THE HOCHSCHILD COHOMOLOGY RING OF A CLASS OF SPECIAL BISERIAL ALGEBRAS

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1 THE HOCHSCHILD COHOMOLOGY RING OF A CLASS OF SPECIAL BISERIAL ALGEBRAS NICOLE SNASHALL AND RACHEL TAILLEFER Abstract. We consider a class of self-injective special biserial algebras Λ N over a field K and show that the Hochschild cohomology ring of Λ N is a finitely generated K-algebra. Moreover the Hochschild cohomology ring of Λ N modulo nilpotence is a finitely generated commutative K-algebra of Krull dimension two. As a consequence the conjecture of [1, concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras. Introduction Let K be a field. For m 1, let Q be the quiver with m vertices, labeled 0, 1..., m 1, and m arrows as follows: a a a ā ā ā ā a Let a i denote the arrow that goes from vertex i to vertex i + 1, and let ā i denote the arrow that goes from vertex i + 1 to vertex i, for each i = 0,..., m 1 (with the obvious conventions modulo m). We denote the trivial path at the vertex i by e i. Paths are written from left to right. In this paper, we study the Hochschild cohomology ring of a family of algebras Λ N given by this quiver Q and certain relations. Let N 1, and let Λ N = KQ/I N where I N is the ideal of KQ generated by a i a i+1, ā i 1 ā i, (a i ā i ) N (ā i 1 a i 1 ) N, for i = 0, 1,..., m 1, where the subscripts are taken modulo m. These algebras are all self-injective special biserial algebras and as such play an important role in various aspects of representation theory of algebras. First, consider the case N = 1, so that Λ 1 = KQ/I 1 where I 1 is the ideal of KQ generated by a i a i+1, ā i 1 ā i and a i ā i ā i 1 a i 1, for i = 0,..., m 1. In the case where m is even, this algebra occurred in the presentation by quiver and relations of the Drinfeld double of the generalised Taft algebras studied in [6. It also occurs in the study of the representation theory of U q (sl ); see [0,, 3; see also [5. In the case N = 1, the Hochschild cohomology ring of the algebra Λ 1 with m = 1 is discussed in [1, where q-analogues of Λ 1 were used to answer negatively a question of Happel, in that they have finite dimensional Hochschild cohomology ring but are of infinite global Date: June 5, Mathematics Subject Classification. Primary: 16E40, 18G10, 16D40, 16D50. Secondary: 16S37. 1

2 dimension when q K is not a root of unity. The more general algebras Λ N occur in [10, 9, in which the authors determine the Hopf algebras associated to infinitesimal groups whose principal blocks are tame when K is an algebraically closed field of characteristic p 3; among the principal blocks that are obtained in this classification are the algebras Λ p n with m = p r vertices, for some integers n and r. This paper describes explicitly the structure of the Hochschild cohomology ring of the algebras Λ N, for all N 1 and m 1, and in all characteristics. In particular, we determine the Hochschild cohomology ring of the algebras Λ 1 of [6 and the algebras of [10. The main results show that the Hochschild cohomology ring, HH (Λ N ), is a finitely generated K-algebra. We give an explicit basis for each cohomology group HH n (Λ N ) together with generators of the algebra HH (Λ N ). We then determine explicitly the Hochschild cohomology ring modulo nilpotence, HH (Λ N )/N, where N is the ideal of HH (Λ N ) generated by all homogeneous nilpotent elements. Furthermore, we show that the conjecture of [1 holds for all Λ N, that is, that HH (Λ N )/N is a finitely generated K-algebra. Moreover, we show that HH (Λ N )/N is a commutative ring of Krull dimension. It is to be expected that the results in this paper will give some insight into the more general problem of verifying the conjecture of [1 for all self-injective special biserial algebras. It is known that the conjecture is not true in general (see [4 in which the author gives a counter-example which is a 7-dimensional Koszul algebra that is not self-injective). It is therefore important to find out for which algebras the conjecture is true in order to apply the results of [1, 7. For comparison, we recall that the conjecture has been verified for finite-dimensional monomial algebras A in [13, where it was shown that HH (A)/N is a finitely generated commutative algebra of Krull dimension at most 1. It has also been proved for many classes of finite-dimensional self-injective algebras, including self-injective algebras of finite representation type (see [14), cocommutative Hopf algebras (see [10), and, more recently, pointed Hopf algebras with abelian group of grouplike elements (see [19) and Hopf algebras of rank one (see [). Note that in this last paper there are general results describing the structure of the Hochschild cohomology ring of a smash product of an algebra with a Hopf algebra. Although these results could, in principle, be applied to the algebras in the current paper, this would require further significant structural information to be determined. We now outline the structure of the paper. Section 1 describes the minimal projective resolution (P, ) of the algebra Λ N as a bimodule, together with an explanation of the construction of this resolution. This construction is of more general interest and applicability. Section describes the methods used to find the dimensions of the kernel and image of the induced maps in the complex Hom(P, Λ N ) and hence gives the dimension of each Hochschild cohomology group HH n (Λ N ) in the case m 3. In calculating HH (Λ N ), we first consider the general case m 3, and start with a description in section 3 of the centre of Λ N, that is, of HH 0 (Λ N ), before giving the structure of the Hochschild cohomology ring for m 3 in sections 4 and 5. Section 6 deals with the case m = and section 7 with the case m = 1. The final section summarises the paper, and remarks that the finiteness conditions (Fg1) and (Fg) of [7 hold when N = 1, thus enabling one to describe the support varieties for finitely generated modules over the algebras Λ A minimal projective bimodule resolution Let N 1, and let Λ N = KQ/I N where I N is the ideal of KQ generated by a i a i+1, ā i 1 ā i and (a i ā i ) N (ā i 1 a i 1 ) N, for i = 0, 1,..., m 1. Where there is no confusion, we label the arrows of Q generically by a and ā. We write o(α) for the trivial path corresponding to the origin of the arrow α, so that o(a i ) = e i and o(ā i ) = e i+1. We write t(α) for the trivial path corresponding to

3 the terminus of the arrow α, so that t(a i ) = e i+1 and t(ā i ) = e i. Recall that a non-zero element r KQ is said to be uniform if there are vertices v, w such that r = vr = rw. Let S i denote the simple module at the vertex i, so that S 0,..., S m 1 } is a complete set of non-isomorphic simple modules for Λ N. The Hochschild cohomology ring of a K-algebra Λ is given by HH (Λ) = Ext Λe(Λ, Λ) = n 0 Ext n Λ e(λ, Λ) with the Yoneda product, where Λe = Λ op K Λ is the enveloping algebra of Λ. Since all tensors are over the field K we write for K throughout. We now describe a minimal projective bimodule resolution (P, ) of Λ N. By [16, we know that the multiplicity of Λe i e j Λ as a direct summand of P n is equal to the dimension of Ext n Λ(S i, S j ). Thus we have, for n 0, that P n = m 1 i=0 [ n r=0λ N e i e i+n r Λ N. We wish to label the summands of P n by certain elements in the path algebra KQ. The description given here is motivated by [1 where the first terms of a minimal bimodule resolution of a finite-dimensional quotient of a path algebra were determined explicitly from the early terms of the minimal right Λ-resolution of Λ/r, where r is the Jacobson radical of Λ, using [15. We start by recalling briefly the theory of projective resolutions developed in [15 and [1. In [15, the authors give an explicit inductive construction of a minimal projective resolution of Λ/r as a right Λ-module, for a finite-dimensional algebra Λ over a field K. For Λ = KQ/I and finitedimensional, they define g 0 to be the set of vertices of Q, g 1 to be the set of arrows of Q, and g to be a minimal set of uniform relations in the generating set of I, and then show that there are subsets g n, n 3, of KQ, where x g n are uniform elements satisfying x = y g n 1 yr y = z g n zs z for unique r y, s z KQ, which can be chosen in such a way that there is a minimal projective Λ-resolution of the form Q 4 Q 3 Q Q 1 Q 0 Λ/r 0 having the following properties. (1) For each n 0, Q n = x g t(x)λ. n () For each x g n, there are unique elements r j KQ with x = j gn 1 j r j. (3) For each n 1, using the decomposition of (), for x g n, the map Q n Q n 1 is given by t(x)λ r j t(x)λ. j where the elements of the set g n are labeled by g n = gj n }. Thus the maps in this minimal projective resolution of Λ/r as a right Λ-module are described by the elements r j which are uniquely determined by (). In [1, these same sets g n are used to give an explicit description of the first three maps in a minimal projective resolution of Λ as a Λ, Λ-bimodule, thus connecting the minimal projective resolution of Λ/r as a right Λ-module with a minimal projective resolution of Λ as a Λ, Λ-bimodule. We use the same ideas here to give a minimal projective bimodule resolution for the algebra Λ N, for all N 1. In the case where N = 1, the algebra Λ 1 is Koszul, and we refer to [11 which uses this approach and gives a minimal projective bimodule resolution for any Koszul algebra. We start with the case N = 1 and define sets g n in the path algebra KQ which we will use to label the generators of P n The bimodule resolution for Λ 1. Consider the algebra Λ 1. For each vertex i = 0,..., m 1 and for each r = 0,..., n, we define elements gr,i n in KQ as follows. Let g n r,i = p ( 1) s p 3

