isomorphisms in the ho- motopic c ate gory K − ( A − bimo d) is lo c alizing. Pr o of. W e show ed already that the class of relativ e quasi-isomorphisms is closed under the comp osition of maps. T o conclude this class is lo calizing w e need to justify t wo facts: 1) The extension conditions: F or ev ery f ∈ M or K − ( A − bimod) and s relativ e quasi-isomorphism there exist g ∈ M o r K − ( A − bimod) and t r elat ive quasi-isomorphism suc h that the follow ing square ON THE COHOMOLO GY COMP ARISO N THEOREM 19 N • f / / t   M • s   K • g / / L • (resp. L • g / / s   K • t   M • f / / N • is comm uta tiv e. 2) Giv en f , g tw o morphisms from N • to M • , the existence of s rel- ativ e quasi-isomorphism with sf = sg is equiv a len t to t he existence of t relativ e quas i-isomorphism with f t = g t . The pro of of theorem 4, c ha pter 3 in [5 ], wh ic h states that the class of quasi-isomorphisms (not relativ e) in the homotopic catego ry of an ab elian category is lo calizing, can b e used en tirely so w e will not re- pro duce it here. One needs to note fo r 1) that the cone of t he map t constructed there is the same, in K − ( A − bimo d), as the cone of s ; and for 2) that the cone of the map t constructed the re is the cone of s shifted b y 1. Th us in b oth cases t is a relativ e quasi-isomorphism.  Remark that the same result is tr ue for K − ( A ! − bimo d) and K − ( A ! − albimo d). 20 ALIN ST ANCU W e now define the relativ e deriv ed categories by f ormally in v erting all relativ e quasi-isomorphisms . Definition 3.4. Let A b e an y of the categories A -bimo d, A !- bimo d o r A !-albimo d and P the appropriate class of r elative quasi-isomorphisms. D − k ( A ) := K − ( A )(Σ − 1 ) , where K − is the corresp onding homoto p y category . Because P is lo calizing w e may rega r d the morphisms, in an y of the re lative deriv ed categories defined ab ov e, as equiv alence classes of diagrams U t ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ g   ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ X Y The maps t and g are morphisms in the homotop y category with t ∈ P . These diagrams are usually called ro ofs and we adopt this terminology . In addition, b ecause P is a lo calizing class the relative deriv ed cate- gories defined ab ov e are triangula ted. W e b egin studying the ob jects of D − k ( A − bimo d ) with the complexes of relativ e pro jectiv e bimo dules. ON THE COHOMOLO GY COMP ARISO N THEOREM 21 Lemma 3.5. L et P • b e a c omplex of r e lative pr oje ctive A - bimo dules and R • f / / P • a r elative quasi-isomorphism. We have M or K − ( A − bimod ) ( P • , C ( f ) • ) = 0 . Pr o of. Because f is a relativ e quasi-isomorphism the cone C ( f ) i is acyclic and allow able ( ∀ ) i ∈ C . Giv en g ∈ M o r K − ( A − bimod ) ( P • , C ( f ) • ) w e show that g = ( g ) i : P i − → C ( f ) i , i ≥ 0 is homotopic to 0 in- ductiv ely . Since P 0 is a complex of relativ e pro jectiv e A -bimo dules w e obtain that the map g 0 from P 0 to C ( f ) 0 can b e lifted to a map δ 0 : P 0 − → C ( f ) 1 suc h that d C ( f ) 1 δ 0 = g 0 . The image of g 1 − δ 0 d P 1 is con tained in the image of d C ( f ) 1 so it has a lif t ing δ 1 : P 1 − → C ( f ) 2 suc h that d C ( f ) 2 δ 1 = g 1 − δ 0 d P 1 . Now, the image of g 2 − δ 1 d P 2 is con tained in the image of d C ( f ) 2 and the conclusion follows inductiv ely .  Prop osition 3.6. L et P • b e a c omplex of r elative pr oje ctive bimo d ules in K om − ( A − bimo d) . The c anonic al map M or K − ( A − bimod) ( P • , M • ) can / / M or D − k ( A − bimod) ( P • , M • ) is an isomorphism for al l M • ∈ K om − ( A − bimo d) . 22 ALIN ST ANCU Pr o of. T o pro v e the injectivit y let P • α / / M • and P • β / / M • suc h that their corresp onding ro ofs: P • id ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ α ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ P • M • and P • id ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ β ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ P • M • are equiv alent in D − k ( A − bimo d). Th us, w e ha v e the comm utative diagram in K − ( A − bimo d) X • a ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ b ❆ ❆ ❆ ❆ ❆ ❆ ❆ P • id ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ α * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ P • id t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ β ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ P • M • W e o btain a = b and αa = β b . T o chec k that α = β , apply M or K − ( A − bimod ) ( P • , − ) to the distin- guished tria ngle X • a / / P • / / C ( a ) • / / X • [1] and use pre- vious