Hochschild homology and cohomology of Generalized Weyl algebras: the quantum case

Hochschild homology and cohomology of Generalized W eyl algebras: the quantum case Andrea Solotar Mariano Suár ez-Alvarez Quimey V iv as ∗ June 27, 2011 Abstract W e determine the Hochschild h omology and cohomolog y of the generalized W eyl algebras of rank one which are of ‘quantum’ type in all but a few exceptional cases. 2010 MSC: 16E40, 16E65, 16U80, 16W50, 16W70. 1 Introduction The Hochschild cohomology HH ∗ ( A ) and homology HH ∗ ( A ) of a k -algebra A are invariants which are usually hard to compute. For a long time it has be en known that they a re related to the smooth- ness of the algebra . For example, if A is a commutative a lgebra A essentially of finite type — i.e. , a quotient of a polynomial algebra on a finite number of variables by an ide al, or a localization of one of these algebras — several authors [ 2 ] [ 5 ] [ 12 ] [ 14 ] [ 1 5 ] have obtained results which ca n be summarized in the statement If k is a field, gldim ( A ) < ∞ if and only if there exists n such that HH i ( A ) = 0 , for all i > n . Some years ago, L. A vramov a nd S. Iyeng ar [ 1 ] proved a cohomological version of this property: if k is a field, gldim ( A ) < ∞ if a nd only if there exists n such t h at HH i ( A ) = 0 , for all i > n . The non commutative case is different. After D. Happel a sked in [ 11 ] given a fi nit e dim ensional k - algebra A , is it true th at t he v anishing of HH i ( A ) for all large i implies t h at gldim ( A ) < ∞ ? several articles have been devoted to trying to provide a n a ffirmative answer . However , in [ 8 ] a counterexample was given, the “ small” a lgebr a k h x , y i / ( x 2 , y 2 , xy + qyx ) , with q ∈ k ∗ not a root ∗ This work has b e en supported by the projects UBA CYTX212, PIP-CONI CET 112-200801 -00487, PICT -2007 -02182, UBA- CYT 2002009030 0102 IJ and MA THAMSUD-NOCOMALRET . The first and second authors are research me mbers of CON- ICET (Argentina). A . Solotar thanks Unive rsidad de V alparaíso (Project MECESUP UV A0806). 1 of unity . Sub sequently , Y . Han [ 10 ] showed that the Hochschild homology of this algebra does not vanish in infinitely many degrees, p roposin g thus what is now known a s Han’ s conjecture : If a ll the h igher Hochschild h omology groups of a finite dimensional algebra vanish, then the global d imension of the algebra is finite. This conjecture has been proved to be true for commutative a lgebras e ssentially of finite type, not necessarily finite dimensional [ 2 , 5 ], for finite d imensional graded local algebr as [ 7 ], f or finite di- mensional monomial algebras [ 10 ], for finite dimensional graded cellular algebras in char acteristic zero [ 7 ], for finite dimensional Koszul algebras in chara c te ristic zero [ 7 ], for quantum complete in- tersections [ 6 ], for finite dimensional graded local a lgebras satisfying the hypotheses of Theorem II of [ 17 ], and for algebras satisfying the hypotheses of Theorem I of [ 1 7 ]. The genera l answer is, however , still unknown. The proof of this last case makes use of the fact that Hochschild homology is f unctorial, which is not valid f or Hochschild cohomology . The results of Theorem I of [ 17 ] led us to consider the conjecture without the hypothesis of A being finite dimensional. It is worth to notice that the proof of the conjecture — homolo gical and cohomol ogical — in the commutative ca se, uses the existence of a model , that is, a d ifferential graded algebr a quasi- isomorphic to the inital one, a nd having thus isomorphic Hochschild homology and cohomology . The importance of the model, stated informally , is that it allows, in a certain way , to treat mor e easily the singularities of the algebra. In other words, the difficulty is no longer in the algebra itself, but in the differentials of the model. This kind of model, coming from algebra ic topology , a lways exists in the commutative essentially of finite type case, but usually not in the non commutative case. One e xample of a situation where it e x ists is treated in Theorem II of [ 17 ]. Also, for Koszul algebras, it is clea r that the complex which can be used to compute Hochschild (co)homology is similar to the one constructed from a model in the commutative c ase. So, in our opinion, and although the methods used in [ 7 ], [ 1 0 ], [ 6 ] are different, Han’s c onjecture ha s been proven, up to now , for algebr a s which have some kind of “model”. Following this point of view , in this a rticle we prove it for a family of non commutative algebra s A q , the quantum generalized W eyl algebras , which we shall call simply Bavula algebras . For this we compute their Hochschild cohomology a nd homology , completing in this way the results of [ 9 ], leaving out only a few cases. W e get the following two results: Theorem 1.1. Let A = A ( σ q , a ) be a Bavula algebra with q ∈ k × not a root of 1 . Th en HH p ( A ) =              k N ⊕ M r ∈ Z \ 0 k if p = 0 ; k M ⊕ M r ∈ Z \ 0 k if p = 1 ; k M if p ≥ 2 ; HH p ( A ) =        k if p = 0 , 1 ; k N if p = 2 ; k M if p ≥ 3 ; 2 where N = deg a and M = deg ( a : a ′ ) . Theorem 1.2. Let A = A ( σ q , a ) be a Bavula algebra with q ∈ k × such t hat q e = 1 . Then HH p ( A ) =                        k η ( a ) ⊕ M r ∈ Z \ 0 S if p = 0 ; k η ( c ) ⊕ M r ∈ Z \ e Z  k [ h ] / ( h )  ⊕ M r ∈ e Z S 2 if p = 1 ; k [ h ] / ( c ) ⊕ M r ∈ e Z S if p = 2 ; k [ h ] / ( c ) if p ≥ 3 . HH p ( A ) =                      M r ∈ e Z S if p = 0 ; M r ∈ e Z S 2 if p = 1 ; k η ( a / c ) ⊕ k [ h ] / c ⊕ M r ∈ e Z S if p = 2 ; k [ h ] / c if p ≥ 3 . where, for a polynomial f ∈ k [ h ] , we write η ( f ) = deg f − 1 e deg N ( f ) with N the operator defined in section 2 below , N = deg a , c = ( a : a ′ ) and and M = deg c . Whether the ‘quantum parameter ’ q appear ing in the definition of these Bavula algebra s is a root of unity or not is a fact that plays a fundamental role, since the c omputations differ substan- tially in both cases. The a r ticle is organized as follows. In Section 2 we fix the notations and state some a uxiliar results that will be necessary in the rest of the a rticle. In Section 3 we recall form [ 3 ] the d efinition of these algebras a nd we study their global dimension. In Se c tion 4 we compute a projective resolution of our algebr a A as an A -bimodule. In Section 5 we compute the Hochschild homology and, finally , in Section 6 we compute the Hochschild cohomology . 2 Notations and some generalities Let k be a field of characteristic ze ro. If λ ∈ k and n ≥ 0 , we write [ n ] λ = 1 + λ + · · · + λ n − 1 ; in particular , if λ = 1 , then [ n ] λ = n . W e fix a scala r q ∈ k \ { 0 , 1 } and a monic polynomial a = P N i = 0 α i h i ∈ k [ h ] of degree deg a = N > 1 . Througho ut the paper , A = A ( a , q ) will denote the k -algebra f reely generated by letters y , h and x subject to the relations xh = qhx , yx = a ( h ) , hy = qyh , xy = a ( qh ) . It is easy to see that the set { y i h j : i , j ≥ 0 } ∪ { h j x k : j ≥ 0 , k ≥ 1 } is a basis of A as a k -module. 