4 where the sum is over all paths p of length n, written p = α 1 α α n where the α i are arrows in Q, such that (i) p e i KQ, (ii) p contains r arrows of the form ā and n r arrows of the form a, and (iii) s = α j=ā j. It follows that g n r,i e i(kq)e i+n r, for i = 0,..., m 1 and r = 0,..., n. Since the elements g n r,i are uniform elements, we may define o(g n r,i ) = e i and t(g n r,i ) = e i+n r. Then We set P n = m 1 i=0 [ n r=0λ 1 o(g n r,i) t(g n r,i)λ 1. g n = gr,i n r = 0,..., n}. m 1 i=0 It is easy to see, for the cases n = 0, 1,, that we have g 0 0,i = e i, g 1 0,i = a i and g 1 1,i = ā i 1, whilst g 0,i = a ia i+1, g 1,i = a iā i ā i 1 a i 1 and g,i = ā i 1ā i. Thus g 0 = e i i = 0,..., m 1}, g 1 = a i, ā i i = 0,..., m 1}, g = a i a i+1, a i ā i ā i 1 a i 1, ā i 1 ā i for all i}, so that g is a minimal set of uniform relations in the generating set of I 1. To describe the map n : P n P n 1, we need to write the elements of the set g n in terms of the elements of the set g n 1. The proof of the next lemma is straightforward, and is left to the reader. Lemma 1.1. For the algebra Λ 1, for i = 0, 1,..., m 1 and r = 0, 1,..., n, we have: g n r,i = g n 1 r,i a i+n r 1 + ( 1) n gr 1,iāi+n r n 1 with the conventions that g n 1,i = 0 and gn 1 n,i = ( 1) r a i g n 1 r,i+1 + ( 1)r ā i 1 g n 1 r 1,i 1 = 0 for all n, i. Thus g n 0,i = g n 1 0,i a i+n 1 = a i g n 1 0,i+1 and gn n,i = ( 1) n g n 1 n 1,iāi n+1 = ( 1) n ā i 1 g n 1 n 1,i 1. Since Λ 1 is Koszul, we now use [11 to give a minimal projective bimodule resolution (P n, n ) of Λ 1. We define the map 0 : P 0 Λ 1 to be the multiplication map. However, to define n for n 1, we first need to introduce some notation. In describing the image of o(gr,i n ) t(gn r,i ) under n in the projective module P n 1, we use subscripts under to indicate the appropriate summands of the projective module P n 1. Specifically, let r denote a term in the summand of P n 1 corresponding to gr, n 1, and r 1 denote a term in the summand of P n 1 corresponding to gr 1, n 1, where the appropriate index of the vertex may always be uniquely determined from the context. Indeed, since the relations are uniform along the quiver, we can also take labeling elements defined by a formula independent of i, and hence we omit the index i when it is clear from the context. Recall that nonetheless all tensors are over K. Now, for n 1, keeping the above notation and using [11, we define the map n : P n P n 1 for the algebra Λ 1 as follows: n : o(g n r,i ) t(gn r,i ) (e i r a i+n r 1 + ( 1) n e i r 1 ā i+n r ) +( 1) n (( 1) r a i r e i+n r + ( 1) r ā i 1 r 1 e i+n r ). Using our conventions that g 1,i n = 0 and gn 1 n,i = 0 for all n, i, the degenerate cases (r = 0, r = n) simplify to n : o(g0,i) n t(g0,i) n e i 0 a i+n 1 + ( 1) n a i 0 e i+n 4

5 where the first term is in the summand corresponding to g n 1 0,i and the second term is in the summand corresponding to g n 1 0,i+1, whilst n : o(g n n,i) t(g n n,i) ( 1) n e i n 1 ā i n + ā i 1 n 1 e i n, with the first term in the summand corresponding to g n 1 n 1,i and the second term in the summand corresponding to g n 1 n 1,i 1. The following result is now immediate from [11, Theorem.1. Theorem 1.. With the above notation, (P n, n ) is a minimal projective resolution of Λ 1 as a Λ 1, Λ 1 -bimodule. 1.. The bimodule resolution for Λ N with N 1. We now consider the general case N 1. In this case we use the approach of [11, 1 to construct a minimal projective resolution for Λ N. We keep the conventions that g 1,i n = 0 and gn 1 n,i = 0, for all n, i, throughout the paper. Definition 1.3. For the algebra Λ N (N 1), for i = 0,..., m 1 and r = 0,..., n, we define g 0 0,i = e i, and, inductively for n 1, gr,i n = g n 1 r,i a + ( 1) n g n 1 r 1,iā(aā) if n r > 0; g n 1 r,i a(āa) + ( 1) n gr 1,iā n 1 if n r < 0; g n 1 r,i a(āa) + g n 1 r 1,iā(aā) if n = r. Remark. (1) If N = 1, the above definition reduces to that given for Λ 1. () We have g 1 0,i = a i and g 1 1,i = ā i 1, for all i. Also g 0,i = a ia i+1, g 1,i = (a iā i ) N (ā i 1 a i 1 ) N and g,i = ā i 1ā i, for all i. Hence g is a minimal set of uniform relations in the generating set of I N. (3) The projectives in a minimal projective bimodule resolution of Λ N are given by P n = m 1 i=0 [ n r=0λ N o(g n r,i) t(g n r,i)λ N. The next result is an analogue of Lemma 1.1 for the cases n r > 0 and n r < 0. The proof of the Lemma is easy to verify and is omitted. Lemma 1.4. For the algebra Λ N (N 1), for i = 0,..., m 1, n 1, and r = 0,..., n, we have: gr n 1 a + ( 1) n gr 1 n 1 = ( 1) r agr n 1 + ( 1) r ā(aā) gr 1 n 1 if n r > 0; gr,i n = gr n 1 a(āa) + ( 1) n gr 1 n 1 a(āa) gr n 1 + ( 1) r āgr 1 n 1 if n r < 0. We now define the maps n. Definition 1.5. For the algebra Λ N (N 1), and for n 0, we define maps n as follows. For n = 0, the map 0 : P 0 Λ N is the multiplication map. For n 1, and i = 0, 1,..., m 1, the map n : P n P n 1 is given by n : e i r e i+n r 5