lemma to see that M or K − ( A − bimod ) ( P • , C ( a ) • ) = 0. This implies the existence of a map c such that ac = id P • in K − ( A − bimo d) and the injectivit y follo ws from here. ON THE COHOMOLO GY COMP ARISO N THEOREM 23 F or a morphism in M or D − k ( A − bimod ) ( P • , M • ) represen ted b y t he r o of R • α ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ f ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ P • M • the distinguished triangle R • f / / P • / / C ( f ) • / / R • [1] in- duces a long exact sequence b y applying M or K − ( A − bimod ) ( P • , ( − )) t o it. Again, by the previous lemma M or K − ( A − bimod ) ( P • , C ( f ) • ) = 0, th us the map M or K − ( A − bimod) ( P • , R • ) f / / M or K − ( A − bimod) ( P • , P • ) is on to, so ( ∃ ) a map P • s / / R • suc h that f s = id P • in K − ( A − bimo d). Since f is a relativ e quasi-isomorphism s is a relativ e quasi- isomorphism, so w e hav e the comm utative diagram: P • s ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ id ❅ ❅ ❅ ❅ ❅ ❅ ❅ R • f ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ α * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ P • id t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ αs ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ P • M • Th us, the ro ofs R • f ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ α ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ P • M • and P • id ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ αs ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ P • M • 24 ALIN ST ANCU are equiv alent and since the sec ond is the image of αs the surjec tivity is pro v ed.  Note that relativ e pro jectiv e complexes in K om − ( A ! − bimo d) and K om − ( A ! − albimo d) satisfy the same prop ert y . W e pro v e now t ha t eac h complex of A -bimo dules is relativ e quasi- isomorphic t o a comp lex of relative pro jectiv e bimo dules. F or this we need the follo wing Prop osition 3.7. L et A b e a k algeb r a and assume that we have a double c omplex of A bimo d ules . . . d 1   . . . d 0   . . . d M   · · · d 2 / / X 12 d 2 / / d 1   X 02 ε 2 / / d 0   M 2 d M   / / 0 · · · d 1 / / X 11 d 1 / / d 1   X 01 d 0   ε 1 / / M 1 d M   / / 0 · · · d 0 / / X 10 d 0 / / X 00 ε 0 / / M 0 / / 0 such that: a) Each r ow is k c ontr ac tible. ( i.e. Th er e exi st k -bimo dule maps X ( k − 1) i t k i / / X k i such that d i t k +1 i + t k i d i = id X ki .) b) The fol lowing dia gr ams ar e c o mmutative: ON THE COHOMOLO GY COMP ARISO N THEOREM 25 X k i d k   X ( k − 1) i d k − 1   t k i o o X k ( i − 1) X ( k − 1)( i − 1) t k i − 1 o o X 0 i d 0   M i d M   t 0 i o o X 0( i − 1) M i − 1 t 0 i − 1 o o for al l k , i ≥ 0 , Then 1. M • t 0 • / / ( T otX •• ) and ( T otX •• ) ε • / / M • ar e maps of c o mplexes of k -bimo dules, wher e ε i = 0 o n X j k , j + k = i if j > 0 . 2. ε • t 0 • = id M • and t 0 • ε • ∼ id T otX •• in K om − ( k − bimod ) , wher e ∼ =homotopy e quivalenc e. Pr o of. 1. The map t 0 • is a map of complex es b y b) and ε • is a map o f complexes b ecause d M ε i +1 = d 0 ε i and ε i d i = 0. 2. The only thing to pro v e here is t 0 • ε • ∼ id T otX •• in K om − ( k − bimod ) . F o r n ≥ 0 we define the map ( T otX •• ) n h n / / ( T otX •• ) n +1 b y h n := ( t n +1 0 , t n 1 , . . . , t 1 n , 0). It is a simple exercis e to c hec k that h • d T otX •• + d T otX •• h • = id − t 0 • ε • .  Theorem 3.8. F or e ach M • ∈ D − k ( A − bimo d) ther e exist U M • ∈ D − k ( A − bimo d) a nd U M • ε / / M • a r elative quasi-isomorphism such that U M • is a c omplex of r elative pr oje ctive A -bimo dules. 26 ALIN ST ANCU Pr o of. W e describ ed in section 2 a metho d of constructing a relativ e pro jectiv e allo w able resolution T ot S • B • ( M ) − → M , for each M ∈ A - bimo d. W e use this for eac h term M i of the complex M • , i ≥ 0. W e obtain a double complex with augmen ted column M • . In addition, eac h ro w is con t ractible a nd for a ll p ∈ C w e obtain a do uble complex of A p -bimo dules whic h satisfies the conditions of the previous prop o- sition. Thu s, b y taking the total complex of the double comple x with augmen ted column M • w e obtain the desire d complex of relativ e pro- jectiv e A - bimo dules, U M • , together with a relative quasi-isomorphism U M • ε / / M • .  