3 W e le t σ = σ q : k [ h ] → k [ h ] be the algebra automorphism such that σ ( h ) = qh . Then xr = σ ( r ) x and ry = yσ ( r ) for all r ∈ k [ h ] , and xy = σ ( a ) . Moreover , the algebra A is Z -graded in such a way that the genera tors have degrees | y | = 1 , | h | = 0 and | x | = − 1 ; we refer to the d egree of a n element homogeneous with respect to this grad ing a s its weight . W e remark that there is an algebra isomorphism Φ : A ( a , q ) → A ( σ q ( a ) , q − 1 ) such that Φ ( x ) = y , Φ ( y ) = x a nd Φ ( h ) = h . This isomorphism maps the homogeneous component of weight r ∈ Z of A ( a , q ) to the component of weight − r of its codomain. This observa tion will allow us to carry out homol ogical computations just in weights r ≥ 0 , since all arguments will be transfera ble to negative d egrees using Φ . Given polynomials p , t ∈ k [ h ] , we shall write ( p : t ) their greatest common divisor and p ′ the derivative of p and we make the convention that the d egree of the zero polynomial is − ∞ . W e let c = ( a : a ′ ) and M = de g ( c ) . If q is a root of 1 , we le t e be its order; if q is not a root of unity we let e = 0 . If r ∈ Z , we say that r is singular if e | r , and that it is regular otherwise. The subring of k [ h ] fixed by σ is S = ker ( σ − 1 ) , generate d by h e . W e sa y that a polynomial p ∈ k [ h ] is singular if p ∈ S . More genera lly , when e > 0 we have ke r ( σ − q l ) = h l k [ h e ] for e ach l ∈ { 0 , . . . , e − 1 } . If e > 0 , for each f ∈ k [ h ] such tha t f ( 0 ) 6 = 0 we define N ( f ) = lcm  f : σ ( f ) : · · · : σ e − 1 ( f )  and f = N ( f ) f . Clearly σ ( N ( f )) is a scalar multiple of N ( f ) ; evaluating both at 0 shows then they a re in fact equal, so that N ( f ) ∈ S . The reason which motivates our interest in the operator N is the followin g propositio n: Proposition 2.1. Let f , g ∈ k [ h ] and supp ose f ( 0 ) 6 = 0 . (i) If fg ∈ S , then f | g . (ii) If g ∈ S and f | g , then there exists s ∈ S such that g = N ( f ) s . Proof. S ince fg ∈ S , we know that σ i ( fg ) = fg , so σ i ( f ) | fg f or all i . The first statement f ollows now from the definition of N ( f ) . The second one is an immediate consequence. W e end this section with two technical lemmas which will be of use in the computation of Sections 5 and 6 . Lemma 2.2 . Let f ∈ k [ h ] and suppose that f ( 0 ) 6 = 0 and th at q is a root of unity of order e . If π : k [ h ] → k [ h ] / ( f ) be the canonical p rojection, t hen for each l ≥ 0 we hav e dim π ( h l S ) = deg N ( f ) e . Proof. S ince multiplication by π ( h ) on k [ h ] / ( f ) is an isomorphism, it is enough to prove this when l = 0 . Let us consider the following commutative d ia gram, in which the morphisms are the 4 obvious ones: k [ h ] π   π ′ / / / / k [ h ] / ( N ( f )) ρ x x p p p p p p p p p p p k [ h ] / ( f ) Let G be a cyclic group of order e generated by an element g ∈ G . W e endow k [ h ] with the action of G such that g acts as σ . Since N ( f ) is G -invariant there is an induced action on k [ h ] / ( N ( f )) . T he map π ′ is surjective, so the restriction ( π ′ ) G : S →  k [ h ] / ( N ( f ))  G is surjective too. The situation is described by the following commutative dia gra m S π | S   ( π ′ ) G / / ( k [ h ] / ( N ( f ))) G ρ w w o o o o o o o o o o o k [ h ] / ( f ) If s ∈ S is such that π | S ( s ) = 0 , then there exists a b ∈ k [ h ] such that fb = s ∈ S a nd it follows from the previous proposotion that b = ¯ fs 1 for some s 1 ∈ S : we see that s = N ( f ) s 1 and ( π ′ ) G ( s ) = 0 . As ( π ′ ) G is surjective, this implies that the map ρ is injective and, as a consequnce, that dim π ( S ) = dim ( k [ h ] / ( N ( f ))) G . Now , k [ h ] / ( N ( f )) has { h i : 0 ≤ i < deg N ( f ) } as a basis and the action of G is diagonal with respect to it. It is immediate, then, that dim ( k [ h ] / ( N ( f ))) G = 1 e deg N ( f ) Lemma 2.3. Let f ∈ k [ h ] such th at f ( 0 ) 6 = 0 , q ∈ k a root of unity of ord er e , l ≥ 0 and consider t h e S -linear m ap ψ f , l : p ∈ k [ h ] 7 → ( σ − q l )( fp ) ∈ k [ h ] . Then coker ψ f , l ∼ = h l S ⊕ k η ( f ) with η ( f ) = deg f − 1 e deg N ( f ) . Proof. W e decompose k [ h ] ∼ = S ⊕ h S · · · ⊕ h e − 1 S as S -module. Since ker ( σ − q l ) = h l S , the map σ − q l induces an injective map k [ h ] / h l S → k [ h ] , still denoted σ − q l . Consider the following diagram k [ h ] f / / k [ h ] σ − q l / / p   k [ h ] k [ h ] / h l S σ − q l : : u u u u u u u u u Because σ − q l is injective, it is immediate that coker ψ f , l ∼ = h l S ⊕ coker ( p ◦ f ) and, since coker p ◦ f ∼ = k [ h ] h l S + ( f ) ∼ = k [ h ] ( f ) . π ( h l S ) with π the map defined in Lemma 2.2 , we see that dim coker p ◦ f = η ( f ) . 5 W e remark that the isomorphism in the statement of this lemma is actually an isomorphism of S -modules, if we identify the summand k η ( f ) with the quotient k [ h ] / ( h l S + ( f )) appearing in the proof. 3 Global dimension Given a noetherian algebra R which is an integral domain, a non zero central element a ∈ R a nd an algebra automorphism σ ∈ Aut k ( R ) , the Bavula algebra Λ = Λ ( R , σ , a ) is the k -algebra generated by R and two varia b le s x , y subject to the relations yx = a , xy = σ ( a ) , xr = σ ( r ) x , ry = yσ ( r ) for all r ∈ R . It was introduced by V . Bavula in [ 3 ] with the name of g eneralized Weyl algebra . The algebra A introduced in S ection 2 is a special case of this construction . The algebra A is a noetherian domain and there is a Z -grading on Λ with a ll elements of R in degree 0 , and x a nd y in degrees − 1 and 1 , respectively; we de note | u | the degree of a n homoge- neous e lement u ∈ Λ and call it its weigh t . Using the e a sily obta ined description of automorphisms of k [ h ] , one can obtain the following classification of the algebras of the form Λ ( k [ h ] , σ , a ) up to isomorphism, a s in [ 13 ]: Proposition 3.1. The algebra Λ = Λ ( k [ h ] , σ , a ) is isomorphic to exact ly one of th e following list: 1. Λ ( k [ h ] , Id, a ) for some a ∈ k [ h ] ; 2. Λ ( k [ h ] , σ cl , a ) with σ cl ( h ) = h − 1 and a ∈ k [ h ] ; 3. Λ ( k [ h ] , σ q , a ) with q ∈ k \ { 0 , 1 } , σ q ( h ) = qh and a ∈ k [ h ] . W e refer t o case 2 as the classical case and t o case 3 as the quantum case . If b ∈ R , let I ( x , b ) = Λx + Λb ⊆ Λ . Bavula proved in [ 4 ] the following result concerning the global dimension of his algebras: Theorem 3.2. [ 4 , Thm. 3.5] If R is a commutativ e Noetherian dom ain of finite global dim ension n and a 6 = 0 , then the following t wo co nd itions are equivalent: • gldim Λ < ∞ • pdim Λ Λ / I ( x , p) < ∞ for all prime ideals p of R which c o ntain a . When R = k [ h ] , the hypotheses of this theorem are satisfied and we ca n give a chara c terization of Bavula algebras of finite global dimension. Theorem 3.3. Let R = k [ h ] , a ∈ R , σ ∈ Aut k ( R ) and Λ = Λ ( R , σ , a ) . Then gldim Λ < ∞ ⇐ ⇒ ( a : a ′ ) = 1 . Proof. The “only if ” part has be e n proved by Bavula in [ 4 ], so we only have to prove the “ if ” part. 6 Let p ∈ R be a prime element which d ivides a , so that there is a b ∈ R with a = pb . The canonical short exact sequence of left Λ -modules 0 → I ( x , p ) → Λ → Λ / I ( x , p ) → 0 tells us that pdim Λ Λ / I ( x , p ) < pdim Λ I ( x , p ) + 2 . W e shall prove that if ( a : a ′ ) = 1 , then I ( x , p ) is a projective Λ - module. W e start by showting that Λx ∩ Λp = I ( x , b ) p . Fix f ∈ Λx ∩ Λp ; we may assume that f is homogeneous with respect to the weight and that | f | = r ≥ 0 : the case in which the weight of f is negative is similar . Since f ∈ Λx ∩ Λp , there exist u , v ∈ R such that f = y r + 1 ux = y r vp . As y r + 1 ux = y r aσ − 1 ( u ) = y r pbσ − 1 ( u ) = y r σ − 1 ( u ) bp and Λ is a domain, σ − 1 ( u ) b = v and, in consequence, f ∈ I ( x , b ) p . The other inclusion is e asy . Consider now the short sequence of left Λ -modules 0 / / I ( x , b ) γ / / Λ ⊕ Λ φ / / I ( x , p ) / / 0 (1) where φ ( α , β ) = αx − βp and γ ( w ) = ( wpx − 1 , w ) ; this last expression makes sense because for all p ∈ I ( x , b ) we have wp ∈ Ax = xA and A is a domain. It is clear that γ is a monomorphism, φ is an epimorphism and that im γ ⊆ ker φ . The se- quence ( 1 ) is in fact exact: to check the other inclusion suppose that ( α , β ) ∈ Λ ⊕ Λ is such that αx = βp . This element belongs to Λx ∩ Λp = I ( x , b ) p , and it follows that α = βpx − 1 . If ( a : a ′ ) = 1 , then ( p : b ) = 1 a nd there e xist s , t ∈ R such that 1 = sp + tb . W e define the map ψ : Λ ⊕ Λ → Λ by ψ ( α , β ) = αxs + βbt . It is easy to verify that im ψ ⊆ I ( x , b ) and that ψ ◦ γ = Id I ( x , b ) . As a consequence, the sequence ( 1 ) splits and I ( x , p ) is a projective Λ -module. In particular , for the algebras introduced in Section 2 we have the following: Corollary 3.4. For all q ∈ k \ { 0 , 1 } and all a ∈ k [ h ] we have gldim Λ ( k [ h ] , σ q , a ) < ∞ = ⇒ gldim Λ ( k [ h ] , σ q , a ) = 2 . Proof. It follows from [ 4 , Thm. 2.7] that if the global d imension of Λ ( R , σ , a ) is finite, it equals either gldim R or gldim R + 1 . In the situation of the corollary , then, gldim Λ ( k [ h ] , σ , a ) ∈ { 1 , 2 } if it is finite. Moreover , using [ 4 , T hm. 3.7] , we see that gldim Λ ( k [ h ] , σ , a ) = 2 if and only if either (i) there is a maximal ideal of k [ h ] of height 1 with finite orbit under σ , or (ii) if there a re maximal ideals p , q of k [ h ] of height 1 such that σ i (p) = q for some i 6 = 0 , i ∈ Z and a ∈ p ∩ q . Since the idea l ( h ) of k [ h ] is obviously fixed by σ a nd it is of height 1 , we a re always in case (i) , and the corollary follows. The conditions (i) a nd (ii) mentioned in the proof of this corollary are not exclusive. Indeed, most of the complication encountered in the computations that follow ar ises when the algebra A satisfies condition (ii) or , in other words, when the polynomial a has two roots in the same orbit under σ q . 7 4 A projective resolution The purpose of this section is to construct a projective resolution of the Bavula algebra A . W e do this in two steps, using an algebra B l as a n intermediate step, as in [ 9 ]. 4.1 Sm ith algebras Fix a polynomial l = P m i = 0 λ i H i ∈ k [ H ] , with m > 0 and λ m 6 = 0 . W e consider the k -a lgebra B l , or simply B , with generators Y , H and X subject to the relations HY = qYH , [ X , Y ] = l , XH = qHX . This algebra was considered by P . Smith in [ 16 ], observing that it is in many aspects similar to the enveloping algebra U (sl 2 ) ; we will call it a Smit h a lgebra . The set { Y i H j X k : i , j , k ≥ 0 } is a basis of B a s a k -module. Let V = kY ⊕ kH ⊕ kX ⊂ B . S etting | X | = | Y | = 1 and | H | = 0 we obtain a grading on TV , which induces an increasing filtration on B ; let us write Y , H and X for the pr incipal symbols of Y , H and X , respectively , in B = gr B . Then B is the k -algebra generated by Y , H and X , subject to the relations HY = qY H , [ X , Y ] = 0 , XH = qH X . Of course, V ∼ = gr V is spa nned by X , Y and H , and these elements are k -linearly independent. W e will use f requently the following notation: given a function f of two integer a rguments, and i ∈ N 0 , we will write R i f ( s , t ) = X s + t + 1 = i 0 ≤ s , t f ( s , t ) . In par ticula r , in such an ”integral” ex p ression, the indices s and t are not free. W e note that the identity R i f ( s + 1 , t ) − R i f ( s , t + 1 ) = f ( i , 0 ) − f ( 0 , i ) holds f or all f and i : we will make use of it repeatedly . Consider now the complex of B e -projective modules over B 0 / / B | V 3 V | B / / B | V 2 V | B d / / B | V | B d / / B | B µ / / / / B (2) with differentials given by d ( 1 | v | 1 ) = 1 | v − v | 1 , ∀ v ∈ V ; d ( 1 | H ∧ X | 1 ) = 1 | X | H − qH | X | 1 − q | H | X + X | H | 1 ; d ( 1 | Y ∧ X | 1 ) = 1 | X | Y − Y | X | 1 − 1 | Y | X + X | Y | 1 − P i R i λ i H s | H | H t ; d ( 1 | Y ∧ H | 1 ) = 1 | H | Y − qY | H | 1 − q | Y | H + H | Y | 1 ; 8 d ( 1 | Y ∧ H ∧ X | 1 ) = 1 | H ∧ X | Y − qY | H ∧ X | 1 − q | Y ∧ X | H + qH | Y ∧ X | 1 + q | Y ∧ H | X − X | Y ∧ H | 1 . The verification that d 2 = 0 is a routine computation. The filtra tions on B and on V de termine a filtra tion on the complex ( 2 ), whose associated graded complex is 0 / / B | V 3 V | B d / / B | V 2 V | B d / / B | V | B d / / B | B µ / / / / B with B e -linear differentials determined by the conditions d ( 1 | v | 1 ) = 1 | v − v | 1 , ∀ v ∈ V ; d ( 1 | H ∧ X | 1 ) = 1 | X | H − qH | X | 1 − q | H | X + X | H | 1 ; d ( 1 | Y ∧ X | 1 ) = 1 | X | Y − Y | X | 1 − 1 | Y | X + X | Y | 1 ; d ( 1 | Y ∧ H | 1 ) = 1 | H | Y − qY | H | 1 − q | Y | H + H | Y | 1 ; d ( 1 | Y ∧ H ∧ X | 1 ) = 1 | H ∧ X | Y − qY | H ∧ X | 1 − q | Y ∧ X | H + qH | Y ∧ X | 1 + q | Y ∧ H | X − X | Y ∧ H | 1 . This complex is exact. Indeed, there is a left B -linear contraction given by s ( 1 ) = 1 | 1 ; s ( 1 | Y i H j X k ) = P i R i Y s | Y | Y t H j X k + P i R j Y i H s | H | H t X k + P i R k Y i H j X s | X | X t ; s ( 1 | Y | Y i H j X k ) = 0 ; s ( 1 | H | Y i H j X k ) = P i R i q s Y s | Y ∧ H | Y t H j X k ; s ( 1 | X | Y i H j X k ) = P i R i Y s | Y ∧ X | Y t H j X k + P i R j q s Y i H s | H ∧ X | H t X k ; s ( 1 | H ∧ X | Y i H j X k ) = P i R i q s Y s | Y ∧ H ∧ X | Y t H j X k ; s ( 1 | Y ∧ X | Y i H j X k ) = 0 ; s ( 1 | Y ∧ H | Y i H j X k ) = 0 . It follows that the complex ( 2 ) is a B e -projective resolution of B . 4.2 Bavula algebras Next we construct a resolution of our Ba vula a lgebra as a bimodule over itself. Let l = σ ( a ) − a ; then deg a ≥ deg l and l = P N i = 0 λ i h i with λ i = ( q i − 1 ) α i . W e consider the Smith algebra B = B l corresponding to the polynomial l , and the element Ω = YX − a ∈ B . A simple computation shows that Ω = XY − σ ( a ) and that Ω is central in B . In pa r ticular , BΩ = ΩB is a two-sided ide al of B and the quotient B / ΩB is isomorphic to A via an isomorphism which sends the classes of Y , H and X to y , h and x respectively . W e will identify A with the quotient. 9 Let π : B → A denote the canonical projection. Since Ω is not a z e ro divisor in B , the complex 0 − → B Ω − → B π − → A − → 0 (3) is a p rojective resolution of A as a B -module both on the left and on the right; here the first arrow is simply the multiplication by Ω . On the other ha nd, b y a p plying the functor (−) ⊗ B A to the resolution ( 2 ) of B a s B e -module given in the previous subsection, we obtain the complex 0 / / B | V 3 V | A d / / B | V 2 V | A d / / B | V | A d / / B | A µ / / A / / 0 (4) with B ⊗ A op -linear differentials given by d ( 1 | v | 1 ) = 1 | v − v | 1 , ∀ v ∈ V ; d ( 1 | H ∧ X | 1 ) = 1 | X | h − qH | X | 1 − q | H | x + X | H | 1 ; d ( 1 | Y ∧ X | 1 ) = 1 | X | y − Y | X | 1 − 1 | Y | x + X | Y | 1 − P i R i λ i H s | H | h t ; d ( 1 | Y ∧ H | 1 ) = 1 | H | y − qY | H | 1 − q | Y | h + H | Y | 1 ; d ( 1 | Y ∧ H ∧ X | 1 ) = 1 | H ∧ X | y − qY | H ∧ X | 1 − q | Y ∧ X | h + qH | Y ∧ X | 1 + q | Y ∧ H | x − X | Y ∧ H | 1 . The homology of this complex is T or B • ( B , A ) , so that it is in f act acyclic. This means that ( 4 ) is a projective resolution of A as a left B -module. There exist morphisms between the two resolutions ( 3 ) and ( 4 ) of the left B -module A lifting the identity map of A : 0 / / B Ω / / f 1   B π / / / / f 0   A 1 A   · · · / / B | V | A d / / g 1 O O B | A µ / / / / g 0 O O A 1 A O O (5) given by f 0 ( 1 ) = 1 | 1 ; f 1 ( 1 ) = − Y | X | 1 − 1 | Y | x + P i R i α i H s | H | h