6 e i r e i+(n 1) r a + ( 1) n+r ae i+1 r e i+1+(n 1) r +( 1) n+r ā(aā) e i 1 r 1 e i 1+(n 1) (r 1) +( 1) n e i r 1 e i+(n 1) (r 1) ā(aā) if n r > 0, e i r e i+(n 1) r a(āa) + ( 1) n+r a(āa) e i+1 r e i+1+(n 1) r +( 1) n+r āe i 1 r 1 e i 1+(n 1) (r 1) +( 1) n e i r 1 e i+(n 1) (r 1) ā if n r < 0, k=0 (āa)k [e i n e i 1a + ( 1) n āe i 1 n e i (āa) N k 1 +(aā) k [( 1) n ae i+1 n e i + e i n e i+1 ā(aā) N k 1 if n r = 0. In order to prove that (P n, n ) is a minimal projective resolution of Λ N as a Λ N, Λ N -bimodule, we use an argument which was given in [1. To do this, we note that Λ N /r ΛN P n = m 1 i=0 n r=0 t(g n r,i)λ N as right Λ N -modules and that the map id n : Λ N /r ΛN P n Λ N /r ΛN P n 1 is equivalent to the map m 1 i=0 n r=0 t(gr,i n )Λ N m 1 i=0 n 1 r=0 t(gn 1 r,i )Λ N given by t(g n 1 r,i )a + ( 1) n t(g n 1 r 1,i )ā(aā) if n r > 0, t(gr,i) n t(g n 1 r,i )a(āa) + ( 1) n t(g n 1 r 1,i )ā if n r < 0, t(g n 1 n,i )a(āa) + t(g n 1 n if n r = 0.,i)ā(aā) It is then straightforward (with Lemma 1.4) to see that (Λ N /r ΛN P n, id n ) is a minimal projective resolution of Λ N /r as a right Λ N -module, which satisfies the conditions of [15 as explained above, for all N 1. For completeness in the proof of Theorem 1.6, we explain in full the strategy used in [1, Proposition.8. Theorem 1.6. With the above notation, (P n, n ) is a minimal projective resolution of Λ N as a Λ N, Λ N -bimodule, for all N 1. Proof The first step is to verify that we have a complex by showing that = 0; this is straightforward and the details are left to the reader. Now suppose for contradiction that Ker n 1 Im n for some n 1. Then there is some nonzero map Ker n 1 Ker n 1 / Im n. Hence there is a simple Λ N, Λ N -bimodule U V and epimorphism Ker n 1 / Im n U V such that the composition f : Ker n 1 Ker n 1 / Im n U V is a non-zero epimorphism. We can verify directly that (Λ N /r ΛN P n, id n ) is a minimal projective resolution of Λ N /r as a right Λ N -module, so we have that Λ N /r ΛN Im n = Im(id n ) = Ker(id n 1 ) = Λ N /r ΛN Ker n 1. Applying the functor Λ N /r ΛN to the map n : P n P n 1 gives: Λ N /r ΛN P n id n Λ N /r ΛN P n 1 id n Λ N /r ΛN Im n = Λ N /r ΛN Ker n 1 id f Λ N /r ΛN U V. Now, U is a simple left Λ N -module (and V is a simple right Λ N -module) so Λ N /r ΛN U V is non-zero. Since the functor Λ N /r ΛN preserves epis, the map id f is non-zero, so the composition Λ N /r ΛN P n Λ N /r ΛN U V is non-zero. However, this is simply the functor Λ N /r ΛN applied to the composition P n n Ker n 1 f U V. Now the composition P n n 6

7 Ker n 1 f U V is zero since Im n Ker f. This gives the required contradiction, so the complex is exact. Finally, minimality follows since we know that the projectives are those of a minimal projective resolution of Λ N as a Λ N, Λ N -bimodule from [16. We note that the degenerate cases (r = 0, r = n) in the maps n may be written as: n : e i 0 e i+n e i 0 e i+n 1 a + ( 1) n ae i+1 0 e i+n and n : e i n e i n āe i 1 n 1 e i n + ( 1) n e i n 1 e i n+1 ā Conventions on notation. So far, we have tried to simplify notation by denoting the idempotent o(g n r,i ) t(gn r,i ) of the summand Λ No(g n r,i ) t(gn r,i )Λ N of P n uniquely by e i r e i+n r where 0 i m 1. However, even this notation with subscripts under the tensor product symbol becomes cumbersome in computations and in describing the elements of the Hochschild cohomology ring. Thus we make some conventions (by abuse of notation) which we keep throughout the paper. First, we note that, in the language of [3, the n-th projective module P n in a projective bimodule resolution of Λ N is denoted S S V n S S where S is the semisimple K-algebra generated by the idempotents e 0, e 1,..., e m 1 } and V n is the S, S-bimodule generated by the set g n r,i i = 0, 1,..., m 1, r = 0, 1,..., n}. (Note that here S denotes a tensor over S, that is, over a finite sum of copies of K; we continue to reserve the notation exclusively for tensors over K.) There is a bijective correspondence between the idempotent o(gr,i n ) t(gn r,i ) and the term e i S gr,i n S e i+n r in V n, for i = 0, 1,..., m 1 and r = 0, 1,..., n. Since e i+n r e 0, e 1,..., e m 1 }, it would be usual to reduce the subscript i + n r modulo m. However, to make it explicitly clear to which summand of the projective module P n we are referring and thus to avoid confusion, whenever we write e i e i+k for an element of P n, we will always have i 0, 1,..., m 1} and consider i + k as an element of Z, in that r = (n k)/ and e i e i+k = e i n k e i+k and thus lies in the n k -th summand of P n. We do not reduce i + k modulo m in any of our computations. In this way, when considering elements in P n, our element e i e i+k corresponds uniquely to the idempotent o(g n r,i ) t(gn r,i ) of P n (or, equivalently, to e i S g n r,i S e i+n r ) with r = (n k)/, for each i = 0, 1,..., m 1. Since we take 0 i m 1 we clarify our conventions in Definition 1.5 in the cases i = 0 and i = m 1, to ensure that we always have 0 j m 1 in the first idempotent entry of each tensor (e j ). Specifically, we have: n : e 0 e n r e 0 e (n 1) r a + ( 1) n+r ae 1 e 1+(n 1) r +( 1) n+r ā(aā) e m 1 e m 1+(n 1) (r 1) +( 1) n e 0 e (n 1) (r 1) ā(aā) if n r > 0, e 0 e (n 1) r a(āa) + ( 1) n+r a(āa) e 1 e 1+(n 1) r +( 1) n+r āe m 1 e m 1+(n 1) (r 1) +( 1) n e 0 e (n 1) (r 1) ā if n r < 0, k=0 (āa)k [e 0 e 1 a + ( 1) n āe m 1 e m (āa) N k 1 +(aā) k [( 1) n ae 1 e 0 + e 0 e 1 ā(aā) N k 1 if n r = 0, 7

8 and n : e m 1 e m 1+n r e m 1 e m 1+(n 1) r a + ( 1) n+r ae 0 e (n 1) r +( 1) n+r ā(aā) e m e m +(n 1) (r 1) +( 1) n e m 1 e m 1+(n 1) (r 1) ā(aā) if n r > 0, e m 1 e m 1+(n 1) r a(āa) + ( 1) n+r a(āa) e 0 e (n 1) r +( 1) n+r āe m e m +(n 1) (r 1) +( 1) n e m 1 e m 1+(n 1) (r 1) ā if n r < 0, k=0 (āa)k [e m 1 e m a + ( 1) n āe m e m 1 (āa) N k 1 +(aā) k [( 1) n ae 0 e 1 + e m 1 e m ā(aā) N k 1 if n r = 0. Our notation e i e i+k is used without further comment throughout the rest of the paper. We end this section by determining the dimension of each space Hom Λ e N (P n, Λ N ) for each n The complex Hom Λ e N (P n, Λ N ). All our homomorphisms are Λ e N-homomorphisms and so we write Hom(, ) for Hom Λ e N (, ). Lemma 1.7. If m 3, write n = pm + t where p 0 and 0 t m 1. Then dim K Hom(P n (4p + )mn if t m 1, Λ N ) = (4p + 4)mN if t = m 1. If m = 1 or m = then dim K Hom(P n, Λ N ) = 4N(n + 1). Proof An element f in Hom(P n, Λ N ) is determined by the elements f(e i e i+n r ), which can be arbitrary elements of e i Λ N e i+n r. Suppose first that m 3. Then, for vertices i, j, we have dim e i Λ N e j = N if i = j, N if i j ±1 (mod m), 0 otherwise. For a finite subset X Z and 0 s m 1, we define ν s (X) = x X : x s (mod m)}. With this we have dim Hom(P n, Λ N ) = mn(ν 0 (I n ) + ν 1 (I n ) + ν 1 (I n )) where I n = n r : 0 r n}. We observe that the disjoint union I n I n 1 is equal to [ n, n, the set of all integers x with n x n. Clearly ν 0 ([ n, n) = p + 1 and p if t = 0 ν 1 ([ n, n) = p + 1 if 1 t m p + if t = m 1. Moreover ν 1 ([ n, n) = ν 1 ([ n, n), by symmetry. The lemma now follows by induction on n, with the case n = 0 being clear. For the inductive step, note that dim Hom(P n, Λ N ) = mn(ν 0 (I n ) + ν 1 (I n ) + ν 1 (I n )) = mn(ν 0 ([ n, n) + ν 1 ([ n, n)) dim Hom(P n 1, Λ N ). The statement for dim Hom(P n, Λ N ) now follows for m 3. 8