Note that the same argumen t sho ws tha t for each complex M • ! ∈ D − k ( A ! − bimo d ) the tot a l complex, T ot T • M • , of the double complex T • M • obtained b y taking the allow able resolution of eac h M i ! described in section 2, giv es a relativ e quasi-isomorphism ( T ot T • M • ) ε / / M • ! . In addition, b y theorem 2.4., the left adjoint ¡ to ! has the prop erty that ( T ot T • M • ε / / M • ! ) ¡ is isomorphic to T ot S • M • ε ¡ / / M • , so ε ¡ is a relativ e quasi-isomorphism. T o see ho w the relativ e deriv ed categories defined earlier relate to Ho c hsc hild cohomology recall that giv en a presheaf of k - algebras A the relativ e Ho c hsc hild cohomolog y of A , denoted H • ( A , ( − )), is the same as the relativ e Y oneda cohomology E xt • A − A ( A , ( − )) of the category of ON THE COHOMOLO GY COMP ARISO N THEOREM 27 A -bimo dules. The w ord relativ e app ears as an indication that k is not necess arily a field, in g eneral only a comm utat ive ring. Th us, the relativ e Ho c hsc hild cohomology of a presheaf of alge- bras, with co efficie nts in an arbitrary A -bimo dule M , is computed b y ta king an y relative pro jectiv e allo w able resolution of A , applying H om A − A (( − ) , M ) to it and then taking the homology of the resulting complex. Theorem 3.9. Ext i A − A ( M , N ) ≃ M or D − k ( A − bimod ) ( M • , N • [ i ]) . In p ar- ticular, H i ( A , N ) ≃ M or D − k ( A − bimod) ( A • , N • [ i ]) . Pr o of. Let T ot B • S • M the relativ e allow able pro jectiv e resolution de- scrib ed in section 2. ( same as U M • in this case since M i = 0, ( ∀ ) i 6 = 0 .) Using prop osition 3.6. and theorem 3.8 . w e obtain the isomorphisms Ext i A − A ( M , N ) = H i ( H om A − A ( U M • , N )) = M or K − ( A − bimod ) ( U M • , N • [ i ]) ∼ = M or D − k ( A − bimod) ( U M • , N • [ i ]) ∼ = M or D − k ( A − bimod ) ( M • , N • [ i ]) .  4. Functors be tween derived c a tegories The functor A − bimo d ! / / A ! − bimo d is exact and pres erv es allo w ability so it induces a functor b etw een the cor r espo nding relativ e deriv ed categories. In this section we pro v e t he following prop ert y of the induced functor. 28 ALIN ST ANCU Theorem 4.1. The functor D − k ( A − bimo d) ! / / D − k ( A ! − bimo d) is ful l and faithful. That is, M or D − k ( A − bimod) ( M • , N • ) ! / / M or D − k ( A ! − bimo d) ( M • ! , N • !) is an isomorphism of sets for al l M • , N • ∈ D − k ( A − bimo d) . The difficulties in proving the theorem reside in t w o places. First, since the morphisms in D − k ( A − bimo d) and D − k ( A ! − bimod ) are equiv- alence classes of ro ofs, it is no t clear how one can find ancestors in D − k ( A − bimo d) fo r arbitrary ro ofs in D − k ( A ! − bimo d). A go o d sign for that w ould b e the existenc e of a left adjoin t for !, but there is none. F ortunately , a left adjoin t exists bet w een A -bimo d and the full sub catego r y of A !-bimo d of aligned bimo dules. Second, left adjoin ts do not necessarily preserv e all relativ e quasi-isomorphisms. Ho w ev er, this left adjoin t preserv es some that can b e used to t r ace bac k ancestors for any ro o f in M or D − k ( A ! − bimo d) ( M • ! , N • !) . W e will pro v e that D − k ( A − bimo d) ! / / D − k ( A ! − a lbimo d) and the inclusion D − k ( A ! − albimo d) inc / / D − k ( A ! − bimo d) are full and faithful. Prop osition 4.2. The functor D − k ( A − bimo d) ! / / D − k ( A ! − albimo d) ON THE COHOMOLO GY COMP ARISO N THEOREM 29 is ful l and faithful. Pr o of. T o pro v e the prop osition w e need to sho w t hat M or D − k ( A − bimod ) ( M • , N • ) ! / / M or D − k ( A ! − albimo d) ( M • ! , N • !) is an isomorphism for all M • and N • ∈ D − k ( A − bimo d) . Since for all M • ∈ D − k ( A − bimo d) there exist U M • ε / / M • relativ e quasi-isomorphism in D − k ( A − bimo d) suc h that U M i is relativ e pro jectiv e f o r all i , w e may assume that M • is a complex of relative pro jectiv e A bimo dules . This is b ecause of the comm utative diagram M or D − k ( A − bimod ) ( M • , N • ) ! / / ε   M or D − k ( A ! − albimo d) ( M • ! , N • !) ε !   M or D − k ( A − bimod) ( U M • , N • ) ! / / M or D − k ( A ! − albimo d) ( U M • ! , N • !) where ε and ε ! are isomorphisms. Because ( M i ) p is a relativ e pro jectiv e A p -bimo dule, ( ∀ ) p ∈ C , eac h M i ! admits a resolution of relativ e pro jectiv e aligned A !-bimo dules obtained using T • . The total complex of the double complex ob- tained b y ta king the resolution of eac h M i ! giv es a relativ e quasi- isomorphism T ot ( T • M • ) ε / / M • ! , where eac h T ot ( T • M • ) i is a relativ e pro jectiv e aligned A ! bimo dule. Moreo v er, the left adjo in t ¡ has the prop erty that ( T ot T • M • ε / / M • ! ) ¡ is isomorphic to T ot S • M • ε ¡ / / M • and ε ¡ is a r elat ive quasi-isomorphism. 