t ; g 0 ( 1 | y i h j ) = Y i H j ; g 0 ( 1 | h j x k ) = H j X k ; g 1 ( 1 | Y | y i h j ) = 0 ; g 1 ( 1 | Y | h j x k + 1 ) = − q − j H j X k g 1 ( 1 | H | y i h j ) = 0 ; g 1 ( 1 | H | h j x k ) = 0 ; g 1 ( 1 | X | y i + 1 h j ) = − Y i H j ; g 1 ( 1 | X | h j x k ) = 0 . Using ( 3 ), the computation of T or B • ( A , A ) is immediate because the only relevant differential vanishes, and we see that T or B p ( A , A ) =        A ⊗ B B , p = 0 ; A ⊗ B B , p = 1 ; 0 , p ≥ 2 . (6) 10 Since T or B • ( A , A ) ca n be ca lcula ted from any resolution of A a s left B -module, the complex ob- tained by applying the functor A ⊗ B (−) to the resolution ( 4 ), that is 0 / / A | V 3 V | A d / / A | V 2 V | A d / / A | V | A d / / A | A (7) with A e -linear differential: d ( 1 | v | 1 ) = 1 | π ( v ) − π ( v ) | 1 , ∀ v ∈ V ; d ( 1 | H ∧ X | 1 ) = 1 | X | h − qh | X | 1 − q | H | x + x | H | 1 ; d ( 1 | Y ∧ X | 1 ) = 1 | X | y − y | X | 1 − 1 | Y | x + x | Y | 1 − P i R i λ i h s | H | h t ; d ( 1 | Y ∧ H | 1 ) = 1 | H | y − qy | H | 1 − q | Y | h + h | Y | 1 ; d ( 1 | Y ∧ H ∧ X | 1 ) = 1 | H ∧ X | y − qy | H ∧ X | 1 − q | Y ∧ X | h + qh | Y ∧ X | 1 + q | Y ∧ H | x − x | Y ∧ H | 1 , has homolog y isomorphic to T or B • ( A , A ) . U sing the morphisms f • and g • from ( 5 ), we see that the homolo gy of the complex ( 7 ) is freely generate d as left A -module by the classes of the cycles 1 | 1 ∈ A ⊗ A and y | X | 1 + 1 | Y | x − P i R i α i h s | H | h t ∈ A ⊗ V ⊗ A , of degrees 0 and 1 , respectively . 4.3 The resolution Next we consider the third-quadrant double complex X • , • depicted in the following diagram 0 / / A | V 3 V | A d / / A | V 2 V | A d / / A | V | A d / / A | A 0 / / A | V 3 V | A d / / O O A | V 2 V | A d / / δ O O A | V | A d / / δ O O A | A δ O O . . . . . . O O . . . δ O O . . . δ O O . . . δ O O so tha t X p , q = A | V p − q V | A if q ≥ 0 and X p , q = 0 otherwise, with horizontal A e -linear differ- entials d , of bide gree (− 1 , 0 ) , given as in ( 7 ), a nd v e rtical differentials δ , of bidegree ( 0 , 1 ) , given by δ ( 1 | 1 ) = y | X | 1 + 1 | Y | x − P i R i α i h s | H | h t ; δ ( 1 | Y | 1 ) = − y | Y ∧ X | 1 + P i R i α i q t h s | Y ∧ H | h t ; δ ( 1 | H | 1 ) = 1 | Y ∧ H | x − y | H ∧ X | 1 ; δ ( 1 | X | 1 ) = 1 | Y ∧ X | x − P i R i α i q s h s | H ∧ X | h t ; δ ( 1 | Y ∧ H | 1 ) = y | Y ∧ H ∧ X | 1 ; 11 δ ( 1 | Y ∧ X | 1 ) = P i R i α i q i − 1 h s | Y ∧ H ∧ X | h t ; δ ( 1 | H ∧ X | 1 ) = 1 | Y ∧ H ∧ X | x . A direct computation shows that it is indeed a complex with anti-commuting differentials. T o compute the homology of the total complex T ot X • , • we use the spectral sequence E which arises f rom the filtration by rows. The differential on the first page E 0 of this spe c tral sequence is the horizontal differential d on X • , • , a nd we have essentially computed the corresponding homol- ogy in ( 6 ): we see from this that the second page E 1 of E is, up to isomorphism, as in the following diagram: 0 0 A A 0 0 A A d 1 O O 0 0 A A d 1 O O . . . . . . . . . . . . Consequently , the only components of the differential d 1 which can possibly be non zero are the maps d 1 p , p : E 1 p , p → E 1 p , p − 1 , with p ≥ 1 , and they are induced by the vertical differentials δ in X • , • . W e know that E 1 p , p and E 1 p , p − 1 are f ree left A -modules on the horizontal homology classes of 1 | 1 ∈ X p , p and ω = y | X | 1 + 1 | Y | x − P i R i α i h s | H | h t ∈ X p , p − 1 , respectively . In view of the definition of δ , d 1 ([ 1 | 1 ]) = [ ω ] , and, since d 1 is A -linear , this shows that all components of d 1 which are not trivially zero are isomorphisms. It follows that the c omplex T ot X • , • is acyclic over A , with augmentation given by the multipli- cation map µ : X 0 , 0 = A ⊗ A → A and, since its components are free A e -modules, it is in fa ct a projective resolution of A as A e -module. W e consider the grading V such that Y , H and X are homogeneous of degrees 1 , 0 and − 1 , respectively . This, together with the grad ing of A by weights, induces a grading on the complex X • , • such the differentials a re homogeneous. It follows that the complexes obtained by applying the functors A ⊗ A e (−) and hom A e (− , A ) below will also be graded b y weights in a natural way . 5 Hochschild homology In this section we will compute the Hochschild homolog y of A using the resolution described in the previous section a nd a spectra l sequence a rgument. Applying the f unctor A ⊗ A e − to X • , • and identifying A ⊗ A e ( A ⊗ ∧ p V ⊗ A ) with A ⊗ ∧ p V in the natural way , we get a double complex such that the homology of its total complex is HH ∗ ( A ) , 12 the Hochschild homology of A with coefficients in itself. This double complex is 0 / / A | V 3 V d / / A | V 2 V d / / A | V d / / A 0 / / A | V 3 V d / / δ O O A | V 2 V d / / δ O O A | V d / / δ O O A δ O O . . . . . . δ O O . . . δ O O . . . δ O O . . . δ O O with differentials given by d ( u | Y ) = [ y , u ] , (8a) d ( u | H ) = [ h , u ] , (8b) d ( u | X ) = [ x , u ] , (8c) d ( u | Y ∧ H ) = [ y , u ] q | H + [ u , h ] q | Y , (8d) d ( u | Y ∧ X ) = [ y , u ] | X + [ u , x ] | Y − P i λ i R i h t uh s | H , (8e) d ( u | H ∧ X ) = [ h , u ] q | X + [ u , x ] q | H , (8f) d ( u | Y ∧ H ∧ X ) = [ y , u ] q | H ∧ X + q [ u , h ] | Y ∧ X − [ u , x ] q | Y ∧ H , (8g) and δ ( u ) = uy | X + xu | Y − P i α i R i h t uh s | H , (9a) δ ( u | Y ) = − uy | Y ∧ X + P i α i R i q t h t uh s | Y ∧ H , (9b) δ ( u | H ) = xu | Y ∧ H − uy | H ∧ X , (9c) δ ( u | X ) = xu | Y ∧ X − P i α i R i q s h t uh s | H ∧ X , (9d) δ ( u | Y ∧ H ) = uy | Y ∧ H ∧ X , (9e) δ ( u | Y ∧ X ) = P i α i R i q i − 1 h t uh s | Y ∧ H ∧ X , (9f) δ ( u | H ∧ X ) = xu | Y ∧ H ∧ X . (9g) W e will use the filtration by columns on this complex a nd de note E the corresponding spectral sequence, which, as the complex A ⊗ A e X • , • itself, is grad e d by weights. W e are going to write HH • ( A ) ( r ) and E ( r ) the components of weight r in HH • ( A ) = H ( A ⊗ A e X • , • ) and E . 5.1 First Pa ge Let X be the complex 0 / / A δ / / A | V δ / / A | V 2 V δ / / A | V 3 V / / 0 (10) graded so that A and A | V 3 V are in degrees 0 and 3 , respectively , and with differentials as in ( 9a )– ( 9g ). It is clear that E 1 p , q = H p − q ( X ) for all q > 0 and that the E 1 p , 0 can be seen a s cokernels of the differentials of X . 13 For each r ∈ Z , let X ( r ) be the homogeneous component of weight r . In this subsection, we compute H • ( X ) = L r ∈ Z H • ( X ( r ) ) . Proposition 5 .1. If r ∈ Z is non zero, then the complex X ( r ) is exact. O n the ot her h and, th ere are isomorphisms of S -modules H p ( X ( 0 ) ) ∼ =  k [ h ] / ( c ) , if 2 ≤ p ≤ 3 ; 0 , otherwise. Proof. One wa y to organize the computation is as follows: • If u = p ∈ X ( 0 ) 0 , with p ∈ k [ h ] , then δ ( u ) = yσ ( p ) | X + σ ( p ) x | Y − a ′ p | H . (11) As A is a domain, it follows immediately that δ is a monomorphism and that H 0 ( X ( 0 ) ) = 0 . • Let u = p 1 x | Y + p 2 | H + yp 3 | X ∈ X ( 0 ) 1 , with p 1 , p 2 , p 3 ∈ k [ h ] . W e know that δ ( u ) = ( p 1 σ ( a ′ ) + σ ( p 2 )) x | Y ∧ H + σ ( a )( p 3 − p 1 ) | Y ∧ X − y ( p 3 σ ( a ′ ) + σ ( p 2 )) | H ∧ X . (12) Since A is a domain, we see that δ ( u ) = 0 if and only if p 1 = p 3 and p 2 = − σ − 1 ( p 1 ) a ′ . This description of cyles together with the e xpression ( 11 ) of b oundar ies imply that H 1 ( X ( 0 ) ) = 0 . • Let u = p 1 x | Y ∧ H + p 2 | Y ∧ X + yp 3 | H ∧ X ∈ X ( 0 ) 2 . A computation shows that