9 In the case m = 1, we have P n = n r=0λ N Λ N and dim Λ N = 4N. Thus dim Hom(P n, Λ N ) = 4N(n + 1). Finally, for m =, suppose first that n is even. Then P n has n + 1 summands equal to Λ N e 0 e 0 Λ N and n + 1 summands equal to Λ N e 1 e 1 Λ N. By symmetry, dim Hom(P n, Λ N ) = (n + 1) dim e 1 Λ N e 1 = 4N(n + 1). For n odd, P n has n + 1 summands equal to Λ N e 0 e 1 Λ N and n+1 summands equal to Λ N e 1 e 0 Λ N. Hence, again using symmetry, we have dim Hom(P n, Λ N ) = (n + 1) dim e 0 Λ N e 1 = 4N(n + 1). This completes the proof.. The dimensions of the Hochschild cohomology spaces for m 3. We give here the main arguments we used to compute the kernel and the image of the differentials, and leave the details to the reader. We denote both the map P n P n 1 and its induced map Hom(P n 1, Λ N ) Hom(P n, Λ N ) by n. Then we have that HH n (Λ N ) = Ker( n+1 )/ Im( n ). We assume without further comment that m 3 throughout this section..1. Explicit description of maps. In order to compute the Hochschild cohomology groups, we need an explicit description of the image of each e i r e i+n r = e i e i+n r under f Hom(P n, Λ N ) for i = 0, 1,..., m 1. We write n = pm + t with 0 t m 1, p 0. The image f(e i e i+n r ) lies in e i Λ N e j where j Z m with j i + n r (mod m). Since e i Λ N e j = (0) if the vertices i and j are sufficiently far apart in the quiver of Λ N (at least for m 3), for a non-zero image f(e i e i+n r ) we are only interested in examining the cases where i + n r i, i 1, i + 1 (mod m). This leads to the consideration of terms of the form f(e i e i+sm ), f(e i e i+sm 1 ) and f(e i e i+sm+1 ) where s Z, s 0. For any s 0 and 0 i m 1, we have that f(e i e i+sm ) is a linear combination of the elements (aā) k, 0 k N 1, and (āa) k, 1 k N; f(e i e i+sm 1 ) is a linear combination of the ā(aā) k, 0 k N 1; and f(e i e i+sm+1 ) is a linear combination of the a(āa) k, 0 k N 1. For m 3, we give here a list of the terms which must be considered in giving a possible nonzero image of f; these are used without further comment throughout the paper. (We assume m 3; the cases m = 1 and m = are different and are considered separately later.) For each i = 0, 1,..., m 1: m even, m t even f(e i e i+αm ), p α p, t odd, t m 1 f(e i e i+βm 1 ), p β p, f(e i e i+γm+1 ), p γ p, t = m 1 f(e i e i+βm 1 ), p β p + 1, f(e i e i+γm+1 ), p 1 γ p, m odd, m 1 t even, t m 1 f(e i e i+(p α)m ), 0 α p, f(e i e i+(p β 1)m 1 ), 0 β p 1, f(e i e i+(p γ 1)m+1 ), 0 γ p 1, t odd f(e i e i+(p α 1)m ), 0 α p 1, f(e i e i+(p β)m 1 ), 0 β p, f(e i e i+(p γ)m+1 ), 0 γ p, t = m 1 f(e i e i+(p α)m ), 0 α p, f(e i e i+(p β 1)m 1 ), 1 β p 1, f(e i e i+(p γ 1)m+1 ), 0 γ p... Kernel of pm+t+1 : Hom(P pm+t, Λ N ) Hom(P pm+t+1, Λ N ). An element f in Hom(P pm+t, Λ N ) is determined by f(e i e i+n r ) for appropriate i 0, 1,..., m 1}. Each term f(e i e i+n r ) can be written as a linear combination of the basis elements in e i Λ N e i+n r, and we need to find the conditions on the coefficients in this linear combination when f is in Ker pm+t+1. To do this, we separate the study into several cases: when m is even, we consider 9

10 the three cases (i) t = m 1, (ii) t even, and (iii) t odd with t m 1, and, when m is odd, we look at the four cases (i) t = m 1, (ii) t = m, (iii) t even with t m 1, and (iv) t odd with t m. Note that in some cases a factor N appears so that the results depend on the characteristic of K. Note also that, when m is odd, the case char K = needs to be considered separately. f(e i e i+αm ) = k=0 σα i,k (a iā i ) k + N k=1 τ i,k α (ā i 1a i 1 ) k, For f Hom(P pm+t, Λ N ), we write f(e i e i+βm 1 ) = k=0 λβ i,k (āa)k ā i 1, f(e i e i+γm+1 ) = k=0 µγ i,k (aā)k a i, with coefficients σi,k α, τ i,k α, λβ i,k and µγ i,k in K, for i = 0, 1,..., m 1. We shall now do two of the cases above, in order to illustrate the general method. In both these cases, for consistency with our convention on the writing of elements e i e i+k with i 0, 1,..., m 1} and k Z, and to simplify, we shall use the following notation: if n Z, n denotes the representative of n modulo m in 0, 1,..., m 1}. In particular, m = 0, and this must be borne in mind when computing kernels and images. We write f (rather than pm+t+1 f) for the image of f under the map pm+t+1 : Hom(P pm+t, Λ N ) Hom(P pm+t+1, Λ N ). Case m even and t even. In this case, f is entirely determined by the f(e i e i+α m) for p α p and i 0, 1,..., m 1} and f is determined by f(e i e i+βm 1 ) for p β p (or p β p + 1 if t = m ) with 0 i m 1. So f f(e i e i+γm+1 ) for p γ p (or p 1 γ p if t = m ) is in the kernel of if and only if f(e i e i+βm 1 ) = 0 = f(e i e i+γm+1 ) for all i, β and γ. Now f(e [ i e i+βm 1 ) = ( 1) (p β) m σ β i 1,0 σβ i,0 ā(aā) if β > 0 [ ( 1) (p β) m σ β i 1,0 σβ i,0 ā + [ k=1 ( 1) (p β) m σ β i 1,k τ β i,k (āa) k ā if β 0 f(e [ i e i+γm+1 ) = σ γ i,0 ( 1)(p γ) m σ γ i+1,0 a + [ k=1 σ γ i,k ( 1)(p γ) m τ γ i+1,k a(āa) k if γ 0 [ σ γ i,0 ( 1)(p γ) m σ γ i+1,0 a(āa) if γ > 0. For any α with p α p, we can see that σ0,0 α determines the other σi,0 α (setting β or γ to be α ). Combining the cases β 0 and γ 0 tells us that the τi,k α for k = 1,..., N 1 are determined by the σi,k α α. Finally the τi,n are arbitrary, and the σα i,k for k = 1,..., N 1 are arbitrary. Therefore the dimension of Ker pm+t+1 in this case is (p + 1) + (p + 1)m + (p + 1)m(N 1). Case m even and t odd, t m 1. In this case, f is entirely determined by f(e i e i+β m 1) for p β p, f(e i e i+γ m+1) for p γ with i 0, 1,..., m 1} p, and f Hom(P (p+1)m, Λ N ) is determined by f(e i e i+αm ) for p α p. Now [ N λ 0 i,0 + ( 1)(p+1) m λ 0 i+1,0 + ( 1)(p+1) m µ 0 i 1,0 + µ0 i,0 (aā) N if α = 0 [ λ α i, + ( 1)(p+1+α) m λ α i+1, + ( 1)(p+1+α) m µ α i 1,0 + µα i,0 (aā) N f(e i e i+αm ) = + N k=0 λα i,k (āa)k+1 + ( 1) (p+1+α) m N k=0 λα i+1,k (aā)k+1 if α > 0 [ λ α i,0 + ( 1)(p+1+α) m λ α i+1,0 + ( 1)(p+1+α) m µ α i 1, + µα i, (aā) N +( 1) (p+1+α) m N k=0 µα i 1,k (āa)k+1 + N k=0 µα i,k (aā)k+1 if α < 0. So for α = 0, λ 0 i,0, for i 0,..., m 1} and µ0 0,0 determine the other µ 0 i,0, unless the characteristic of K divides N. For α > 0, µ α i,0 and λα 0, determine the other λα i,, and for α < 0, 10