30 ALIN ST ANCU No w, giv en any ro of X • s } } ④ ④ ④ ④ ④ ④ ④ ④ f ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! in M or D − k ( A ! − albimo d) ( M • ! , N • !) tak e T ot ( T • M • ) ε / / M • ! as ab ov e. By a pplying M or D − k ( A ! − albimo d) ( T ot ( T • M • ) , ( − )) to the distinguished triangle X • s / / M • ! / / C ( s ) • / / X • [1] w e o btain a long exact sequence. In this sequence M or D − k ( T ot ( T • M • ) , C ( s ) • ) = 0 b ecause C ( s ) • is con- tractible, as a complex of k - bimo dules, and T ot ( T • M • ) is a complex of relativ e pro jectiv e aligned A ! bimo dules, so the map M or D − k ( T ot ( T • M • ) , X ) s / / M or D − k ( T ot ( T • M • ) , M • !) is on to. Because ε ∈ M or D − k ( A ! − albimo d) ( T ot ( T • M • ) , M • !) , there exist q ∈ M or D − k ( A ! − albimo d) ( T ot ( T • M • ) , X • ) suc h that the diagram T ot ( T • M • ) q z z ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ε   X • s / / M • ! ON THE COHOMOLO GY COMP ARISO N THEOREM 31 comm utes. The map q is a relative quasi-isomorphism because both s and ε are and w e ha v e the equiv alence of ro ofs X • s } } ④ ④ ④ ④ ④ ④ ④ ④ f ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! and T ot ( T • M • ) ε y y t t t t t t t t t t f q $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ M • ! N • ! b ecause the diagram T ot ( T • M • ) q z z ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ id & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f , , ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ T ot ( T • M • ) ε s s ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ f q $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ M • ! N • ! is comm uta tiv e. Since ( T ot T • M • ε / / M • ! ) ¡ is isomorphic to T ot S • M • ε ¡ / / M • and ε ¡ is a relativ e quasi-isomorphism, the ro of ( T ot ( T • M • )) ¡ ε M • ε ¡ y y s s s s s s s s s s s ε N • f ¡ q ¡ % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ M • N • exists in D − k ( A − bimo d) . Here, ε M • and ε N • are the maps of complexes induced b y the counit o f the adjunction A − bimo d ! / / A ! − albimo d . ¡ o o The image of this 32 ALIN ST ANCU ro of via ! is [( T ot ( T • M • ) ¡ ])! ε M • ! ε ¡ ! x x q q q q q q q q q q q ε N • ! f ¡ ! q ¡ ! & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ M • ! N • ! and is equiv alen t to T ot ( T • M • ) ε y y t t t t t t t t t t f q $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ M • ! N • ! . This results from the comm utativ e dia gram T ot ( T • M • ) id w w ♥ ♥ ♥ ♥ ♥ η T o t ( T • M • ) ( ( ❘ ❘ ❘ ❘ ❘ T ot ( T • M • ) ε y y r r r r r f q - - ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ ❩ [( T ot ( T • M • )) ¡ ]! ε M • ! ε ¡ ! r r ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ε N • ! f ¡ ! q ¡ ! ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ M • ! N • ! (1) ε M • !( ε ¡ )! η T ot ( T • M • ) = ε and (2) ε N • ![ f ¡ q ¡ ]! η T ot ( T • M • ) = f q T o ch ec k (1) observ e t hat w e ha v e ε M • ! η M • ! = id M • ! b y the adjunction. In addition, the functorialit y of η induces the comm utative square T ot ( T • M • ) η T o t ( T • M • )   ε / / M • ! η M • !   [( T ot ( T • M • )) ¡ ]! ( ε ¡ )! / / [( M • !) ¡ ]! ON THE COHOMOLO GY COMP ARISO N THEOREM 33 Th us, w e ha v e ( ε ¡ )! η T ot ( T • M • ) = η M • ! ε and b y comp osing with ε M • ! w e obtain (1). Similarly one may c hec k (2). T o pro v e injectivity , let R • ! r ! } } ④ ④ ④ ④ ④ ④ ④ ④ f ! ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ M • ! N • ! and S • ! s ! } } ④ ④ ④ ④ ④ ④ ④ ④ g ! ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! b e equiv alen t ro ofs in D − k ( A ! − albimo d) . One may assume that R • is a complex o r relative pro jectiv e A bimo dules. T o see this, let U M • ε / / M • the relative quasi-isomorphism with U M i relativ e pro jectiv e A - bimo dules. Again, applying M or D − k ( A − bimod ) ( U M • , ( − )) to the distinguished tri- angle R • r / / M • / / C ( r ) • / / R [1] • , in D − k ( A − bimo d) , w e obtain a long exact sequence where M or D − k ( U M • , C ( r ) • ) = 0 . This implies the existence of a map t suc h that the follow ing diagram U M • t | | ③ ③ ③ ③ ③ ③ ③ ③ ε   R • r / / M • comm utes. In addition, t is a relative qu asi-isomorphism, since r and ε are and w e ha ve the equiv alent ro ofs R • r ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ M • N • and U M • ε | | ② ② ② ② ② ② ② ② f t " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ M • N • 34 ALIN ST ANCU in D − k ( A − bimo d) b ecause of the following comm utative diagram U M • t | | ③ ③ ③ ③ ③ ③ ③ ③ id # # ● ● ● ● ● ● ● ● R • r ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f + + ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ U M • ε s s ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ f t " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ M • N • This implies the the equiv alence of R • ! r ! } } ④ ④ ④ ④ ④ ④ ④ ④ f ! ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ M • ! N • ! and U M • ! ε ! { { ① ① ① ① ① ① ① ① f ! t ! " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ M • ! N • ! in D − k ( A ! − albimo d) . So, w e ma y assume that R • is a complex of relativ e pro jectiv e A -bimo dules. The equiv alence of R • ! r ! } } ④ ④ ④ ④ ④ ④ ④ ④ f ! ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ M • ! N • ! and S • ! s ! } } ④ ④ ④ ④ ④ ④ ④ ④ g ! ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! translates in to the existence of a comm utative diagram X • x } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ p ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ R • ! r ! } } ④ ④ ④ ④ ④ ④ ④ ④ f ! * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ S • ! s ! t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ g ! ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! ON THE COHOMOLO GY COMP ARISO N THEOREM 35 Here, X • ∈ D − k ( A ! − albimo d) and x is a relative quasi-isomorphism suc h tha t f ! x = g ! p , (1) and s ! p = r ! x , (2). Since x is a relativ e quasi- isomorphism and T ot T • R • is a complex of a ligned relativ e pro jectiv e A !-bimo dules there exist j suc h that the diagr a m T ot T • R • j { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ε   X • x / / R • ! is comm uta t ive. Moreo ver, j is a relativ e quasi-isomorphism b ecause ε and x are. W e obta in t he commutativ e diagram T ot T • R • xj z z ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ pj $ $ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ R • ! r ! } } ④ ④ ④ ④ ④ ④ ④ ④ f ! + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ S • ! s ! s s ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ g ! ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! b ecause f ! xj = g ! pj , by (1) and s ! pj = r ! xj , b y (2). Because ! is full and faithful w e ha v e the isomorphism ( T • !) ¡ ε T • / / T • for all T • in D − k ( A − bimo d) and so ( r !) ¡ and ( s !) ¡ are relative quasi-isomorphisms in D − k ( A − bimo d) . In addition, ε ¡ is a relative quasi-isomorphism and w e g et the comm utativ e diagra m 36 ALIN ST ANCU T ot S • R • ( xj ) ¡ = ε ¡ y y t t t t t t t t t ( pj ) ¡ $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ( R • !) ¡ ( r ! ) ¡ { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ( f !) ¡ + + ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ( S • !) ¡ ( s !) ¡ s s ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ( g ! ) ¡ # # ● ● ● ● ● ● ● ● ● ( M • !) ¡ ( N • !) ¡ Finally , we obtain the equiv a lence of R • ! r ! } } ④ ④ ④ ④ ④ ④ ④ ④ f ! ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ M • ! N • ! and S • ! s ! } } ④ ④ ④ ④ ④ ④ ④ ④ g ! ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • ! N • ! b y constructing T ot S • R • ε R • ε ¡ { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ε S • ( pj ) ¡ # # ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ R • r ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f + + ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ S • s s s ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ g ❆ ❆ ❆ ❆ ❆ ❆ ❆ M • N • This is b ecause f ε R • ε ¡ = ε N • ( f !) ¡ ε ¡ = ε N • ( g !) ¡ ( pj ) ¡ = g ε S • ( pj ) ¡ and sε S • ( pj ) ¡ = ε M • ( s !) ¡ ( pj ) ¡ = ε M • ( r !) ¡ ε ¡ = r ε R • ε ¡ .  W e show no w that the inclusion D − k ( A ! − albimo d) inc / / D − k ( A ! − bimo d) ON THE COHOMOLO GY COMP ARISO N THEOREM 37 is full and faithful. The lac k of an adjoin t in this case requires a tw o step pro cess of replacing the top of eac h ro of by a complex of aligned bimo dules. F or X ∈ A ! − bimod , let X + := Q i ∈C ϕ ii X . This defines an exact functor A ! − bimo d + / / A ! − bimo d that preserv es allo w a- bilit y , s o a lso relativ e quasi-isomorphisms. W e also hav e the natural maps X β X / / X + , x / / < ϕ ii x > and X al γ X / / X + , < x ij > / / < P j ≥ i x ij > . Also, if X is a ligned b oth β X and γ X are isomorphisms and β X = γ X α X , where α is the natural isomorphism α : I d A ! − albimod − → ( − ) al ◦ inc . Prop osition 4.3. The functor D − k ( A ! − albimo d) inc / / D − k ( A ! − bimo d) is ful l and faithful. Pr o of. W e ha v e t o pro v e that M or D − k ( A ! − albimo d) ( M • , N • ) inc / / M or D − k ( A ! − bimo d) ( M • , N • ) is an isomorphism o f sets for all M • and N • ∈ D − k ( A ! − albimo