δ ( u ) = ( p 1 σ ( a ) + p 2 σ ( a ′ ) + σ ( a ) p 3 ) | Y ∧ H ∧ X . (13) Suppose that u ∈ ker δ , so p 1 σ ( a ) + p 2 σ ( a ′ ) + σ ( a ) p 3 = 0 . It follows immediate ly from this that σ ( a c )( p 1 + p 3 ) = − σ ( a ′ c ) p 2 . Since a / c and a ′ / c are coprime, there exists g ∈ k [ h ] such that p 1 + p 3 = − σ ( a ′ c ) g and p 2 = σ ( a c ) g . If v , r ∈ k [ h ] are such that g = vσ ( c ) + r and deg r < deg c , then u is homologous to u − δ ( σ − 1 ( p 1 ) | H + yv | X ) = rσ ( a c ) | Y ∧ X − yrσ ( a ′ c ) | H ∧ X . It follows from this that e very homology class of degree 2 in X ( 0 ) is represented by a cycle of the form rσ ( a c ) | Y ∧ X − yrσ ( a ′ c ) | H ∧ X with r ∈ k [ h ] with d eg r < deg c = M . In view of the formula ( 12 ), one of these cycles is a boundary if and only if it is zero, and we can then conclude that H 2 ( X ( 0 ) ) ∼ = k [ h ] / ( σ ( c )) ∼ = k [ h ] / ( c ) . • It follows immediately from ( 13 ) tha t δ ( X ( 0 ) 2 ) = σ ( c ) k [ h ] | Y ∧ H ∧ X , so H 3 ( X ( 0 ) ) ∼ = k [ h ] / ( c ) . W e fix now r > 0 , and show that X ( r ) is exact. • Let u ∈ X ( r ) 0 , so that u = y r p f or some p ∈ k [ h ] . Then δ ( u ) = y r − 1 σ r ( a ) p | Y − y r p P i α i [ i ] q r h i − 1 | H + y r + 1 σ ( p ) | X , ( 14) and we see immediately that this is zero if and only if p = 0 , so H 0 ( X ( r ) ) = 0 . 14 • Let u = y r − 1 p 1 | Y + y r p 2 | H + y r + 1 p 3 | X ∈ X ( r ) 1 with p 1 , p 2 , p 3 ∈ k [ h ] . A s δ ( u ) = y r − 1  p 1 P i α i [ i ] q r h i − 1 + σ r ( a ) p 2  | Y ∧ H + y r (− σ ( p 1 ) + σ r + 1 ( a ) p 3 ) | Y ∧ X − y r + 1  σ ( p 2 ) + p 3 P i α i q i − 1 [ i ] q r h i − 1  | H ∧ X , we have that u is a cycle if and only if p 1 P i α i [ i ] q r h i − 1 + σ r ( a ) p 2 = 0 , σ r + 1 ( a ) p 3 = σ ( p 1 ) , σ ( p 2 ) + p 3 P i α i q i − 1 [ i ] q r h i − 1 = 0 . The first one follows from the other two, so we can drop it, and we c a n replace the remaining ones b y p 2 = − σ − 1 ( p 3 ) P i α i [ i ] q r h i − 1 , p 1 = σ r ( a ) σ − 1 ( p 3 ) . W e thus obtain a description of all 1 -cycles in X ( r ) and comparing it with ( 14 ), we see that they are all boundaries: it follows that H 1 ( X ( r ) ) = 0 . • For u = y r − 1 p 1 | Y ∧ H + y r p 2 | Y ∧ X + y r + 1 p 3 | H ∧ X ∈ X ( r ) 2 with p 1 , p 2 , p 3 ∈ k [ h ] , we have δ ( u ) = y r  σ ( p 1 ) + p 2 P i α i q i − 1 [ i ] q r h i − 1 + σ r + 1 ( a ) p 3  | Y ∧ H ∧ X . If u is a cycle, then p 1 = − σ − 1 ( p 2 P i α i q i − 1 [ i ] q r h i − 1 + σ r + 1 ( a ) p 3 ) so that, in fact, u = − δ ( y r − 1 σ − 1 ( p 2 ) | Y + y r σ − 1 ( p 3 ) | H ) . It follows f rom this that H 2 ( X ( r ) ) = 0 . • For each p ∈ k [ h ] , we have that δ ( y r − 1 σ − 1 ( p ) | Y ∧ H ) = y r p | Y ∧ H ∧ X . This means that δ ( X r 2 ) = X r 3 , so H 3 ( X ( r ) ) = 0 . At this point, we know most of the second pa ge of our spectr al sequence: Corollary 5.2. Let r ∈ Z a weight. The dimensions of the v ect or spaces a p pearing in th e hom ogeneous component of weight r of E 1 are M ? ? ? M M 0 0 M M 0 0 . . . . . . . . . . . . or 0 ? ? ? 0 0 0 0 0 0 0 0 . . . . . . . . . . . . depending on wheth er r = 0 o r not. Th e question marks d enote vecto r spaces for which we still do not know the dimension. 15 5.2 Second page In vie w of the shape of E 1 , we have E ∞ = E 2 . The following proposition takes care of the latter , except f or its first row , and the rest of this section will be de voted to the computation of the few remaining vector spaces. Proposition 5. 3. For each p ≥ 0 , the differential d 1 p + 3 , p : E 1 p + 3 , p → E 1 p + 2 , p vanishes. In consequence, except for the vector sp aces denoted wit h question marks in the d iagrams of Cor ollary 5.2 , the E ∞ page coincides with the page E 1 . Proof. A simple computation shows that if f ∈ k [ h ] then d ( f | Y ∧ H ∧ X ) = y ( 1 − qσ )( f ) | H ∧ X − ( 1 − qσ )( f ) x | Y ∧ H = δ (( q − σ − 1 )( f ) | H ) . (15) It follows that a ll the differentials d 2 p + 3 , p are zero, as claimed, and the computation of E ∞ is im- mediate except for E ∞ 0 , 0 , E ∞ 1 , 0 and E ∞ 2 , 0 . Corollary 5.4. For all p ≥ 3 and all r ∈ Z th ere are isomorphisms of S -m odules HH p ( A ) ( r ) ∼ =  k [ h ] / ( c ) if r = 0 ; 0 if r 6 = 0 . Notice that this result is independent of q . Proof. A c cording to the proposition and in v ie w of the shape of the E 1 page of the spectral se- quence, this is a consequence of convergence. T o finish the computation, we need to take care of the spots in the spectral sequence tagged with question marks in the diagra ms of Corollary 5.2 . W e do this in the following two propositio ns, first for weight zero a nd then for the remaining ones. Proposition 5.5. When q is a root of unity , we have isomorphisms of S -m odules HH p ( A ) ( 0 ) ∼ = E 2 ( 0 ) p , 0 ∼ =        k η ( a ) , if p = 0 ; S ⊕ S ⊕ k η ( c ) , if p = 1 ; S ⊕ k [ h ] / ( c ) , if p = 2 ; with η ( f ) = N − 1 e deg N ( f ) for f ∈ k [ h ] as in Lemma 2.3 . On the ot her hand , if q is of infinite order we have isomorphisms HH p ( A ) ( 0 ) ∼ = E 2 ( 0 ) p , 0 ∼ =        k N , if p = 0 ; k M , if p = 1 ; k M , if p = 2 . Proof. W e write E 1 p , 0 instead of E 1 ( 0 ) p , 0 througho ut this proof, to lighten the notation. 16 Homology at E 1 2 , 0 . Suppose u = p 1 x | Y ∧ H + p 2 | Y ∧ X + yp 3 | H ∧ X ∈ E 0 2 , 0 , with p 1 , p 2 , p 3 ∈ k [ h ] , lives to E 2 , so that there exists an f ∈ k [ h ] such that d ( u ) = δ ( f ) . This means that ( 1 − σ )( p 2 ) = σ ( f ) , aσ − 1 ( p 1 + p 3 ) − qσ ( a )( p 1 + p 3 ) − p 2 ( qσ ( a ′ ) − a ′ ) = − a ′ f . Since σ is a automorphism, we can eliminate f obtaining the equivalent equation aσ − 1 ( p 1 + p 3 ) − qσ ( a )( p 1 + p 3 ) − p 2 ( qσ ( a ′ ) − a ′ ) = − a ′ σ − 1 (( 1 − σ )( p 2 )) , which we can rewrite more compactly as ( 1 − qσ )( aσ − 1 ( p 1 + p 3 ) + a ′ σ − 1 ( p 2 )) = 0 . (16) It will be necessary to treat two ca ses separately , since the result dep e nds on whether q is a root of unity or not. • Suppose first that q is not a root of 1 . In this ca se, the map 1 − qσ is a monomorphism, so ( 16 ) is the same as aσ − 1 ( p 1 + p 3 ) + a ′ σ − 1 ( p 2 ) = 0 . From this it follows that there e x ists g ∈ k [ h ] such that p 2 = − σ ( a c ) g , p 1 + p 3 = σ ( a ′ c ) g . Let b , r ∈ k [ h ] be such that g = bσ ( c ) + r with deg r < d eg c . Then u is homologous to u + δ  yb | X − σ − 1 ( p 1 ) | H  = σ ( a c ) r | Y ∧ X + yσ ( a ′ c ) r | H ∧ X , and we see that every homology class in E 2 2 , 0 is represented by a cycle of the form σ ( a c ) r | Y ∧ X + yσ ( a ′ c ) r | H ∧ X (17) with r ∈ k [ h ] with deg r < M = deg c . Conversely , each element of this form lives to E 2 . Using ( 15 ) we see tha t the image of d contains the image of δ . On the other hand, the coefficient of Y ∧ X in e very non zero element of δ ( X ( 0 ) 1 ) is multiple of σ ( a ) , so in par ticula r it ha s degree at least N : comparing with ( 17 ) we see that u is not in the image of δ . W e can therefore conclude that these e lements a re non zero in E 2 , so that dim E 2 2 , 0 = M . • Suppose now that q is a root o f 1 . In this case the condition ( 16 ) is equivalent to the existence of a singular polynomial s ∈ S such that aσ − 1 ( p 1 + p 3 ) + a ′ σ − 1 ( p 2 ) = h e − 1 s . (18) As a ( 0 ) 6 = 0 , c divides s and it follows from Proposition 2.1 (ii) that there exists s 1 ∈ S such that s = N ( c ) s 1 . Let α , β ∈ k [ h ] be such that a c α + a ′ c β = 1 ; eac h solution of the equation ( 18 ) is of the f orm p 3 = σ  h e − 1 cs 1 α + a ′ c g  − p 1 , 17 p 2 = σ  h e − 1 cs 1 β − a c g  for some g ∈ k [ h ] . Let b , r ∈ k [ h ] be such g = bc + r and deg r < M . W ithout c hanging its class in E 2 , we can replace u by u − δ ( σ − 1 ( p 1 ) | H − yσ ( b ) | X ) , and then we see that we may assume that u = σ ( h e − 1 cs 1 β − a c r ) | Y ∧ X + yσ ( h e − 1 cs 1 α + a ′ c r ) | H ∧ X . (19) If u represents the zero cla ss in E 1 , then there exist v 1 , v 2 , v 3 ∈ k [ h ] such that u = δ ( v 1 x | Y + v 2 | H + yv 3 | X ) = ( v 1 σ ( a ′ ) + σ ( v 2 )) x | Y ∧ H + σ ( a )( v 3 − v 1 ) | Y ∧ X − y ( v 3 σ ( a ′ ) + σ ( v 2 ) | H ∧ X . Equating coefficients and eliminating v 2 , we see that aσ − 1 ( v 3 − v 1 ) = h e − 1 ¯ cs 1 β − a c r , − a ′ σ − 1 ( v 3 − v 1 ) = h e − 1 ¯ cs 1 α + a ′ c r . Solving now for s 1 and then for r , we see that u must be zero. Let us show now u represents a non z ero e le ment of E 2 . Inde ed, if there exists a p ∈ k [ h ] such that u = d ( p | Y ∧ H ∧ X ) = y ( 1 − qσ )( p ) | H ∧ X − ( 1 − qσ )( p ) x | Y ∧ H , then we must have ( 1 − qσ )( p ) = 0 and a c r = h e − 1 cs 1 β , a ′ c r = − h e − 1 cs 1 α . Solving these equations for s 1 and r , recalling the way α a nd β were chosen, a nd using that h e − 1 ¯ c 6 = 0 , we see that s 1 = r = 0 . W e conclude in this way tha t every element of E 2 2 , 0 is represented uniquely by a cycle of the form ( 19 ). In particular , we have a vector space isomorphism E 2 2 , 0 ∼ = S ⊕ k [ h ] / ( c ) . Homology at E 1 1 , 0 . Let u = p 1 x | Y + p 2 | H + yp 3 | X ∈ E 0 1 , 0 , with p 1 , p 2 , p 3 ∈ k [ h ] , an element which survives to E 2 . As u is homologous to u − δ ( σ − 1 ( p 1 )) = ( p 2 + a ′ σ − 1 ( p 1 )) | H + y ( p 3 − p 1 ) | X , we can suppose that p 1 = 0 . If u is a boundary , so that u = d ( f 1 x | Y ∧ H + f 2 | Y ∧ X + yf 3 | H ∧ X ) + δ ( f 4 ) , for some f i ∈ k [ h ] , looking a t the coefficient of Y on both sides of this equality we find that ( 1 − σ )( f 2 ) + σ ( p 4 ) = 0 . This implies that p 3 = 0 and that p 2 ∈ ( 1 − qσ )(( c )) . On the other ha nd, since d ( u ) = σ ( a ) p 3 − aσ − 1 ( p 3 ) = ( σ − 1 )( aσ − 1 ( p 3 )) = 0 , (20) we see that aσ − 1 ( p 3 ) ∈ S . 18 • Suppose first that q is a root of 1 . Then p 3 = σ ( a ) s for some s ∈ S , according to Propositio n 2.1 , and thus we have u = p 2 | H + yσ ( a ) s | X . In view of the description given a bove for the boundaries, we conclude that E 2 1 , 0 ∼ = k [ h ] ( 1 − qσ )(( c )) | H ⊕ yσ ( ¯ a ) S | X . Using Lemma 2.3 we see that the first summand is isomorphic to k η ( c ) ⊕ S . • Suppose next that q is not a root of 1 . In this ca se , since a is not c onstant, e quation ( 20 ) implies that p 3 = 0 . Using again the description of boundarie s, we ha ve E 2 1 , 0 ∼ = k [ h ] ( 1 − qσ )(( c )) | H , a vector spa ce of dimension M . Homology at E 1 0 , 0 . W e have to compute the cokernel of the map d : A | V → A . One sees at once that its image coincides with the image of the map ψ a , 0 : f ∈ k [ h ] 7 → ( σ − 1 )( af ) ∈ k [ h ] from Lemma 2.3 . If q is not a root of unity , it is immediate that the cla sses of 1 , . . . , h N − 1 freely span coker ψ a , 0 , so that dim E 2 0 , 0 = N . On the other hand, if q is a root of unity , then Lemma 2.3 tells us that the dimension of the cokernel of ψ a , 0 , equal to that of E 2 0 , 0 , is η ( a ) = N − 1 e deg N ( a ) . Proposition 5.6. Let r 6 = 0 . Acc o rding to wheth er r is regular or not, there are isomorphisms of S -mod ules HH p ( A ) ( r ) ∼ = E 2 ( r ) p , 0 ∼ =        S , if p = 0 ; k , if p = 1 ; 0 , if p = 2 . or HH p ( A ) ( r ) ∼ = E 2 ( r ) p , 0 ∼ =        S , if p = 0 ; S ⊕ S , if p = 1 ; S , if p = 2 . Proof. By symmetry , we ca n consider just the case where r > 0 . Homology at E 1 ( r ) 2 , 0 . Let u ∈ E 0 ( r ) 2 , 0 be an element representi ng a cycle in E 1 . It follows that u = y r − 1 p 1 | Y ∧ H + y r p 2 | Y ∧ X + y r + 1 p 3 | H ∧ X with p 1 , p 2 , p 3 ∈ k [ h ] . W ithout loss of generality , we can a ssume that p 2 = p 3 = 0 ; if that is not the case, we ca n replace u by u + δ  y r − 1 σ − 1 ( p 2 ) | Y + y r σ − 1 ( p 3 ) | H  without changing the class of u in E 1 . Computing, we find then that d ( u ) = y r − 1 ( 1 − q r ) p 1 h | Y + y r ( p 1 − qσ ( p 1 )) | H . (21) Comparing with equation ( 9a ) we see that, since d ( u ) is in the image of δ , ( 1 − q r ) p 1 = 0 a nd p 1 − qσ ( p 1 ) = 0 . If r is a regular weight, it follows that p 1 = 0 , so E 2 ( r ) 2 , 0 = 0 . On the other hand, if r is singular , these equations are satisfied if a nd only if p 1 ∈ h e − 1 S : in this case we have E 2 ( r ) 2 , 0 ∼ = h e − 1 S . 19 Homology at E 1 ( r ) 1 , 0 . Let u = y r − 1 p 1 | Y + y r p 2 | H + y r + 1 p 3 | X ∈ E 0 ( r ) 1 , 0 , with p 1 , p 2 , p 3 ∈ k [ h ] , an element which lives to E 2 . U p to replacing u by u − δ ( y r σ − 1 ( p 3 )) , we ca n assume that p 3 = 0 , so that d ( u ) = y r ( p 1 − σ ( p 1 ) + ( q r − 1 ) hp 2 ) = 0 . (22) Assume that r is singular . It f ollows that p 1 ∈ S ; moreover , in view of formula ( 21 ), we ca n reduce p 2 modulo the image of 1 − qσ , so that we c a n suppose that p 2 ∈ h e − 1 S . From equations ( 8d ), ( 8e ), ( 8 f ) and ( 9a ) we see then that u is not a boundary and we conclude that E 2 ( r ) 1 , 0 ∼ = S ⊕ S in this case, freely genera te d a s a S -module by the classes of y r − 1 | Y and y r h e − 1 | H . Finally , let us a ssume that r is regular . Using aga in ( 21 ), we see that we can now replace u by an homologous e lement of the same form but now with p 1 ∈ k and then, beca use of ( 22 ), we must have p 2 = 0 . In this way , we see that u must be a scala r multiple of y r − 1 | Y . If such a n e lement is a boundary , looking at the constant term in the f ormulas ( 8d ), ( 8 e ), ( 8f ) and ( 9a ), we infer that u is zero, therefore E 2 ( r ) 1 , 0 is one-dimensional. Homology at E 1 ( r ) 0 , 0 . Let u = y r p ∈ E 0 ( r ) 0 , 0 . W e can add to u e lements in the image of d without changing its homology class; d oing so, we can assume that p ∈ S . Moreover , u itself is then not in the image of d : this means that E 1 ( r ) 0 , 0 ∼ = S , freely generated by the class of 1 . 6 Hochschild cohomolog y In this section we compute the Hochschild cohomology of A using, as before, a spectral sequence. W rite ^ V = hom k ( V , k ) , and let { ^ Y , ^ H , ^ X } be the ba sis of ^ V dual to { Y , H , X } . W e id e ntify in the usual way hom k ( V p V , k ) with V p ^ V . Applying the f unctor hom A e (− , A ) to the resolution constructed in 4.3 we obtain a double complex whose cohomology is the Hochschild cohomology HH • ( A ) of A . After we identify hom A e ( A | V p V | A , A ) with A | V p ^ V in the natural way , this double complex is 0 δ   A | V 3 ^ V d o o δ   A | V 2 ^ V d o o δ   A | ^ V d o o δ   A d o o 0 δ   A | V 3 ^ V d o o δ   A | V 2 ^ V d o o δ   A | ^ V d o o δ   A d o o . . . . . . . . . . . . . . . with differentials given by d ( u ) = [ u , y ] | ^ Y + [ u , h ] | ^ H + [ u , x ] | ^ X ; (23a) d ( u | ^ Y ) = [ h , u ] q | ^ Y ∧ ^ H − [ u , x ] | ^ Y ∧ ^ X ; (23b) d ( u | ^ H ) = [ x , u ] q | ^ H ∧ ^ X − P i λ i R i h s uh t | ^ Y ∧ ^ X + [ u , y ] q | ^ Y ∧ ^ H ; (23c) d ( u | ^ X ) = [ u , h ] q | ^ H ∧ ^ X + [ u , y ] | ^ Y ∧ ^ X ; (23d) 20 d ( u | ^ Y ∧ ^ H ) = −[ x , u ] q | ^ Y ∧ ^ H ∧ ^ X ; (23e) d ( u | ^ Y ∧ ^ X ) = q [ h , u ] | ^ Y ∧ ^ H ∧ ^ X ; (23f) d ( u | ^ H ∧ ^ X ) = [ u , y ] q | ^ Y ∧ ^ H ∧ ^ X ; (23g) and δ ( u | ^ Y ) = ux ; (24a) δ ( u | ^ H ) = − P i α i R i h s uh t ; (24b) δ ( u | ^ X ) = yu ; (24c) δ ( u | ^ Y ∧ ^ H ) = P i α i R i q t h s uh t | ^ Y + ux | ^ H ; (24d ) δ ( u | ^ Y ∧ ^ X ) = ux | ^ X − yu | ^ Y ; (24e) δ ( u | ^ H ∧ ^ X ) = − yu | ^ H − P i α i R i q s h s uh t | ^ X ; (24f) δ ( u | ^ Y ∧ ^ H ∧ ^ X ) = ux | ^ H ∧ ^ X + P i α i R i q i − 1 h s uh t | ^ Y ∧ ^ X + yu | ^ Y ∧ ^ H ; (24g) W e consider the spectral sequence E which a rises from the filtration of this double complex by columns. 6.1 First Pa ge In this section we dea l with the first page of the spectral sequence. Let Y be the c omplex 0 / / A | V 3 ^ V δ / / A | V 2 ^ V δ / / A | ^ V δ / / A (25) with differentials a s in ( 24a )–( 24g ). As before, we have E p , q 1 ∼ = H p − q ( Y ) for a ll q > 0 a nd the vector spaces E p , 0 1 are isomorphic to the kernels of the differentials of Y . For each r ∈ Z we denote Y ( r ) the component of weight r in Y , a nd extend this notation to related objects. Proposition 6.1. If r ∈ Z is non zero, th en the com plex Y ( r ) is exact. O n th e ot h er hand, th ere are S -m odule isomorphisms H p ( Y ( 0 ) ) ∼ =  k [ h ] / ( c ) , if 0 ≤ p ≤ 1 ; 0 , otherwise. Proof. W e prove this by computing the relevant homology groups: • If u = p | ^ Y ∧ ^ H ∧ ^ X ∈ Y 3 ( 0 ) with p ∈ k [ h ] , then δ ( u ) = px | ^ H ∧ ^ X + pσ ( a ′ ) | ^ Y ∧ ^ X + yp | ^ Y ∧ ^ H . (26) It is clea r then that H 3 ( Y ( 0 ) ) = 0 . • Let u = yp 1 | ^ Y ∧ ^ H + p 2 | ^ Y ∧ ^ X + p 3 x | ^ H ∧ ^ X ∈ Y 2 ( 0 ) with p 1 , p 2 , p 3 ∈ k [ h ] . One can see that δ ( u ) = y ( p 1 σ ( a ′ ) − p 2 ) | ^ Y + ( aσ − 1 ( p 1 − p 3 )) | ^ H + (− p 3 σ ( a ′ ) + p 2 ) x | ^ X . (27) In particula r , if u is a cycle, p 2 = σ ( a ′ ) p 1 and p 3 = p 1 . Comparing with the e xpression ( 26 ) for 2 -boundar ies in Y , we see at once that H 2 ( Y ( 0 ) ) = 0 . 21 • Finally , let u = yp 1 | ^ Y + p 2 | ^ H + p 3 x | ^ X ∈ Y 1 ( 0 ) , with p 1 , p 2 , p 3 ∈ k [ h ] , a 1 -cycle. Since we ca n replace u for u + δ ( p 1 | ^ H ) , without changing the homology class it represents, we can assume that p 1 = 0 , and then δ ( u ) = aσ − 1 ( p 3 ) − p 2 a ′ = 0 . It follows that there e xists g ∈ k [ h ] such that p 3 = σ ( a ′ c g ) and p 2 = a c g . Let b , r ∈ k [ h ] such that g = bc + r and deg r < M . Then u + δ ( σ ( b ) x | ^ H ∧ ^ X ) = a c r | ^ H + σ ( a ′ c r ) x | ^ X This means that all classes in H 1 ( Y ( 0 ) ) can be represented by a element of the form a c r | ^ H + σ ( a ′ c r ) x | ^ X with r ∈ k [ h ] and d eg r < M and, moreover , such a n element represents the zero class only when it is itself zero: this ca n be seen by looking at the degree of the c oefficient of ^ H a p pearing the formula ( 27 ) f or 1 -boundaries. Conversely , e v e ry such element is a cycle. W e conclude that H 1 ( Y ( 0 ) ) ∼ = k [ h ] / ( c ) . • If u = yp 1 | ^ Y + p 2 | ^ H + p 3 x | ^ X ∈ Y 1 ( 0 ) , with p 1 , p 2 , p 3 ∈ k [ h ] , then δ ( u ) = aσ − 1 ( p 1 + p 3 ) − p 2 a ′ , so H 0 ( Y ( 0 ) ) ∼ = k [ h ] / ( c ) . It remains to check, in these last two items, tha t the obtained isomorphisms are S -linear : this is just a matter of following the computation, and we omit the details. Let us now fix r > 0 . • If u = y r p | ^ Y ∧ ^ H ∧ ^ X ∈ Y 3 ( r ) , with p ∈ k [ h ] , then δ ( u ) = y r + 1 p | ^ Y ∧ ^ H + y r p P i α i q i − 1 [ i ] q r h i − 1 | ^ Y ∧ ^ X + y r − 1 aσ − 1 ( p ) | ^ H ∧ ^ X . (28) Looking at the coefficient of ^ Y ∧ ^ H we see that u is a cycle if and only if u is z ero, so H 3 ( Y ( r ) ) = 0 . • Let u = y r + 1 p 1 | ^ Y ∧ ^ H + y r p 2 | ^ Y ∧ ^ X + y r − 1 p 3 | ^ H ∧ ^ X ∈ Y 2 ( r ) , with p 1 , p 2 , p 3 ∈ k [ h ] . S ince δ ( u ) = y r + 1 ( p 1 P i α i q i − 1 [ i ] q r h i − 1 − p 2 ) | ^ Y + y r ( aσ − 1 ( p 1 ) − p 3 ) | ^ H + y r − 1 ( aσ − 1 ( p 2 ) − p 3 P i α i [ i ] q r h i − 1 ) | ^ X , it is easy to see that u is a cycle if a nd only if p 2 = p 1 X i α i q i − 1 [ i ] q r h i − 1 , p 3 = aσ − 1 ( p 1 ) , and in tha t case, according to ( 28 ), we have u = δ ( y r p 1 | ^ Y ∧ ^ H ∧ ^ X ) . W e conclude that H 2 ( Y ( r ) ) = 0 . • Let u = y r + 1 p 1 | ^ Y + y r p 2 | ^ H + y r − 1 p 3 | ^ X ∈ Y 1 ( r ) , with p 1 , p 2 , p 3 ∈ k [ h ] a cycle. W ithout changing its homology class, we ca n replace u by u + δ ( y r p 1 | ^ Y ∧ ^ X + y r − 1 p 2 | ^ H ∧ ^ X ) , and hence we can suppose that p 1 = p 2 = 0 . In that case δ ( u ) = y r p 3 , a nd we see that u = 0 . It follows that H 1 ( Y ( r ) ) = 0 . • Finally , for each p ∈ k [ h ] , δ ( y r − 1 p | ^ X ) = y r p , so that δ ( Y 1 ( r ) ) = Y 0 ( r ) and H 0 ( Y ( r ) ) = 0 . 22 Corollary 6.2. If r ∈ Z , the dimensions of th e vecto r spaces ap pearing in the component E 1 ( r ) of E 1 are 0 ? ? ? 0 0 M M 0 0 M M . . . . . . . . . . . . or 0 ? ? ? 0 0 0 0 0 0 0 0 . . . . . . . . . . . . if r = 0 or r 6 = 0 , respectively . The question marks denote vector spaces for wh ich we still do not know the dimension. Proof. This follows f rom the proposition a nd the isomorphisms E p , q 1 ( r ) ∼ = H p − q ( Y ( r ) ) . 6.2 The second page Proposition 6 .3. For each p ≥ 0 , the differe ntial d p , p 1 : E p , p 1 → E p + 1 , p 1 vanishes. The page E ∞ then coincides with E 1 , except at the p laces marked with question marks in t h e diagrams of Corollary 6.2 , and we have HH p ( A ) ( r ) ∼ =  k [ h ] / ( c ) , if r = 0 ; 0 , if r 6 = 0 . Proof. The set of homology classes of the e lements of { h l : 0 ≤ l < M } is a basis of the space E p , p 1 , and d ( h l ) = ( q l − 1 ) yh l | ^ Y − ( q l − 1 ) h l x | ^ X = δ (− ( q l − 1 ) h l | ^ Y ∧ ^ X ) . It follows that d p , p 1 is indeed zero, as claimed . The rest of the proposition is then a consequence of the fact that the spectral sequence E converges to HH • ( A ) . Proposition 6.4. If q is a root of unity, th en E p , 0 2 ( 0 ) ∼ =        S , if p = 0 ; S ⊕ S , if p = 1 ; S ⊕ k η ( a / c ) , if p = 2 , where, as in Lemma 2.3 , η ( a / c ) = N − M − deg N ( a / c ) / e , and if q has infinite order , E p , 0 2 ( 0 ) ∼ =        k , if p = 0 ; k , if p = 1 ; k N − M , if p = 2 . Proof. W e write, during this proof, E p , q r instead of E p , q r ( 0 ) for simplicity . 23 Homology at E 0 , 0 1 . If u = p ∈ E 0 , 0 1 , so that in fac t p ∈ k [ h ] , we have d ( p ) = y ( σ ( p ) − p ) | ^ Y − ( σ ( p ) − p ) x | ^ X . (29) It follows that E 0 , 0 1 = ker ( σ − 1 ) = S . Homology at E 1 , 0 1 . If u ∈ E 1 , 0 1 , there e xist p 1 , p 2 ∈ k [ h ] such that u = yp 1 | ^ Y + a d p 2 | ^ H + ( σ ( a ′ d p 2 ) − p 1 ) x | ^ X ; this is a c onsequence of the formulas ( 24a ), ( 2 4b ) and ( 2 4c ) using the same reasoning as in the third step of the proof of Propositio n 6.1 . Moreover , there exist s 1 ∈ S and b ∈ k [ h ] such that p 1 = s 1 + ( σ − 1 )( b ) and we ca n replace u by u − d ( b ) so, in the end, we can assume that p 1 = s 1 ∈ S . In that case, u is boundary only if it zero: this f ollows by comparing with the coefficient of ^ Y in ( 29 ). Computing, we find that d ( u ) = ( σ − q )( a c p 2 ) x | ^ H ∧ ^ X + y ( σ − q )( a c p 2 ) | ^ Y ∧ ^ H +  ( σ − 1 )( aa ′ c p 2 ) − a c p 2 ( qσ ( a ′ ) − a ′ )  | ^ Y ∧ ^ X . If d ( u ) = 0 , then ( σ − q )( a c p 2 ) = 0 and a c p 2 ∈ h S ; conversely , if a c p 2 ∈ h S , then u is a cycle. we treat separately two cases, according to whether q is a root of unity or not. • Suppose first that q is not a root of 1 . As a c p 2 ∈ h S and S = k , then p 2 ∈ k . Evalua ting a c p 2 at zero, a nd using the hypothesis that a ( 0 ) 6 = 0 , we see that p 2 = 0 . In this case, then, u is a scalar multiple of y | ^ Y − x | ^ X . Since all such non zero multiples are cycles a nd not boundaries, we conclude tha t E 1 , 0 2 is one dimensional, generated by the class of y | ^ Y − x | ^ X . • Suppose now that q is a root o f 1 . As h ∤ a , we must have h | p 2 and a c p 2 h ∈ S . There exists then, by Propositio n 2.1 (i) , s 2 ∈ S such that p 2 = hs 2  a c  . This gives us a description of homology: it is the free S -module of rank 2 genera ted by the classes of y | ^ Y − x | ^ X and N ( a c ) h | ^ H + σ ( a ′ c a c h ) x | ^ X . Homology at E 2 , 0 1 . Let u ∈ E 2 , 0 1 , so in fact u ∈ E 2 , 0 0 and δ ( u ) = 0 . In view of ( 27 ), there exists p ∈ k [ h ] such that u = yp | ^ Y ∧ ^ H + pσ ( a ′ ) | ^ Y ∧ ^ X + px | ^ H ∧ ^ X . The element u is a boundary if there e x ist f 1 , f 2 ∈ k [ h ] such that u = d ( yf 1 | ^ Y + a c f 2 | ^ H + ( σ ( a ′ c f 2 ) − f 1 ) x | ^ X ) or , making this explicit, p = ( σ − q )( a c f 2 ) , σ ( a ′ ) p = D q ( aa ′ c f 2 ) − a c f 2 ( qσ ( a ′ ) − a ′ ) . The second e quation follows from the first, and we conclude that u is a boundary if and only if p ∈ im ψ a / c , 1 with ψ a / c , 1 defined as in Lemma 2.3 . In other words, there is a n isomorphism E 2 , 0 2 ∼ = coker ψ a / c , 1 . W e have two cases: • First, suppose that q is not a root of 1 . If deg ( a c ) > 1 , then deg ψ a / c , 1 ( f ) = deg ( a c ) + de g ( f ) for f ∈ k [ h ] \ 0 . It follows then that coker ψ a / c , 1 is freely spa nned by the classes of 1 , h , . . . , h N − M − 1 , because im ψ a / c , 1 is spanned by a set of p olynomials of each degree greater or equal to N − M . W e conclude tha t dim ( coker ( ψ a / c , 1 )) = N − M . On the other hand, if d eg ( a c ) = 1 , we have d e g ψ a / c , 1 ( f ) = 1 + deg ( f ) for all non-constant f ∈ k [ h ] and deg ψ a / c , 1 ( f ) = 0 f or f ∈ k \ 0 , so that the cokernel is f reely spanned by the class of h . In particular , dim coker ( ψ a / c , 1 ) = 1 = N − M . 24 • Suppose now that q is a root of 1 . W e computed the d imension of coker ψ a / c , 1 in Lemma 2.3 , so that the the d imension of E 2 , 0 2 ( 0 ) is η ( a / c ) , as claimed in the statement of the proposition. Corollary 6.5. If q is a root of unity, th en there are isomorp hisms of S -modules HH p ( A ) ( 0 ) ∼ =        S , if p = 0 ; S ⊕ S , if p = 1 ; S ⊕ k η ( a / c ) ⊕ k [ h ] / ( c ) , if p = 2 . If, on the other hand , q has infinite order , HH p ( A ) ( 0 ) ∼ =        k , if p = 0 ; k , if p = 1 ; k N − M ⊕ k M , if p = 2 . Proof. This follows f rom the proposition a nd the c onvergence of the spectra l sequence. Remark 6 .6 . In the computation of the Hochschild c ohomology the fact tha t a ( 0 ) 6 = 0 is only used in the proof of the Proposition 6.4 . In the case when q is not a root of 1 , using a n analogous reasoning one can prove that if a ( 0 ) = 0 and a 6 = h N then the same result holds. If instead a = h N then E p , 0 2 ( 0 ) ∼ =        k , if p = 0 ; k 2 , if p = 1 ; k N − M + 1 , if p = 2 . On the other hand, if q is a root of 1 then E p , 0 2 ( 0 ) ∼ =        S , if p = 0 ; S ⊕ S , if p = 1 ; S ⊕ k η ( a / ( ch ))+ 1 if p = 2 . This difference is to be expected because, for example, when a = h N we have grad ings on A such that deg h = 1 and deg x + deg y = N . The eulerian derivation induced by one of these gradings is a non zero cla ss in HH 1 ( A ) , which is not cohomologous to the induced by the weight. Proposition 6.7. Let r 6 = 0 . Acc o rding to wheth er r is regular or not, there are isomorphisms of S -mod ules E p , 0 2 ( r ) ∼ =        0 , if p = 0 ; 0 , if p = 1 ; 0 , if p = 2 . or E p , 0 2 ( r ) ∼ =        S , if p = 0 ; S ⊕ S , if p = 1 ; S , if p = 2 . 25 Proof. Homology at E 0 , 0 1 ( r ) . Let u ∈ E 0 , 0 0 ( r ) , so that u = y r p for some p ∈ k [ h ] . Since d ( u ) = y r + 1 ( σ ( p ) − p ) | ^ Y + ( 1 − q r ) ph | ^ H + y r − 1 ( aσ − 1 ( p ) − σ r ( a ) p ) | ^ X , (30) u is a non zero c ycle if and only if r is a singular weight and p ∈ S . Homology at E 1 , 0 1 ( r ) . If u ∈ E 1 , 0 1 ( r ) , then there exist p 1 , p 2 , p 3 ∈ k [ h ] such that u = y r + 1 p 1 | ^ Y + y r p 2 | ^ H + y r − 1 p 3 | ^ X a nd δ ( u ) = 0 . This condition implies immediately , using ( 24a ), ( 24b ) and ( 2 4 c ), that p 3 = p 2 P i α i [ i ] q r h i − 1 − aσ − 1 ( p 1 ) . Let us suppose now that d ( u ) = 0 . • If r is regular , we can replace u by u − d (( 1 − q r ) − 1 y r ( p 2 − p 2 ( 0 )) / h ) without changing its homology class, and this amounts to a ssuming initially that p 2 ∈ k . In that case, it is easy to see that the coefficient of ^ Y ∧ ^ H in d ( u ) is y r + 1 ( q ( q r − 1 ) hp 1 + ( 1 − q ) p 2 ) = 0 and, then, p 1 = p 2 = 0 . Similarly , looking at the coefficient of ^ H ∧ ^ X , we can conclude that p 3 = 0 . • If r is singular , there exist b ∈ k [ h ] a nd s 1 ∈ S such that p 1 = σ ( b ) − b + s 1 ; by replacing u by u − d ( y r b ) , which we ma y do as it d oes not change the homology class, we may assume that p 1 = s 1 ∈ S . Computing, we find that d ( u ) = y r + 1 ( σ − q )( p 2 ) | ^ Y ∧ ^ H + y r σ ( a ′ )( σ − q )( p 2 ) | ^ Y ∧ ^ X + y r − 1 a ( σ − q )( σ − 1 ( p 2 )) | ^ H ∧ ^ X , and it is clear that this vanishes exactly when p 2 ∈ h S . W e see that ev e ry element of E 1 , 0 2 ( r ) is represented by an element in the S -submodule generated by the e lements y r + 1 | ^ Y − y r − 1 a | ^ X y r h | ^ H + y r − 1 a ′ h | ^ X . Comparing with ( 30 ), it is easy to see that this submodule does not contain non zero bound- aries, so E 1 , 0 2 ( r ) is S -free of ra nk 2 . Homology at E 2 , 0 1 ( r ) . Let u = y r + 1 p 1 | ^ Y ∧ ^ H + y r p 2 | ^ Y ∧ ^ X + y r − 1 p 3 | ^ H ∧ ^ X ∈ E 2 , 0 1 ( r ) . • If r is regular , let b i = ( p i − p i ( 0 ))( q ( q r − 1 ) h ) − 1 for i ∈ { 1 , 3 } . W e may replace u by u − d ( b 1 | ^ Y + b 3 | ^ X ) , and a computation using ( 23b ) a nd ( 2 3 d ) shows that this means that we can assume that p 1 , p 3 ∈ k . Using now ( 24d ), ( 2 4e ) and ( 24 f ), we e a sily see that δ ( u ) = 0 if and only if u = 0 . It follows that in this c ase E 2 , 0 2 ( r ) = 0 . • T o finish, suppose next that r is singular . Since δ ( u ) = y r + 1 ( σ ( a ′ ) p 1 − p 2 ) | ^ Y + y r ( aσ − 1 ( p 1 ) − p 3 ) | ^ H + y r − 1 ( aσ − 1 ( p 2 ) − a ′ p 3 ) | ^ X = 0 , we see that p 3 = aσ − 1 ( p 1 ) and p 2 = σ ( a ′ ) p 1 . I f b ∈ k [ h ] and s ∈ S a re such that p 1 = σ ( b ) − qb + hs , we can replace u by u − d ( y r b | ^ H + y r − 1 ba ′ | ^ X ) , which is y r + 1 hs 1 | ^ Y ∧ ^ H + y r σ ( a ′ ) hs 1 | ^ Y ∧ ^ X + y r − 1 q − 1 ahs 1 | ^ H ∧ ^ X without changing its class in E 2 , 0 2 ( r ) . No element of this form is in the image of d , as one can see by looking at the coefficient of ^ Y ∧ ^ H in ( 23 b ), ( 23c ) a nd ( 23 d ), so we can conclude that E 2 , 0 2 ( r ) is a free S -module generated by the class of y r + 1 h | ^ Y ∧ ^ H + y r σ ( a ′ ) h | ^ Y ∧ ^ X + y r − 1 q − 1 ah | ^ H ∧ ^ X . 26 References [1] L . L. A vramov and S. Iyengar, Gaps in H ochschild cohomology imply smoothness for commutative algebras , Math. Res. L ett. 12 (2005), no. 5-6, 789–804. M R 2189239 (2006i:13028) [2] L . L. A vramov and M. V igué-Poirrier, H ochschild homology criteria for smoothness , Internat. Math . Res. Notices 1 (1992), 17–25, DOI 10.1155/S107379289200 0035. MR 1149001 (92m:13020) [3] V . V . 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MR 2566552 (2011a:16021) Departamento de Matemá tica , Facultad de Ciencias Exacta s y Naturale s, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1 1428, Buenos Aires, Argentina. Email: asolo tar@dm .uba. ar , mari ano@d m.uba. ar , q vivas@ dm.ub a.ar 27