11 λ α i,0 and µα 0, determine the other µα i,. Moreover, for α > 0 the λα i,k for k = 0,..., N are all 0, and for α < 0 the µ α i,k for k = 0,..., N are 0. The remaining λα i,k, µα i,k are arbitrary, and there are m(n 1) + pm(n 1) + pm(n 1) = (p + 1)m(N 1) of these coefficients. This gives dimension: (m + 1) + p(m + 1) + p(m + 1) + (p + 1)m(N 1) if char K N and (m + 1) + p(m + 1) + p(m + 1) + (p + 1)m(N 1) + m 1 if char K N. Note that when computing the dimensions in the case m odd and char K, we often need to introduce additional cases depending on the value of p modulo 4 and the value of m modulo 4. We summarise the results in the following two propositions, which separate the cases char K N and char K N. Proposition.1. If char K N, the dimension of Ker pm+t+1 is as follows. If m even (p + 1)(mN + 1) if t even, (p + 1)(mN + 1) + m(n 1) if t odd, t m 1 (p + 1)(mN + 1) + m(n 1) + if t = m 1. If m odd (p + 1)(mN + 1) if t m 1, p + t even, and char K = (p + 1)(mN + 1) + m(n 1) if t m 1, p + t odd, (p + 1)(mN + 1) + if t = m 1, p even, (p + 1)(mN + 1) + m(n 1) + if t = m 1, p odd. If m odd (p + 1)mN + p + 1 if t 0 (mod 4), t m 1, p even, and char K (p + 1)mN + p if t (mod 4), t m 1, p even, (p + 1)mN + m(n 1) + p+1 if m + t 1 (mod 4), t m 1, p odd, (p + 1)mN + m(n 1) + p 1 if m + t 3 (mod 4), t m 1, p odd, (p + 1)mN + m(n 1) + p + 1 if t 1 (mod 4), p even, (p + 1)mN + m(n 1) + p if t 3 (mod 4), p even, (p + 1)mN + p 1 if t m (mod 4), p odd, (p + 1)mN + p+1 if t m + (mod 4), p odd, (p + 1)mN + p + 3 if t = m 1 0 (mod 4), p even, (p + 1)mN + p + if t = m 1 (mod 4), p even, (p + 1)mN + m(n 1) + p+1 if t = m 1, p odd. 11

12 Proposition.. If char K N, the dimension of Ker pm+t+1 is given by the following. If m even (p + 1)(mN + 1) if t even, (p + )mn + p if t odd, t m 1 (p + )(mn + 1) if t = m 1. If m odd (p + 1)(mN + 1) if t m 1, p + t even, and char K = (p + )mn + p if t m 1, p + t odd, (p + 1)(mN + 1) + if t = m 1, p even, (p + )(mn + 1) if t = m 1, p odd. If m odd (p + 1)mN + p + 1 if t 0 (mod 4), p even, and char K (p + 1)mN + p if t (mod 4), p even, (p + )mn + p 1 if t even, p 1 (mod 4) (p + )mn + p+1 if m + t 1 (mod 4), p 3 (mod 4) (p + )mn + p 3 if m + t 3 (mod 4), p 3 (mod 4) (p + )mn + p if t odd, p 0 (mod 4) (p + )mn + p + 1 if t 1 (mod 4), p (mod 4) (p + )mn + p 1 if t 3 (mod 4), p (mod 4) (p + 1)mN + p 1 if t m (mod 4), p odd (p + 1)mN + p+1 if t m + (mod 4), p odd (p + )mn + p+1 if t = m 1, p 3 (mod 4) (p + )mn + p 1 if t = m 1, p 1 (mod 4) (p + 1)mN + p + 3 if t = m 1 0 (mod 4), p even (p + 1)mN + p + if t = m 1 (mod 4), p even..3. Image of pm+t : Hom(P pm+t 1, Λ N ) Hom(P pm+t, Λ N ). We explain here how to compute Im pm+t explicitly (rather than simply compute its dimension with the rank and nullity theorem) on the same cases as for the kernels, in order to be able to give a basis of cocycle representatives for each Hochschild cohomology group HH n (Λ N ). For an element g Hom(P pm+t 1, Λ N ), we consider pm+t g, and find the dimension of the subspace of Ker( pm+t+1 ) spanned by all the pm+t g. We shall again use the notation n for the representative modulo m of the integer n, and write g for pm+t g. Case m even and t even, t 0. The map g is determined by g(e i e i+αm ) for p α p and g is determined by g(e i e i+β m 1) for p β p g(e i e i+γ m+1) for p γ with i 0, 1,..., m 1}. Now p [ N ( 1) p m µ 0 i 1,0 + µ0 i,0 + λ0 i,0 + ( 1)p m λ 0 i+1,0 (aā) N if α = 0 [ ( 1) (p+α) m µ α i 1,0 + µα i,0 + λα i, + ( 1)(p+α) m λ α i+1, (aā) N g(e i e i+αm ) = + N k=0 λα i,k (āa)k+1 + N k=0 λα i+1,k (aā)k+1 if α > 0 [ ( 1) (p+α) m µ α i 1, + µα i, + λα i,0 + ( 1)(p+α) m λ α i+1,0 (aā) N + N k=0 µα i,k (aā)k+1 + ( 1) (p+α) m N k=0 µα i 1,k (āa)k+1 if α < 0. We can check that in each case, the sum or the alternate sum over i 0,..., m 1} of the coefficients of the (aā) N is zero (recall that m = 0 and that m is even), so for each α we add m 1 to the dimension, except when char K N and α = 0, in which case, g(e i e i ) = 0 and this makes no contribution to dim Im pm+t. Moreover, for α 0, we can take arbitrary coefficients 1

13 for the (aā) k+1 with 0 k N, and the coefficients of the (āa) k+1 are completely determined by those of the (aā) k+1. So dim K Im pm+t (p + 1)(m 1) + pm(n 1) if char K N = p(mn 1) if char K N. Case m even and t odd, t m 1. g(e i e i+βm 1 ) for p β p The element g is determined by g(e i e i+γm+1 ) for p γ p g(e i e i+α m) for p α p. Now and g is determined by g(e i [ e i+βm 1 ) = ( 1) (p+β+1) m σ β i 1,0 + σβ i,0 (āa) ā if β > 0 [ ( 1) (p+β+1) m σ β i 1,0 + σβ i,0 ā [ k=1 ( 1) (p+β+1) m σ β i 1,k + τ β i,k (āa) k ā if β 0 g(e [ i e i+γm+1 ) = σ γ i,0 + ( 1)(p+γ+1) m σ γ i+1,0 a + [ k=1 σ γ i,k + ( 1)(p+γ+1) m τ γ i+1,k a(āa) k if γ 0 [ σ γ i,0 + ( 1)(p+γ+1) m σ γ i+1,0 a(āa) if γ < 0. For 0 < α p (resp. p α < 0), the coefficient of (āa) ā (resp. ā) in g(e i e i+α m 1) is equal, up to sign, to the coefficient of a (resp. a(āa) ) in g(e i e i+α m+1). The coefficient of ā in g(e i e i 1 ) (that is, when α = 0) is equal, up to sign, to that of a in g(e i e i+1 ). Moreover, either the sum or the alternate sum over i 0,..., m 1} of these coefficients is zero (depending on the parity of (p + α + 1) m ). They contribute m 1 to the dimension of Im pm+t for each α. The remaining coefficients of the (āa) k ā and a(āa) k, for k = 1,..., N 1, are arbitrary except for the case γ = 0, where the coefficient of a(āa) k is equal, up to sign, to that of (āa) k ā in g(e i e i 1 ) (for β = 0). So dim K Im pm+t = (p + 1)(m 1) + (p + 1)m(N 1)..4. Dimension of HH n (Λ N ). We will now give the dimension of each of the Hochschild cohomology groups of Λ N. Note that the dimensions can be verified using Propositions.1 and., together with the rank and nullity theorem in the form dim Im( pm+t ) = dim Hom(P pm+t 1, Λ N ) dim Ker( pm+t ). Proposition.3. If m is even, or if m is odd and char K =, then dim K HH 0 (Λ N ) = Nm + 1 and, for pm + t 0, 4p + + m(n 1) if t m 1, char K N, dim K HH pm+t 4p m(n 1) if t = m 1, char K N, (Λ N ) = 4p mn if t m 1, char K N, 4p mn if t = m 1, char K N. 13