d). First, w e pr ov e that the map is onto. F or an y r o of X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • 38 ALIN ST ANCU in M or D − k ( A ! − bimo d) ( M • , N • ) w e ha v e the equiv alences X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • and X + • β − 1 M • s + } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ β − 1 N • f + ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • X + • β − 1 M • s + } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ β − 1 N • f + ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • and X • al α − 1 M • s al } } ③ ③ ③ ③ ③ ③ ③ ③ α − 1 N • f al ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ M • N • T o see this, observ e that since β is a nat ura l transfor ma t io n we ha v e s + β X • = β M • s and f + β X • = β N • f . In addition, b ecause M • and N • are aligned β M • and β N • are isomor- phisms and w e obtain β − 1 M • s + β X • = s and β − 1 N • f + β X • = f . This implies the first equiv alence b ecause t he dia gram X • id ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ β X • ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ X • s ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ X + • β − 1 M • s + t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ β − 1 N • f + ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • is comm uta tiv e. F or the second equiv alence, since γ is natural we ha v e s + γ X • = γ M • s al and f + γ X • = γ N • f al . ON THE COHOMOLO GY COMP ARISO N THEOREM 39 Because M • and N • are aligned γ M • , γ N • , α M • , α N • , β M • and β N • are isomorphisms, so we get β − 1 M • s + γ X • = β − 1 M • γ M • s al = α − 1 M • γ − 1 M • γ M • s al = α − 1 M • s al and β − 1 N • f + γ X • = β − 1 N • γ N • f al = α − 1 N • γ − 1 N • γ N • f al = α − 1 N • f al . The diagram X • al id | | ② ② ② ② ② ② ② ② γ X • " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ X • al α − 1 M • s al } } ③ ③ ③ ③ ③ ③ ③ ③ α − 1 N • f al * * ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ X + β − 1 M • s + s s ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ β − 1 N • f + ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • is comm utativ e and implies the second equiv a lence. Now , the surjec- tivit y follow s since t he r o of X • al α − 1 M • s al } } ③ ③ ③ ③ ③ ③ ③ ③ α − 1 N • f al ! ! ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ M • N • exists in M or D − k ( A ! − albimo d) ( M • , N • ) and its image is equiv a lent to X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • in M or D − k ( A ! − bimo d) ( M • , N • ). 40 ALIN ST ANCU T o pro v e the inj ectivity , let X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • and Y • t ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ g ❆ ❆ ❆ ❆ ❆ ❆ ❆ M • N • in M or D − k ( A ! − albimo d) ( M • , N • ) equiv a lent in M o r D − k ( A ! − bimo d) ( M • , N • ). Th us, w e ha v e a commutativ e diag ram Z • r ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ h ❅ ❅ ❅ ❅ ❅ ❅ ❅ X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Y • t t t ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ g ❆ ❆ ❆ ❆ ❆ ❆ ❆ M • N • where r and s are relativ e quasi-isomorphisms. Since the alignmen t functor pres erv es relative quasi-isomorphisms and M • , N • , X • and Y • are complexes of aligned A ! bimo dules w e ha v e the comm utativ e dia- gram Z • al r al } } ④ ④ ④ ④ ④ ④ ④ ④ h al ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f * * ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Y • t t t ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ g ❆ ❆ ❆ ❆ ❆ ❆ ❆ M • N • ON THE COHOMOLO GY COMP ARISO N THEOREM 41 whic h implies t he equiv alence of ro ofs X • s } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ f ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ M • N • Y • t ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ g ❆ ❆ ❆ ❆ ❆ ❆ ❆ M • N • in M or D − k ( A ! − albimo d) ( M • , N • ), and so the injectivit y of inc .  The pro of of theorem 4.1. follow s now easily com bining prop ositions 4.2. and 4.3. In particular, we obtain the following theorem of [2 ], due to M. Gerstenhab er and S. D. Sc hac k. Corollary 4.4. (Sp e cial Cohomolo gy Comp a rison The or em) The functor ! induc es an isomorphism of r elative Y one da c oh omolo gies E xt • A − A (( − ) , ( − )) ∼ = E xt • A ! − A ! (( − )! , ( − )!) . In p a rticular, we have an isomorphi sm of r elative Ho chsc hild c oho molo- gies H • ( A , ( − )) ∼ = H • ( A ! , ( − )!) . Pr o of. E xt i A − A ( M , N ) ∼ = M or D − k ( A − bimod) ( M • , N • [ i ]) ∼ = ∼ = M or D − k ( A ! − bimo d) ( M • ! , N • [ i ]!) ∼ = E xt i A ! − A ! ( M ! , N !) .  