14 Proposition.4. If m is odd, char K and char K N, then dim K HH 0 (Λ N ) = Nm + 1 and, for pm + t 0, 1 if t = 0, p even, 3 if t = 0, p odd, m 1 even, 1 if t = 0, p odd, m 1 odd, 1 if t even, t 0, m 1, p even, 1 if m + t 3 (mod 4) even, t 0, m 1, p odd, dim K HH pm+t (Λ N ) = m(n 1) + p + 1 if m + t 1 (mod 4) even, t 0, m 1, p odd, if t 1 (mod 4), p even, 0 if t 3 (mod 4), p even, 0 if t odd, p odd, 3 if t = m 1, p even, 1 if t = m 1, p odd. Proposition.5. If m is odd, char K and char K N, then dim K HH 0 (Λ N ) = Nm + 1 and, for pm + t 0, 1 if t = 0, p 0, 1 (mod 4) 0 if t = 0, p (mod 4) 3 if t = 0, p 3 (mod 4), m 1 (mod 4) 0 if t = 0, p 3 (mod 4), m 3 (mod 4) 1 if t even, t 0, m 1, p + t 0 (mod 4) 0 if t even, t 0, m 1, p + t (mod 4) 0 if m + t 1 (mod 4), t 0, m 1, p 1 (mod 4), 1 if m + t 3 (mod 4), t 0, m 1, p 1 (mod 4) 1 if m + t 1 (mod 4), t 0, m 1, p 3 (mod 4) dim K HH pm+t if m + t 3 (mod 4), t 0, m 1, p 3 (mod 4) (Λ N ) = mn + p + 1 if t 1 (mod 4), p 0 (mod 4) 0 if t 3 (mod 4), p 0 (mod 4) if t 1 (mod 4), p (mod 4) 1 if t 3 (mod 4), p (mod 4) 1 if t odd, p + m t 1 (mod 4) 0 if t odd, p + m t 3 (mod 4) 3 if t = m 1, p + m 1 (mod 4) if t = m 1, p + m 3 (mod 4) 0 if t = m 1, p 1 (mod 4) 1 if t = m 1, p 3 (mod 4). 3. The centre of the algebra Λ N We start by describing Z(Λ N ), the centre of Λ N, since it is well known that HH 0 (Λ N ) = Z(Λ N ). Note that the next result requires m ; the case m = 1 is dealt with separately in Theorem

15 Theorem 3.1. Suppose that N 1, m. Then dim HH 0 (Λ N ) = Nm + 1 and the set 1, (ai ā i ) N, [(a i ā i ) s + (ā i a i ) s for i = 0, 1,..., m 1 and s = 1,..., N 1 } is a K-basis of HH 0 (Λ N ). Moreover, HH 0 (Λ N ) is generated as an algebra by the set 1, a i ā i for i = 0, 1..., m 1} if N = 1; 1, (ai ā i ) N, [a i ā i + ā i a i for i = 0, 1..., m 1 } if N > 1. The proof is straightforward since, for N > 1, we have [(a i ā i ) s + (ā i a i ) s = ([a i ā i + ā i a i ) s. 4. The Hochschild cohomology ring HH (Λ N ) for m 3 and m even. In this section, we assume that N 1, m 3 and m even (so that, necessarily, m 4) and give all the details of HH (Λ N ). In the subsequent sections we will consider the case N 1, m 3 and m odd, and leave many of the details to the reader since the computations are similar Basis of HH n (Λ N ). For each n 1, we describe the elements of a basis of the Hochschild cohomology group HH n (Λ N ) in terms of cocycles in Hom(P n, Λ N ), that is, we give a set of elements in Ker n+1, each of which is written as a map in Hom(P n, Λ N ), such that the corresponding set of cosets in Ker n+1 / Im n with these representatives forms a basis of Ker n+1 / Im n = HH n (Λ N ). Following standard usage, our notation does not distinguish between an element of HH n (Λ N ) and a cocycle in Hom(P n, Λ N ) which represents that element of HH n (Λ N ). However the precise meaning is always clear from the context. For each cocycle in Hom(P n, Λ N ), we simply write the images of the generators e i e i+n r in P n which are non-zero. We keep to these conventions throughout the whole paper. Proposition 4.1. Suppose that N 1, m 3 and m is even. For each n 1, the following elements define a basis of HH n (Λ N ). (1) For n even, n : (a) For p α p: χ n,α : e i e i+αm ( 1) ( n α m )i e i for i = 0, 1,..., m 1; (b) For p α p: π n,α : e 0 e αm (a 0 ā 0 ) N ; (c) For each j = 0, 1,..., m and each s = 1,..., N 1: e j e j (a j ā j ) s F n,j,s : e j+1 e j+1 ( 1) n (ā j a j ) s ; (d) For each s = 1,..., N 1: e m 1 e m 1 (a m 1 ā m 1 ) s F n,m 1,s : e 0 e 0 ( 1) n (ā 0 a 0 ) s ; (e) Additionally in the case char K N, for each j = 1,..., m 1: θ n,j : e j e j (a j ā j ) N. () For n odd, n 1: (a) For p γ < 0 and, if t = m 1, γ = p 1 also: ϕ n,γ : e i e i+γm+1 ( 1) ( n 1 γm )i a i (ā i a i ) for all i = 0, 1,..., m 1; (b) For 0 γ p: ϕ n,γ : e i e i+γm+1 ( 1) ( n 1 γm )i a i for all i = 0, 1,..., m 1; (c) For 0 < β p and, if t = m 1, β = p + 1 also: ψ n,β : e i e i+βm 1 ( 1) n 1 βm iā i 1 (a i 1 ā i 1 ) for all i = 0, 1,..., m 1; (d) For p β 0: ψ n,β : e i e i+βm 1 ( 1) n 1 βm iā i 1 for all i = 0, 1,..., m 1; (e) For each j = 0, 1,..., m 1 and each s = 1,..., N 1: E n,j,s : e j e j+1 (a j ā j ) s a j ; (f) Additionally in the case char K N, for each j = 1,..., m 1: ω n,j : e j e j+1 a j. 15

16 4.. Computing products in cohomology. In order to compute the products in the Hochschild cohomology ring, we require a lifting of each element in the basis of HH n (Λ N ), that is, for each cocycle f Hom(P n, Λ N ), we give a map L f : P +n P such that (1) the composition P n L0 f P 0 0 Λ N is equal to f, and () the square P q+n L q f P q commutes for every q 1. q+n P q 1+n L q 1 f q P q 1 It should be noted that such liftings always exist but are not unique. It is easy to check that the maps given in this paper are indeed liftings of the appropriate cocycle. For homogeneous elements η HH n (Λ N ) and θ HH k (Λ N ) represented by cocycles η : P n Λ N and θ : P k Λ N respectively, the cup product (or Yoneda product) ηθ HH n+k (Λ N ) is the coset represented by the composition P n+k Ln θ P n η Λ N. The fact that the cup product is well-defined means that the composition ηθ does not depend on the choice of representative cocycles for η and θ or on the choice of the liftings of these cocycles Liftings for N = 1. We continue to assume that m is even in the details that follow. Proposition 4.. For n 1 and the algebra Λ 1, m 3 and m even, the following maps are liftings of the cocycle basis of HH n (Λ 1 ) given in Proposition 4.1. (1) For n even, n, and for p α p: L q χ n,α : e i e i+αm+q l ( 1) ( n α m )i e i e i+q l for all i = 0, 1,..., m 1 and all 0 l q, L q π n,α : e 0 e αm+q l a 0 ā 0 e 0 e q l for all 0 l q, () For n odd: For p γ p if t m 1 p 1 γ p if t = m 1: L q ϕ n,γ : e i e i+γm+1+q l ( 1) ( n 1 γ m )i+q e i e i+q l a i+q l for all i = 0, 1,..., m 1 and all 0 l q, p β p if t m 1 For p β p + 1 if t = m 1: L q ψ n,β : e i e i+βm 1+q l ( 1) ( n 1 β m )i e i e i+q l ā i+q l 1 for all i = 0, 1,..., m 1 and all 0 l q Liftings for N > 1. Proposition 4.3. For n 1 and the algebra Λ N with N > 1, m 3 and m even, the following maps give a lifting of the cocycles in Proposition 4.1 above. Once again, to simplify cases and for consistency with our conventions, we use the notation n 0, 1,..., m 1} for the representative of the integer n modulo m. (1) For n even, n : 16

17 For p α p: L q χ n,α (e i e i+q l+αm ) = ( 1) n αm i e i e i+q l if (q l)α 0 0 if (q l)α < 0 and q l > ( 1) n αm i (aā) e i e i+ (āa) if α < 0 and q l = ( 1) n αm i (āa) [ e i e i (aā) if α > 0 and q l = ( 1) n αm i k=0 (aā)k e i e i+1 (āa) N k 1 +( 1) q+1 N k=0 (aā)k ae i+1 e i+1 1 a(āa) N k if α < 0 and q l = 1 [ ( 1) n αm i k=0 (āa)k e i e i 1 (aā) N k 1 +( 1) q+1 N k=0 (āa)k āe i 1 e i 1 +1 ā(aā) N k if α > 0 and q l = 1 for all i = 0, 1,..., m 1 and all 0 l q. For p α p: (aā) N e 0 e q l if α(q l) 0 L q (aā) N e 0 e 1 (āa) if q l = 1 and α < 0 π n,α (e 0 e q l+αm ) = (aā) N e 0 e 1 (aā) if q l = 1 and α > 0 0 otherwise for all 0 l q. For each j = 0, 1,..., m and each s = 1,..., N 1: L q e j e j+q l (a j ā j ) s e j e j+q l F n,j,s : e j+1 e j+1+q l ( 1) n (ā j a j ) s e j+1 e j+1+q l for all 0 l q. For each s = 1,..., N 1: L q e m 1 e m 1+q l (a m 1 ā m 1 ) s e m 1 e m 1+q l F n,m 1,s : e 0 e q l ( 1) n (ā m 1 a m 1 ) s e 0 e q l for all 0 l q. In the case char K N, for each j = 1,..., m 1: L q θ n,j : e j e j+q l (a j ā j ) N e j e j+q l for all 0 l q. () For n odd: For p γ < 0 and, if t = m 1, γ = p 1 also: 0 if q l > 1 L q ϕ n,γ (e i e i+γm+1+q l ) = ( 1) n 1 γm i (aā) e i e i+1 a(āa) if q l = 1 n 1 γm q+ ( 1) i e i e i+q l a(āa) if q l 0 for all i = 0, 1,..., m 1 and all 0 l q. For 0 γ p : If γ > 0 then L q ϕ n,γ (e i e i+γm+1+q l ) = n 1 γm q+ ( 1) i e i e i+q l a if q l 0 ( 1) n 1 γm i ( 1) n 1 γm [ (āa) k e i e i 1 (aā) N k 1 a+ k=0 ( 1) q+1 N (āa) k āe i 1 e i 1 +1 (āa) N k 1 k=0 if q l = 1 i (āa) e i e i (aā) a if q l = 0 if q l < for all i = 0, 1,..., m 1 and all 0 l q. 17

18 If γ = 0 then L q ϕ n,0 (e i e i+1+q l ) = n 1 q+ ( 1) i e i e i+q l a if q l = q ( 1) [ n 1 q+ i e i e i+q l a ( 1) q (N 1)e i e i+q l+ ā(aā) if 0 q l < q n 1 q+ ( 1) i Ne i e i+q l a(āa) if q l < 1 k 1 [ ( 1) n 1 i (aā) v e i e i+1 ā(aā) N v 1 + k=1 v=0 ( 1) q+1 (aā) v ae i+1 e i+1 1 (aā) N v 1 + ( 1) q+1 (āa) v āe i 1 e i 1 +1 (āa) N v 1 + (āa) v e i e i 1 a(āa) N v 1 } (āa) k e i e i 1 a(āa) N k 1 + k=0 if q l = 1 for all i = 0, 1,..., m 1 and all 0 l q. For 0 < β p and, if t = m 1, β = p + 1 also: 0 if q l < 1 L q ψ n,β (e i e i+βm 1+q l ) = ( 1) n 1 βm i (āa) e i e i 1 ā(aā) if q l = 1 ( 1) n 1 βm i e i e i+q l ā(aā) if q l > 1 for all i = 0, 1,..., m 1 and all 0 l q. For p β 0: If β < 0, then L q ψ n,β (e i e i+βm 1+q l ) = ( 1) n 1 βm i e i e i+q l ā if q l < 1 [ ( 1) n 1 βm i (aā) k e i e i+1 ā(aā) N k 1 + k=0 ( 1) q+1 N k=0 (aā)k ae i+1 e i+1 1 (aā) N k 1 if q l = 1 ( 1) n 1 βm i (aā) e i e i+ (āa) ā if q l = 0 if q l > for all i = 0, 1,..., m 1 and all 0 l q. If β = 0 then L q ψ n,0 (e i e i 1+q l ) = ( 1) n 1 i e i e i+q l ā if q l = q ( 1) [ n 1 i e i e i+q l ā ( 1) q (N 1)e i e i+q l a(āa) if q < q l 0 ( 1) n 1 i Ne i e i+q l ā(aā) if q l > 1 k 1 [ ( 1) n 1 i (āa) v e i e i 1 a(āa) N v 1 + k=1 v=0 + k=0 for all i = 0, 1,..., m 1 and all 0 l q. ( 1) q+1 (āa) v āe i 1 e i 1 +1 (āa) N v 1 +( 1) q+1 (aā) v ae i+1 e i+1 1 aā) N v 1 + (aā) v e i e i+1 ā(aā) } N v 1 (aā) k e i e i+1 ā(aā) N k 1 if q l = 1 18

19 For each j = 0, 1,..., m 1 and each s = 1,..., N 1: L q E n,j,s = e j e j+1 (aā) s e j e j a N s N s [ e j e j (aā) v+s e j e j+1 ā(aā) N v 1 + k=0 v=k ( 1) q+1 (aā) v+s ae j+1 e j+1 1 (aā) N v 1 e j+1 e j+1 ( 1) n 1 N s 1 N s [( 1) q+1 (āa) v+sāe j e j+1 (āa) N v 1 k=0 v=k +(āa) v+s+1 e j+1 e j+1 1 a(āa) N v In the case char K N, for each j = 1,..., m 1: L q ω n,j = e j e j k 1 k=1 v=0 [( 1) q 1 (aā) v ae j+1 e j (aā) N v 1 (aā) v e j e j+1 ā(aā) N v 1 if q is odd k 1 [ e j+1 e j+1 ( 1) n+1 (āa) v+1 e j+1 e j+1 1 a(āa) N v + k=1 v=0 ( 1) q+1 (āa) v āe j e j+1 (āa) N v 1 if q is odd [ e j v e j v +v ( 1) n 1 v q+1 v+ ( 1) ae j v+1 e j v+1 +v 1 + if q is even if q is odd if q is odd e j v e j v +v+1 ā(aā) for 1 v q+1 and q odd e j v e j v +v+1 ( 1) [ n 1 v e j v e j v +v a ( 1) v+ q ā(aā) e j v 1 e j v 1 +v+ for 0 v q and q even The Hochschild cohomology ring HH (Λ 1 ) for N = 1, m 3, m even. Theorem 4.4. For N = 1, m 3 and m even, HH (Λ 1 ) is a finitely generated algebra with generators: 1, a i ā i for i = 0, 1,..., m 1 in degree 0, ϕ 1,0, ψ 1,0 in degree 1, χ,0 in degree, ϕ m 1, 1, ψ m 1,1 in degree m 1, χ m,1, χ m, 1 in degree m. It is known from [1, Proposition 4.4 that those generators in HH (Λ 1 ) whose image is in r are nilpotent, and it may be seen that the remaining generators of HH (Λ 1 ), namely 1, χ,0, χ m,1 and χ m, 1, are not nilpotent. Moreover χ m,0 = χ m,1 χ m, 1. Thus we have the following corollary. Corollary 4.5. For N = 1, m 3 and m even, HH (Λ 1 )/N = K[χ,0, χ m,1, χ m, 1 /(χ m,0 χ m,1 χ m, 1 ) and hence HH (Λ 1 )/N is a commutative finitely generated algebra of Krull dimension. 19

20 4.6. The Hochschild cohomology ring HH (Λ N ) for N > 1, m 3, m even. In this section we describe the main arguments behind the presentation of HH (Λ N ) by generators and relations. The first lemma indicates the method required to show that HH (Λ N ) is a finitely generated algebra whilst the second lemma considers the relations between the generators. Lemma 4.6. Suppose N > 1, m 3 and m even. Then (1) χ,0 χ,0 = χ 4,0 ; () π,0 = (a 0 ā 0 ) N χ,0. Proof (1) By definition, χ,0 χ,0 = χ,0 L χ,0. Now, from Proposition 4.3, L χ,0 (e i e i+ l ) = ( 1) i e i e i+ l for i = 0,..., m 1 and 0 l, and χ,0 (e i e i ) = ( 1) i e i for i = 0,..., m 1. Thus χ,0 L ( 1) χ,0 (e i e i+ l ) = i ( 1) i e i = e i if l = 1, 0 otherwise, for all i = 0,..., m 1. Hence χ,0 χ,0 = χ 4,0. () We know that χ,0 (e i e i ) = ( 1) i e i for i = 0,..., m 1, π,0 : e 0 e 0 (a 0 ā 0 ) N, and (a 0 ā 0 ) N HH 0 (Λ N ). Then ((a 0 ā 0 ) N χ,0 )(e i e i ) = χ,0 ((a 0 ā 0 ) N ( 1) e i e i ) = 0 e 0 (a 0 ā 0 ) N if i = 0, 0 if i = 1,..., m 1. Thus π,0 = (a 0 ā 0 ) N χ,0. We remark that, for n even, n < m and writing n = pm + t as usual, we necessarily have p = 0, and thus, from Proposition 4.1, we have χ n,α p α p} = χ n,0 }. Inductively, Lemma 4.6 (1) gives, for n = k even, n m, that χ n,0 = (χ,0 ) k. Lemma 4.7. Suppose N > 1, m 3 and m even. Then (1) χ m,1 χ m, 1 = 0; () ϕ 1,0 = 0; (3) ϕ 1,0 ψ 1,0 = Nm(a 0 ā 0 ) N χ,0 ; (4) for char K N, S = N k=1 k and ε i = (a i ā i ) N for i = 0, 1,..., m 1, we have Sχ,0 (ε i + ε i+1 ) if char K = and i m 1 ϕ 1,0 ω 1,i = Sχ,0 (ε m 1 + ε 0 ) if char K = and i = m 1 0 if char K. Proof (1) We have χ m,1 χ m, 1 = χ m,1 L m χ m, 1. Since m is even, Proposition 4.3 gives e i e i+m l if m l 0 L m χ m, 1 (e i e i+m l m ) = 0 if m l > 0 and m l > (aā) e i e i+ (āa) if m l = for all i = 0, 1,..., m 1 and all 0 l m. Also χ m,1 (e i e i+m ) = e i for i = 0, 1,..., m 1. Hence χ m,1 L m χ m, 1 (e i e i+m l m ) = 0 and χ m,1 χ m, 1 = 0. () If char K, then ϕ 1,0 = 0 by graded commutativity of HH (Λ N ) so we may assume char K =. Thus we have ϕ 1,0 (e i e i+1 ) = a i for i = 0, 1,..., m 1, and, from Proposition 4.3, 0