42 ALIN ST ANCU Note : By taking a v ery differen t approach W endy Low en a nd Mic hel V a n Den Bergh also prov ed in [8] that the functor ! is full and faithful. Appendix A. t heorem 2.4. Theorem A .1. 1. T he functor ! : A -bimo d − → A ! -albimo d adm its a left adjo int ¡ : A ! -albimo d − → A - bimo d. 2. Ther e ar e natur al isomorphism s T p N ¡ − → S p N which ind uc e a nat- ur al isomorphism of c omple xes ( T • N − → N !) ¡ and ( S • N − → N ) . Pr o of. 1. The left adjoin t is the restriction of a functor ¡ : A !-bimo d − → A -bimo d. F or any A !-bimo dule X and i ∈ C we define X ¡ i as t he colimit of a particular functor o v er the p oset C 1 i whose elemen t s ar e the 1-simplices of C i = { j | j ≥ i } . The ordering is σ ≪ τ ⇔ σ = τ o r σ is degenerate ( d σ = cσ ), τ is not, and either dσ = dτ or d σ = cτ . W e denote b y X pq = ϕ pp X ϕ q q . D efine F i X : C 1 i − → A i -bimo d on each ob ject σ to b e the co equalizer of the A i -bimo dule maps X i,dσ ⊗ k A dσ ⇒ X i,cσ ⊗ A cσ A i giv en by x ⊗ a → xϕ dσ ,cσ ⊗ ϕ i,dσ ( a ) and x ⊗ a → xaϕ dσ ,cσ ⊗ 1. F or σ ≪ τ in C 1 i , the map F i X ( σ τ ) : F i X ( σ ) → F i X ( τ ) is defined by x ⊗ a → xϕ cσ,cτ ⊗ a . Let X ¡ i := colimF i X , ∀ i ∈ C and ι i σ the canonical map F i X ( σ ) → X ¡ i . T o sho w that X ¡ is an A -bimo dule, for each h ≤ i in C , we hav e to define a map T hi X ¡ : X ¡ i → X ¡ h suc h that T hj X ¡ = T hi X ¡ T ij X ¡ if h ≤ i ≤ j . ON THE COHOMOLO GY COMP ARISO N THEOREM 43 First, w e ha v e a natura l t r a nsformation Γ hi X : F i X → hi |−| hi ◦ F h X ◦ inc hi , where inc hi : C 1 i → C 1 h is t he inclusion functor induced b y C i ⊂ C h and hi | − | hi : A h -bimo d → A i -bimo d is the forgetful functor. T o define Γ hi X observ e that, for eac h σ ∈ C 1 , left m ultiplication b y ϕ hi is an A i - A cσ bimo dule map : X i,cσ → X h,cσ , while ϕ hi : A i → A h is an A cσ - A i bimo dule map. The map o f A i -bimo dules ( ϕ hi · − ) ⊗ ϕ hi : X i,cσ ⊗ A cσ A i → hi | X h,cσ ⊗ A cσ A i | hi induces the A i -bimo dule map (Γ hi X ) σ : F i X ( σ ) → hi | F h X ( σ ) | hi giv en by x ⊗ a → ϕ hi x ⊗ ϕ hi ( a ). Second, let T hi X ¡ b e the comp o site of maps X ¡ i = colimF i X → col im ( hi |− | hi ◦ F h X ◦ inc hi ) = hi | col im ( F h X ◦ inc hi ) | hi → hi | col imF h X | hi = hi | X ¡ h | hi . Th us w e ha v e T hi X ¡ ( ι i σ ( x ⊗ a )) = ι h σ ( ϕ hi x ⊗ ϕ hi ( a )). One may easily che c k now the iden t ity T hj X ¡ = T hi X ¡ T ij X ¡ for h ≤ i ≤ j , so X ¡ is an A -bimo dule. So far w e ha v e defined ¡ on the ob jects of A !-bimo d so we need to define it on maps. Let g : X → Y be an A !-bimo dule map. The restriction of g to X ij is an A i - A j bimo dule map, g : X ij → Y ij , and for σ ∈ C 1 , the A i -bimo dule map g ⊗ id : X i,cσ ⊗ A cσ A i → Y i,cσ ⊗ A cσ A i induces the map e g i σ : F i X ( σ ) → F i Y ( σ ) defined b y x ⊗ a → g ( x ) ⊗ a . Its easy to c hec k the naturality of e g i σ since g is a A !-bimo dule map and b y t aking the colimits we obta in a n A i -bimo dule map g ¡ i : X ¡ i → Y ¡ i , giv en b y g ¡ i ( ι i σ ( x ⊗ a )) = ι i σ ( e g i σ ( x ⊗ a )). These are the comp onen ts of 44 ALIN ST ANCU an A - bimo dule map, i.e.T hi Y ¡ ◦ g ¡ i = g ¡ h ◦ T hi X ¡ for h ≤ i because of the comm uta tiv e diagram F i X Γ hi X / / e g i   hi | − | hi ◦ F h X ◦ inc hi id ◦ e g h ◦ id   F i Y Γ hi Y / / hi | − | hi ◦ F h Y ◦ inc hi In a ddition, one has ( g 1 g 2 ) ¡ = ( g 1 ) ¡ ◦ ( g 2 ) ¡ and ( id ) ¡ = id since g g 1 g 2 i = e g 1 i e g 2 i and e id i = id , so ¡ is a functor. W e now pro v e that the functor constructed ab o v e is a left adj o in t to !, when restricted to A -albimo d. Let X b e an alig ned A !- bimo dule. F or i ≤ j w e define η ij X : X ij → ( X ¡ )! ij to b e the A i - A j bimo dule map η ij X = ι i ( j ≤ j ) ( x ⊗ 1) ϕ ij . One may chec k that fo r h ≤ i ≤ j ≤ q , a h ∈ A h , a j ∈ A j and x ∈ X ij w e hav e η hj X ( a h ϕ hi · x ) = a h ϕ hi · η ij X ( x ) and η iq X ( x · a j ϕ j q ) = η ij X ( x ) · a j ϕ j q so the family of maps η ij X determine an A !-bimo dule natural map η X : X → ( X ¡ )!. Let N ∈ A -bimo d. T o define the comp onen ts of the counit ε i N : ( N !) ¡ i = col imF i N ! → N i , we define a f amily of A i -bimo dule maps ε i,σ N : F i N ! ( σ ) → N i suc h tha t ε i,τ N ◦ F i N ! ( σ τ ) = ε i,σ N , for σ ≪ τ in C 1 i and use the univ ersal prop erty of colimits. The A i -bilinear func- tion N i ϕ i,cσ × A i → N i , ( nϕ i.cσ , a ) → na is A cσ -balanced and the in- duced A i -bimo dule map N i ϕ i,cσ ⊗ A cσ A i → N i v anishes on { nϕ dσ ,cσ ⊗ ϕ i.dσ ( a ) − naϕ dσ .cσ ⊗ 1 | n ∈ N i ϕ i,dσ , a ∈ A dσ } so fo r eac h σ ∈ C 1 i , w e ON THE COHOMOLO GY COMP ARISO N THEOREM 45 obtain the A i -bimo dule map ε i,σ N : F i N ! ( σ ) → N i , nϕ i,cσ ⊗ a → na . W e ha v e that ε i,τ N ◦ F i N ! ( σ τ )( nϕ i,cσ ⊗ a ) = ε i,τ N ( nϕ i,cσ · ϕ cσ,cτ ⊗ a ) = na = ε i,σ N ( nϕ i,cσ ⊗ a ) and t hus the map ε i N is g iven by ε i N ( ι i σ ( nϕ i,cσ ⊗ a )) = na . The maps ε i N determine a natural map ε N : ( N !) ¡ → N of A - bimo dules since fo r h ≤ i and σ ∈ C 1 i w e ha v e T hi N ε i,cσ N ( nϕ i,cσ ⊗ a ) = T hi N ( na ) = T hi N ( n ) · a = T hi N ( n ) ϕ hi ( a ) while ε h,σ N (Γ hi N ! ) σ ( nϕ i,cσ ⊗ a ) = ε h,σ N ( ϕ hi · nϕ i,cσ ⊗ ϕ hi ( a )) = ε h,σ N ( T hi N ( n ) ϕ h,cσ ⊗ ϕ hi ( a )) = T hi N ( n ) ϕ hi ( a ) . T o finish the pro of we sho w t hat η and ε fo r m an adjoin t pair. T o see that ε N ! ◦ η N ! = id N ! it is enough to c hec k this on each N i ϕ ij and ε N !( η N ! ( nϕ ij ) = ε N !( ι i ( j ≤ j ) ( nϕ ij ⊗ 1) ϕ ij ) = ε i N ( ι i ( j ≤ j ) ( nϕ ij ⊗ 1)) ϕ ij = ( n · 1) ϕ ij = nϕ ij , as required. Last, for each X ∈ A -albimo d w e need to v erify that ε X ¡ ◦ η X ¡ = id X ¡ . This can b e c heck ed on a set of A i -bimo dule generators for eac h comp onen t X ¡ i and t he set { ι i ( j ≤ j ) ( x ⊗ 1) | j ≥ i, x ∈ X ij } has this prop ert y . Since ( ε i X ¡ ◦ η X ¡ i )( ι i ( j ≤ j ) ( x ⊗ 1))) = ε i X ¡ ( ι i ( j ≤ j ) ( η X ( x ) ⊗ 1)) = ε i X ¡ ( ι i ( j ≤ j ) ( ι i ( j ≤ j ) ( x ⊗ 1) ϕ ij ⊗ 1)) = ι i ( j ≤ j ) ( x ⊗ 1) · 1 w e obtain the required iden tity . 2. Because both T p N and S p N are copro ducts and ¡ , as a left adjoin t, preserv es colimits it is enough to find natural isomorphisms 46 ALIN ST ANCU γ σ : ( T σ p ) ¡ − → S σ p suc h that, for 0 ≤ r ≤ p , t he f ollo wing square ( T σ p N ) ¡ ( d T ,σ r ) ¡ / / γ σ N   ( T σ r p − 1 N ) ¡ γ σ r N   S σ p N d S ,σ r / / S σ r p − 1 N comm utes, where, when p = 0, we in terpret the right column a s the counit ε N : ( N !) ¡ − → N and d T 0 and d S 0 as the augmentations. T o construct the isomorphisms , for p > 0, observ e that, for e ach σ ∈ C [ p ] , the diagram A − bimo d ! / / ( dσ ) ∗   A ! − albimo d ( − ) dσ,cσ   A dσ − bimo d dσ,cσ |−| dσ,cσ   A dσ − mo d − A cσ dσ,cσ |−|   A cσ − bimo d id / / A cσ − bimo d is comm utativ e. Since eac h functor in it admits a left adjo int and dσ ,cσ | − | dσ ,cσ ◦ ( dσ ) ∗ = dσ ,cσ | − | ◦ ( − ) dσ ,cσ ◦ ! w e ha v e the isomorphisms, natural in N , γ σ N : ( L dσ ,cσ ( A dσ ⊗ A cσ N )) ¡ − → ( d σ ) ! ( A dσ ⊗ A cσ N ⊗ A cσ A dσ ) . The A i bimo dule (( L dσ ,cσ ( A dσ ⊗ A cσ N ))) ¡ i is generated by { ι i ( j ≤ j ) ((1 ⊗ n ) ⊗ 1) | j ≥ cσ, n ∈ N } for i ≤ d σ and is 0 if i  d σ . T ra cing through ON THE COHOMOLO GY COMP ARISO N THEOREM 47 the adjunction w e obtain that, for eac h i ∈ C , γ σ ,i N ( ι i ( j ≤ j ) ((1 ⊗ n ) ⊗ 1)) = 1 ⊗ n ⊗ 1 . F or all N and σ w e define γ σ N = γ σ N cσ and, for p > 0, w e hav e tha t ( d T , σ r ) ¡ i ( ι i ( j ≤ j ) ( (1 ⊗ n ) ⊗ 1)) = ι i ( j ≤ j ) ( d T ,σ r (1 ⊗ n ) ⊗ 1) = ι i ( j ≤ j ) ((1 ⊗ T cσ r ,dσ N ( n )) ⊗ 1), while d S ,σ r (1 ⊗ n ⊗ 1 ) = 1 ⊗ T cσ r ,cσ N ( n ) ⊗ 1, so the square is commu- tativ e. When p = 0, we o btain, for σ ∈ C [0] and i ≤ dσ = cσ ≤ j that ε i N ◦ ( ε T ,σ ) ¡ i ( ι i ( j ≤ j ) ((1 ⊗ n ) ⊗ 1)) = ε i N ( ι i ( j ≤ j ) ( ε T ,σ ij (1 ⊗ n ) ⊗ 1)) = ε i N ( ι i ( j ≤ j ) ( T i,cσ N ( n ) ϕ ij ⊗ 1)) = T i,cσ N ( n ) = ε S ,σ ,i (1 ⊗ n ⊗ 1) = ε S ,σ ,i ◦ γ σ ,i N ( ι i ( j ≤ j ) ((1 ⊗ n ) ⊗ 1) ) , so the square commutes in this case to o.  Reference s [1] M. Ger s tenhabe r and S. D. Schac k, “Alg e br aic Cohomo lo gy a nd Deforma- tion Theo ry”, Defor mation Theor y of Algebras and Structures and Applications, Kluw er , Dor drech t (1988) 11-264 . [2] M. Gerstenhab er and S. D. Schac k, “The Cohomolog y of Presheav es of Algebra s : Presheaves over a Partially Or dered Set”, T rans. Amer. Math. Soc. 310 (1988) 135-1 65. [3] C. B. Kullmann, “Adjoints and Cohomology for Presheaves of Alg ebras O ver a Poset ”, SUNY at Buffalo Ph.D. thesis, (19 98). [4] S. MacLa ne, “ Ho mology ”, Spinger-V er lag, Berlin, (1967). [5] S. I. Gelfand a nd Y u. I. Manin, “Methods of Homolo gical Algebra”, Spr inger V erlag (19 96). 48 ALIN ST ANCU [6] B. Ke lle r, “Ho chsc hild Co homology and the Derived Pica rd Gro up” , Jour na l o f Pure and Applied Alg ebra 190 (2004), 177 -196. [7] Alin A. Stancu, “Ho chsch ild Coheomolo gy and Derived Categ ories”, SUNY at Buffalo Ph.D. thesis, (2006). [8] W endy Lowen a nd Michel V an Den Berg h “ A Ho chsc hild Cohomo logy Compar- ison Theorem for Pr estaks”, arXiv:0905 .2354 v1 [math.KT] E-mail addr ess : sta ncu alin1@ columb usstate.edu