21 L 1 ϕ 1,0 (e i e i+1+1 l ) = e i e i+1 a i+1 if l = 0, k 1 [ (aā) v e i e i+1 ā(aā) N v 1 (aā) v ae i+1 e i+1 1 (aā) N v 1 k=1 v=0 (āa) v āe i 1 e i 1 +1 (āa) N v 1 + (āa) v e i e i 1 a(āa) } N v 1 + (āa) k e i e i 1 a(āa) N k 1 k=0 if l = 1, for all i = 0, 1,..., m 1. Hence a i a i+1 if l = 0, ϕ 1,0 L 1 ϕ 1,0 (e i e i+1+1 l ) = k 1 [ (aā) v a i ā(aā) N v 1 + (āa) v āa i 1 (āa) N v 1 if l = 1. k=1 v=0 But a i a i+1 = 0 and (a i ā i ) N +(ā i 1 a i 1 ) N = 0 since char K =. Thus ϕ 1,0 L 1 ϕ 1,0 (e i e i+1+1 l ) = 0 and so ϕ 1,0 = 0. (3) We have ϕ 1,0 (e i e i+1 ) = a i for i = 0, 1,..., m 1, and, from Proposition 4.3, L 1 ψ 1,0 (e i e i 1+1 l ) = e i e i 1 ā i if l = 1, k 1 [ (āa) v e i e i 1 a(āa) N v 1 (āa) v āe i 1 e i 1 +1 (āa) N v 1 k=1 v=0 (aā) v ae i+1 e i+1 1 aā) N v 1 + (aā) v e i e i+1 ā(aā) } N v 1 + (aā) k e i e i+1 ā(aā) N k 1 k=0 if l = 0, for i = 0, 1,..., m 1. Thus ϕ 1,0 L 1 ψ 1,0 (e i e i 1+1 l ) = 0 if l = 1. On the other hand, k 1 [ if l = 0, then ϕ 1,0 L 1 ψ 1,0 (e i e i ) = (āa)vāa i 1 (āa) N v 1 + (aā) v a i ā(aā) N v 1 + k=1 v=0 (aā) k a i ā(aā) N k 1 = (a i ā i ) N = N(a i ā i ) N. k=0 k=0 Now define g Hom(P 1, Λ N ) by e i e i+1 (m 1 i)a i for i = 0, 1,..., m (with our usual convention that all other summands are mapped to zero). Then g Hom(P, Λ N ) and, for i = 1,..., m, we have g(e i e i ) = k=0 [ (āa)k āg(e i 1 e i )(āa) N k 1 + (aā) k g(e i e i+1 )ā(aā) N k 1 = k=0 [ (āa)k ā(m i)a i 1 (āa) N k 1 + (aā) k (m 1 i)a i ā(aā) N k 1 = k=0 [ (m i)(ā i 1a i 1 ) N + (m 1 i)(a i ā i ) N = N(a i ā i ) N. In a similar way, we also have g(e m 1 e m 1 ) = N(a m 1 ā m 1 ) N. For i = 0 we have, g(e 0 e 0 ) = k=0 [ (āa)k āg(e m 1 e m )(āa) N k 1 + (aā) k g(e 0 e 1 )ā(aā) N k 1 = k=0 (aā)k (m 1)a 0 ā(aā) N k 1 = N(m 1)(a 0 ā 0 ) N. 1

22 Moreover, g is zero on all other summands. Hence N(m 1)(a0 ā g(e i e i ) = 0 ) N if i = 0, N(a i ā i ) N if i = 1,..., m 1. So ϕ 1,0 L 1 ψ 1,0 + g is the map given by e 0 e 0 Nm(a 0 ā 0 ) N and thus ϕ 1,0 L 1 ψ 1,0 + g = Nmπ,0. Using Lemma 4.6(), we have that the cocycles ϕ 1,0 ψ 1,0 and Nm(a 0 ā 0 ) N χ,0 represent the same elements in Hochschild cohomology so that ϕ 1,0 ψ 1,0 = Nm(a 0 ā 0 ) N χ,0 as required. (4) Finally, suppose char K N, S = N k=1 k and let ε i = (a i ā i ) N for i = 0, 1,..., m 1. For ease of notation we consider the product ϕ 1,0 ω 1,i where 1 i m ; the cases i = 0 and i = m 1 are similar. We know ϕ 1,0 (e j e j+1 ) = a j for j = 0, 1,..., m 1 and, from Proposition 4.3, we have k 1 [ e i e i (aā) v ae i+1 e i (aā) N v 1 (aā) v e i e i+1 ā(aā) N v 1 k=1 v=0 L 1 ω 1,i : k 1 [ e i+1 e i+1 (āa) v+1 e i+1 e i a(āa) N v (āa) v āe i e i+1 (āa) N v 1 k=1 v=0 e i 1 e i+1 ae i e i+1. Hence ϕ 1,0 L 1 ω 1,i (e i e i ) = k 1 (aā) v a i ā(aā) N v 1 = k=1 v=0 k 1 ϕ 1,0 L 1 ω 1,i (e i+1 e i+1 ) = ( 1) ( 1)(āa) v āa i (āa) N v 1 = k=1 v=0 Finally, ϕ 1,0 L 1 ω 1,i (e i 1 e i+1 ) = 0. Hence ϕ 1,0 L 1 ω 1,i : Now, χ,0 (ε i + ε i+1 ) : e i e i ( 1) i (a i ā i ) N e i+1 e i+1 ( 1) i+1 (a i+1 ā i+1 ) N. k=1 k(a i ā i ) N = S(a i ā i ) N. And k=1 k(ā i a i ) N = S(ā i a i ) N. e i e i S(a i ā i ) N e i+1 e i+1 S(a i+1 ā i+1 ) N. If char K =, then ϕ 1,0 ω 1,i = ϕ 1,0 L 1 ω 1,i = Sχ,0 (ε i + ε i+1 ). However, if char K then, since char K N, we have S = (N 1)N = 0 so that S = 0 and hence ϕ 1,0 ω 1,i = 0. By considering all possible products of the basis elements of HH (Λ N ) from Proposition 4.1, we are now able to give the main result of this section. We do not explicitly list the relations which are a direct consequence of the graded commutativity of HH (Λ N ). Note that we can check that we do indeed have all the relations by comparing dim HH n (Λ N ) from Proposition.3 with the dimension in degree n of the algebra given in Theorem 4.8 below. Theorem 4.8. Suppose N > 1, m 3 and m even. (1) HH (Λ N ) is a finitely generated algebra with generators: 1, (a i ā i ) N, [a i ā i + ā i a i for i = 0, 1,..., m 1 in degree 0 ϕ 1,0, ψ 1,0 in degree 1 ω 1,j for j = 1,..., m 1 if char K N in degree 1 χ,0 in degree ϕ m 1, 1, ψ m 1,1 in degree m 1 χ m,1, χ m, 1 in degree m. () Let ε i = (a i ā i ) N for i = 0, 1,..., m 1 and S = N k=1 k. Then HH (Λ N ) is a finitely generated algebra over HH 0 (Λ N ) with generators: