Support varieties and the Hochschild cohomology ring modulo nilpotence

SUPPOR T V ARIETI E S AND THE HOCH SCHILD COHOMOLOGY R ING MODULO NILPOTENCE NICOLE SNASHALL Abstract. This pap er is bas ed on my talks g iven at the ‘4 1st Sympo sium on Ring Theory a nd Representation Theory’ held at Shizuok a Universit y , Japan, 5-7 Septem b er 2 008. It b egins w ith a brief introductio n to the use of Ho c hschild cohomology in devel- oping the theory of supp or t v ar ieties of [50] for a module o ver an artin alge bra. I then describ e the curre n t status o f resear c h con- cerning the structure of the Ho c hschild cohomology ring mo dulo nilpo tence. A cknowledgement I would like to thank the organisers of the ‘41st Symposium on Ring Theory and Represen tation Theory’ held at Shizuok a Univ ersit y for the in vitation to sp eak a t this conference. The cost of visiting Japan and attending the meeting in Shizuok a w as supp orted by the Japan So ciet y for Promotion o f Science (JSPS) Grant-in Aid for Scien tific Rese arc h (B) 183 40011. I would particularly lik e to thank the head of the gran t, Kiy oichi Oshiro (Y amaguchi Univ ersit y), for providing this financial supp ort a nd in vitation. I would also like to thank Shigeo Koshitani and Katsunori Sanada for their hospitalit y during m y vis it to Ja pan. I also thank the Univ ersit y of Leicester for granting me study lea ve. Intr oduction This surv ey article is based on talks giv en at the ‘41st Symp osium on Ring Theory and Represen tat ion Theory’, Shizuok a Univ ersity in Septem b er 2008, and is or ganised as follows. Section 1 giv es a brief in tro duction to the use of Ho c hsc hild cohomology in dev eloping the theory of supp ort v arieties of [50]. Section 2 considers the Ho c hsc hild cohomology ring of Ω-perio dic algebras. In [50] it had b een conjectured that the Ho chs c hild cohomolo gy ring mo dulo nilp otence of a finite- dimensional algebra is alw a ys finitely generated as an alg ebra. Section 3 describes ma n y classes of algebras where t his holds, that is, that the This pap er is in fina l form and no version of it w ill b e submitted for publication elsewhere. 1 2 SNASH ALL Ho c hsc hild cohomolo gy ring mo dulo nilpotence is finitely generated as an algebra. The final section is devoted to study ing the recen t coun terexample of Xu ([57]) to this conjecture. Throughout this pap er, let Λ be an indecomp osable finite-dimensional algebra ov er an algebraically closed field K , with Jacobson radical r . Denote by Λ e the env eloping algebra Λ op ⊗ K Λ of Λ, so that right Λ e - mo dules correspond to Λ , Λ-bimo dules. The Ho c hsc hild cohomology ring HH ∗ (Λ) of Λ is giv en by HH ∗ (Λ) = Ext ∗ Λ e (Λ , Λ) = ⊕ i ≥ 0 Ext i Λ e (Λ , Λ) with the Y oneda pro duct. W e may conside r an elemen t of Ext n Λ e (Λ , Λ) as an exact seque nce of Λ , Λ-bimo dules 0 → Λ → E n → E n − 1 → · · · → E 1 → Λ → 0 where the Y oneda pro duct is the ‘splicing together’ of exact sequences . The lo w-dimensional Ho c hsc hild cohomology groups are w ell-understoo d via the bar resolution ([41] and see [3 , 39]), and may b e describ ed as follo ws: • HH 0 (Λ) = Z (Λ) , the cen tre of Λ. • HH 1 (Λ) is the space of deriv ations modulo the inner deriv ations. A deriv ation is a K -linear map f : Λ → Λ suc h that f ( ab ) = af ( b ) + f ( a ) b for all a, b ∈ Λ. A deriv a tion f : Λ → Λ is an inner deriv ation if there is some x ∈ Λ suc h tha t f ( a ) = ax − xa for all a ∈ Λ. • HH 2 (Λ) measures the infinitesimal deformat ions of the algebra Λ; in par ticular, if HH 2 (Λ) = 0 then Λ is r igid, that is, Λ has no non- trivial deformations. Recen tly there has b een m uc h w o rk on the structure of the en- tire Ho c hschild cohomolog y ring HH ∗ (Λ) and its connections and ap- plications to the represen tation theory of Λ. One import an t prop- ert y of Ho chs child cohomolo gy in this situation is its inv ariance under deriv ed equiv a lence, pro v ed b y Ric k ard in [48, Proposition 2.5] (see also [39, Theorem 4.2] for a sp ecial case). It is also w ell-kno wn that HH ∗ (Λ) is a graded comm utativ e ring, that is, for ho mogeneous ele- men ts η ∈ HH n (Λ) and θ ∈ HH m (Λ), w e hav e η θ = ( − 1) mn θ η . Th us, when t he ch aracteristic of K is differen t from t w o, then ev ery homoge- neous elemen t of o dd degree squares to zero. Let N denote the ideal of HH ∗ (Λ) whic h is generated b y the homogeneous nilp ot en t elemen ts. Then, for c har K 6 = 2, w e hav e HH 2 k + 1 (Λ) ⊆ N for all k ≥ 0. Hence (in all ch aracteristics) the Ho chs c hild cohomology ring mo dulo nilp otence, HH ∗ (Λ) / N , is a comm utativ e K -a lgebra. Supp ort v arieties for finitely generated mo dules o v er a finite- dimensional algebra Λ w ere in tro duced using Ho c hsc hild cohomology b y Snashall and Solb erg in [50], where it w as also conjectured that the Ho c hsc hild SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 3 cohomology ring modulo nilpo tence is itse lf a finitely generated al- gebra. W e remark that the graded comm utativity of HH ∗ (Λ) im- plies that N is contained in ev ery maximal ideal of HH ∗ (Λ) and so MaxSp ec HH ∗ (Λ) = MaxSpec HH ∗ (Λ) / N . Although the recen t paper [57] prov ides a coun terexample to the conjecture of [50], neve rtheless finiteness conditions play an k ey role in the structure of these supp ort v arieties (see [15]), so it remains of particular importa nce t o determine the structure of the Ho c hsc hild cohomolo gy ring mo dulo nilp otence. 1. Suppor t v arieties One of the mot iv ations f or in tro ducing supp ort v arieties for finitely generated modules ov er a finite-dime nsional algebra came from the ric h theory of supp ort v arieties for finitely generated modules o v er group algebras of finite gro ups. F or a finite group G and finitely generated K G -mo dule M , the v ariet y of M , V G ( M ), w as defined by Carlson [10] to b e the v ariet y of the kerne l of the homo morphism − ⊗ K M : H ev ( G, K ) → Ext ∗ K G ( M , M ) . This map factors thro ugh the Ho c hsc hild cohomology ring o f K G , so that w e hav e the commutativ e diagram H ev ( G, K ) −⊗ K M / / $ $ H H H H H H H H H Ext ∗ K G ( M , M ) HH ∗ ( K G ) −⊗ K G M 9 9 s s s s s s s s s s Linc k elmann consid ered the map −⊗ K G M : HH ev ( K G ) → Ext ∗ K G ( M , M ) when studying v arieties for mo dules for non-principal blo cks ([43]). No w, for any finite-dimensional algebra Λ and finitely g enerated Λ- mo dule M , there is a ring homomorphism HH ∗ (Λ) −⊗ Λ M − → E xt ∗ Λ ( M , M ). This ring homomorphism turns out to provide a similarly fruitful theory of suppor t v arieties for finitely generated mo dules o ver an arbitrary finite-dimensional alg ebra. As usual, let mo d Λ denote the category of all finitely generated left Λ-mo dules. F or M ∈ mo d Λ, the supp ort v ariet y of M , V HH ∗ (Λ) ( M ), w as defined b y Snashall and Solb erg in [50, D efinition 3.3] by V HH ∗ (Λ) ( M ) = { m ∈ MaxSp ec HH ∗ (Λ) / N | Ann HH ∗ (Λ) Ext ∗ Λ ( M , M ) ⊆ m ′ } where m ′ is the preimage in HH ∗ (Λ) of the ideal m in HH ∗ (Λ) / N . W e recall from a b o ve that MaxSp ec HH ∗ (Λ) = MaxSp ec HH ∗ (Λ) / N . Since w e assum ed that Λ is inde comp osable, we know that HH 0 (Λ) is a lo cal ring. Th us HH ∗ (Λ) / N has a unique maximal graded ideal whic h 4 SNASH ALL w e denote b y m gr so that m gr = h rad HH 0 (Λ) , HH ≥ 1 (Λ) i / N . F rom [50, Prop osition 3.4(a)], w e ha ve m gr ∈ V HH ∗ (Λ) ( M ) for all M ∈ mo d Λ. W e sa y that the v ariet y of M is trivial if V HH ∗ (Λ) ( M ) = { m gr } . The follow ing result collects some of the prop erties of v arieties from [50]. F or ease of notation, w e write V ( M ) for V HH ∗ (Λ) ( M ). W e also denote the kernel of the pro jectiv e cov er of M ∈ mo d Λ b y Ω Λ ( M ). Recall that w e assume throughout this pap er that K is an alge- braically closed field. This a ssumption is a necessary assumption in man y of the results in this a rticle. Ho wev er, it is not needed in all of [50], a nd the intereste d reader may refer back to [50] to see precisely what assumptions are required there at eac h stage. Theorem 1.1. ( [50, Prop ositions 3.4, 3.7] ) L et M ∈ mo d Λ . (1) V ( M ) = V (Ω Λ ( M )) i f Ω Λ ( M ) 6 = (0) , (2) V ( M 1 ⊕ M 2 ) = V ( M 1 ) ∪ V ( M 2 ) , (3) If 0 → M 1 → M 2 → M 3 → 0 is an exact se quenc e , then V ( M i 1 ) ⊆ V ( M i 2 ) ∪ V ( M i 3 ) whenev er { i 1 , i 2 , i 3 } = { 1 , 2 , 3 } , (4) If Ext i Λ ( M , M ) = (0) for i ≫ 0 , or the pr oje ctive or the inje ctive dimension of M is finite, then the variety of M is trivia l . (5) If Λ is selfi nje ctive then V ( M ) = V ( τ M ) , wher e τ is the Auslander- R eiten tr anslate. Henc e al l mo dules in a c onne cte d stable c om- p onent of the A uslande r-R eiten quive r h a ve the s a me variety. F or a finitely generated mo dule M o v er a gro up algebra o f a finite group G , it is w ell-known ([10]) that the v ariet y of M is trivial if and only if M is a pro jectiv e mo dule. In con trast, it is still an open que stion as to what are the appropriate nec essary and sufficien t conditions on a mo dule fo r it to hav e trivial v ariet y in the mor e general case where Λ is an arbitra ry finite-dimensional algebra. Th ere are some partial results for a particular class of monomial algebras in [25] (see Section 3). Nev ertheless, the conv erse to Theorem 1.1(4) do es not hold in general, and a coun terexample ma y b e found in [52, Example 4.7]. Ho we v er, this question w as successfully answ ered by Erdmann, Hol- lo wa y , Snashall, Solb erg and T aillefer in [15], b y placing some (reason- able) a dditional assumptions on Λ. (Recall t hat we are already assum- ing that the field K is alg ebraically closed.) Sp ecifically , the follow ing t wo finiteness conditions w ere in tro duced. (Fg1) H is a comm utative No etherian gra ded subalgebra of HH ∗ (Λ) with H 0 = HH 0 (Λ). (Fg2) Ext ∗ Λ (Λ / r , Λ / r ) is a finitely generated H -mo dule. As remarke d in [15], these tw o conditions together imply that b oth HH ∗ (Λ) and Ext ∗ Λ (Λ / r , Λ / r ) are finitely generated K -algebras. In par- ticular, the prop erties (Fg1) and ( Fg2) hold where Λ = K G , G is SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 5 a finite group, and H = HH ev (Λ) ([21, 54]). With conditions (Fg1) and (Fg2) , w e hav e the following results from [15], where we define the v ariet y using the subalgebra H of HH ∗ (Λ), so that V H ( M ) = MaxSp ec( H / Ann H Ext ∗ Λ ( M , M )). Theorem 1.2. ( [15, Theorem 2.5] ) Supp ose that Λ a nd H satisfy (Fg1) and (Fg2) . Then Λ is Gor enstein. Mor e over the fol lowing ar e e quivale n t for M ∈ mo d Λ : (i) The varie ty of M is trivial; (ii) M has finite pr oje ctive dimensio n; (iii) M has finite inje ctive dimension. Theorem 1.3. ( [15, Theorem 4.4] ) Supp ose that Λ a nd H satisfy (Fg1) and (Fg2) . Given a ho m o gene ous ide al a in H , ther e is a mo dule M ∈ mo d Λ such that V H ( M ) = V H ( a ) . Theorem 1.4. ( [15, Theorem 2.5 and Prop ositions 5.2, 5.3] ) Supp ose that Λ and H satisfy (Fg1) and (Fg2) and that Λ is selfi nje ctive. L e t M ∈ mo d Λ b e inde c omp osable. (1) V H ( M ) is trivial ⇔ M is pr oje ctive. (2) V H ( M ) is a lin e ⇔ M is Ω -p erio dic. Our final results in this section concern the represen tation t yp e of Λ and the structure of the Auslander-Reiten quiv er; for more details see [15, 52]. First we recall that Heller sho w ed that if Λ is of finite represen tation t yp e then the complexit y of a finitely g enerated mo dule is at most 1 ([4 0]), and that Ric k ard sho we d that if Λ is of t ame repre- sen tation t yp e then the complexit y of a finitely generated module is at most 2 ([47]). Ho w ev er there are selfinjectiv e prepro jectiv e algebras of wild represen ta tion type where all indecomp osable mo dules ar e either pro jectiv e or p erio dic and so ha ve complexity at most 1. Neve rthe- less, the next result uses the Ho c hsc hild cohomolog y ring to giv e some information on the represen tation t yp e of an algebra. Theorem 1.5. ( [15, Prop o sition 6.1] ) Supp ose that Λ and H sat- isfy (Fg1) and (Fg2) and that Λ is selfinje ctive. S upp ose also that dim H ≥ 2 . The n Λ is of infinite r epr esentation typ e, and Λ has an infinite numb er of inde c omp osable p erio d i c mo dules lying in infinitely many differ ent c omp onents of the stable Au s lander-R eiten quiver. W e end this section with the statemen t of W ebb’s theorem ([55]) for gr oup alg ebras of finite g roups and a generalisation of this theorem from [15]. 6 SNASH ALL Theorem 1.6. ( [55] ) L et G b e a fin ite g r oup and supp ose that c har K divides | G | . Then the orbit gr aph of a c onne cte d c omp onent of the stable A usla n der-R eiten quiver of K G is o ne of the fol lowing: (a) a finite Dynkin diagr am ( A n , D n , E 6 , 7 , 8 ) , (b) a Euclid e an d iagr am ( ˜ A n , ˜ D n , ˜ E 6 , 7 , 8 , ˜ A 12 ) , or (c) an infinite Dynkin di a gr am o f typ e A ∞ , D ∞ or A ∞ ∞ . Theorem 1.7. ( [15, Theorem 5.6] ) Supp ose that Λ a nd H satisfy (Fg1) and (Fg2) and that Λ is selfi nje ctive. Supp ose that the Nakayama functor is of finite o r der o n any inde c omp osable m o dule in mo d Λ . Then the tr e e class of a c omp onent o f the stable Auslander-R eiten quiver of Λ is one o f the fo l lowing: (a) a finite Dynkin diagr am ( A n , D n , E 6 , 7 , 8 ) , (b) a Euclid e an d iagr am ( ˜ A n , ˜ D n , ˜ E 6 , 7 , 8 , ˜ A 12 ) , or (c) an infinite Dynkin di a gr am o f typ e A ∞ , D ∞ or A ∞ ∞ . W e remark that the h yp otheses of Theorem 1.7 are satisfied for all finite-dimensional co comm utativ e Ho pf algebras ([15, Corollary 5.7]). F or more information, the reader should also see the surv ey pap er on support v arieties for mo dules and complex es b y So lb erg [52]. In addition, the pap er b y Bergh [5] in tro duces t he concept of a twis ted supp ort v ariet y for a finitely generated module ov er an artin algebra, where the t wist is induced b y an automorphism of the algebra, and, in [6], Bergh and Solb erg study relativ e supp ort v arieties for finitely generated mo dules o v er a finite-dimensional algebra ov er a field. 2. Ω -pe riodic algebras W e now turn our a tten tion to the structure of the Ho c hsc hild coho- mology ring. One class of algebras where it is relativ ely straightforw ard to determine the structure of the Ho c hsc hild cohomology ring explicitly is the class of Ω-p erio dic algebras. W e recall that Λ is said to b e an Ω-p erio dic algebra if there ex ists some n ≥ 1 suc h that Ω n Λ e (Λ) ∼ = Λ as bimodules. Suc h an algebra Λ has a p erio dic minimal pro jectiv e bimo dule resolution, so that HH i (Λ) ∼ = HH n + i (Λ) for i ≥ 1, and is necessarily self-injectiv e (Butler; see [34 ]). There is an ex t ensiv e surv ey of p erio dic algebras b y Erdmann and Sk ow r o ´ nski in [18]. Examples of suc h algebras include the prepro jective algebras of D ynkin t yp e where Ω 6 Λ e (Λ) ∼ = Λ as bimo dules ([49]; see also [19]), and the deformed mesh algebras of generalized Dynkin t yp e of Bia lk ow ski, Erdmann and Sk o wro ´ nski [7, 18]. F or the selfinjectiv e algebras of finite represen tation type ov er an algebraically closed field, it is kno wn from [34] that there is s o me n ≥ 1 and auto morphism σ o f Λ SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 7 suc h that Ω n Λ e (Λ) is isomorphic as a bimodule to the t wisted bimo dule 1 Λ σ . It ha s no w b een sho wn that all selfinjectiv e algebras of finite represen tation type ov er an algebraically closed field are Ω-p erio dic ([13, 16, 1 7, 18]). The struc ture of the Ho chs child cohomology ring modulo nilp otence of these algebras w as determined b y G reen, Snashall a nd Solb erg in [34]. Theorem 2.1. ( [34, Theorem 1.6] ) L et K b e an algebr aic a l ly close d field. L et Λ b e a finite-dimensional inde c omp osable K -alg ebr a such that ther e is som e n ≥ 1 and some automorphism σ of Λ such that Ω n Λ e (Λ) is isomorphic to the twiste d bimo dule 1 Λ σ . Then HH ∗ (Λ) / N ∼ =  K [ x ] or K. If ther e is some m ≥ 1 such that Ω m Λ e (Λ) ∼ = Λ as bimo dules, then HH ∗ (Λ) / N ∼ = K [ x ] , wher e x is in de gr e e m and m is minima l . Additional informatio n on the ring structure of the Ho c hsc hild coho- mology ring of the prepro jectiv e algebras of Dynkin type A n w as deter- mined in [19 ], of the prepro jectiv e algebras of Dynkin types D n , E 6 , E 7 , E 8 in [20], and of the se lfinjective alg ebras of finite represe ntation type A n o ve r an algebraically closed field in [16, 17 ]. Giv en that the Ho c hsc hild cohomolog y ring of these a lgebras is un- dersto o d, this naturally leads to the study of situations where the Ho c hsc hild cohomology rings of tw o algebras A and B can b e related. This enables us to transfer information ab out the Ho c hsc hild cohomol- ogy ring of, say , an Ω-p erio dic algebra, to other algebras. Apart from p erio dic algebras, there are other algebras where the Ho chs c hild coho- mology ring is kn o wn, and these pro vide additional example s where the transfer of properties betw een Ho c hsc hild cohomology rings ma y also b e studie d. One suc h class of examples is the class of truncated quiv er algebras, whic h has b een extensiv ely studied in t he literature by many authors. Happ el s how ed in [39, The orem 5.3] that if B is a one-p oin t exte nsion of a finite-dime nsional K -algebra A b y a finitely g enerated A -mo dule M , then t here is a long exact sequence connecting the Ho c hsc hild co- homology rings o f A and B : 0 → HH 0 ( B ) → HH 0 ( A ) → Hom A ( M , M ) /K → HH 1 ( B ) → HH 1 ( A ) → Ext 1 A ( M , M ) → · · · · · · → Ext i A ( M , M ) → HH i +1 ( B ) → HH i +1 ( A ) → Ext i +1 A ( M , M ) → · · · 8 SNASH ALL It w a s subsequen tly sho wn b y Green, Marcos a nd Snashall in [29, The - orem 5.1] that there is a gra ded ring homomorphism HH ∗ ( B ) → HH ∗ ( A ) ⊕ K whic h induce s this long exact sequ ence, where K is the g raded K - mo dule with K 0 = K and K n = 0 fo r all n 6 = 0 . These results w ere g eneralized indep enden tly to arbitrary triangu- lar matrix algebras by Cibils [12], by Green and Solb erg [36 ], and by Mic helena and Platzec k [44]. A recen t result of K¨ onig and Nag ase, ([42, 45]), has related the Ho c hsc hild cohomology ring of B to that of B /B eB in the case where B is a n algebra with idemp otent e , suc h that B eB is a stratifying ideal of B . Theorem 2.2. ( [4 2] ) L et B b e an algebr a with id e mp otent e such that B eB is a str atifying id e al of B and let A b e the factor algebr a B / B eB . Then ther e ar e long exac t se quenc es as fol lows: (1) · · · → Ext n B e ( B , B eB ) → HH n ( B ) → HH n ( A ) → · · · ; (2) · · · → Ext n B e ( A, B ) → HH n ( B ) → HH n ( eB e ) → · · · ; and (3) · · · → Ext n B e ( A, B eB ) → HH n ( B ) → HH n ( A ) ⊕ HH n ( eB e ) → · · · . 3. The Hochschild cohomology ring modulo nilpotence The definition of a supp ort v ariet y in [50] led us to consider the structure of HH ∗ (Λ) / N and to conjecture that HH ∗ (Λ) / N is alwa ys finitely generated as a n alg ebra. A counterex ample to this conjecture w as recen tly giv en by Xu in [5 7]; neve rtheless the Ho c hsc hild cohomol- ogy ring mo dulo nilp otence is finitely generated as an algebra fo r man y div erse classes of algebras. The Ho chs c hild cohomology ring mo dulo nilp otence is know n to b e finitely generated as an a lgebra in the follow ing cases. • any blo c k of a gr oup ring o f a finite group ([21, 54]); • any blo c k of a finite-dimensional cocommutativ e Hopf algebra ([24]); • finite-dimensional selfinjectiv e algebras of finite represen tation t yp e ov er an a lgebraically closed field ([34]); • finite-dimensional monomial a lgebras ([35] a nd see [32]); • finite-dimensional alg ebras of finite global dimension (see [39]). F or the last class of examples, if Λ is an algebra of finite g lobal dimension N , then HH i (Λ) = Ext i Λ e (Λ , Λ) = ( 0) for all i > N . Hence HH ∗ (Λ) / N ∼ = K . In [39], Happel ask ed whether or not it was true, SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 9 for a finite-dimensional algebra Γ ov er a field K , that if HH n (Γ) = (0) for n ≫ 0 then the global dimension o f Γ is finite. This question ha s no w b een a nsw ered in the negative b y Buc h w eitz, Green, Madsen a nd Solb erg in [8] b y the follo wing example. Example 3.1. ([8]) Let Λ q = K h x, y i / ( x 2 , xy + q y x, y 2 ) with q ∈ K \ { 0 } . If q is not a ro ot of unity then dim HH i (Λ q ) = 0 for i ≥ 3. Moreo ve r, Λ q is a selfinjectiv e algebra so has infinite global dimension. W e also no te fr om [8] that dim Λ q = 4, dim HH ∗ (Λ q ) = 5 and HH ∗ (Λ) / N ∼ = K . Ho we v er, t he situation for commu tativ e algebras is v ery differen t, as Avramo v and Iy engar hav e shown . Theorem 3.2. ( [1] ) L et R b e a c ommutative finite-dimensional K - algebr a over a fiel d K . I f HH n ( R ) = (0) for n ≫ 0 then R is a (finite) pr o duct of (finite) sep ar able field extension s of K . In p articular, the glob al dimension of R is finite. W e no w turn to a brief discussion of the Ho c hsc hild cohomology ring mo dulo nilp otence for a monomial algebra, w hich w as studied b y Green, Snashall and Solb erg. L et Λ b e a quotien t of a path algebra so that Λ = K Q /I for some quiv er Q and admissible ideal I of K Q . Then Λ = K Q /I is a m onomial algebra if the ide a l I is generated by monomials of length at least tw o. It should be noted that monomial algebras are v ery rarely se lfinjective and so do not usually exhibit the same prop erties as gr oup algebras. Ho we v er, the Ho c hsc hild cohomology ring mo dulo nilp otence of a monomial algebra turns out to ha v e a part icularly nice structure. Theorem 3.3. ( [35, Theorem 7.1] ) L et Λ = K Q /I b e a finite-dimensional inde c omp osable monomi a l algebr a. Th e n HH ∗ (Λ) / N is a c ommutative finitely gener ate d K -algebr a of Krul l di m ension at most on e. F or some sp ecific sub classes of monomial algebras, the structure of the Ho c hsc hild cohomology ring modulo nilpotence w as explicitly de- termined in [32]. One o f the main to ols used w as the minimal pro jectiv e bimo dule resolution of a monomial algebra o f Bardzell [2]. In order to define t he particular class of ( D , A )-stack ed mono mial algebras, w e re- quire the concept of o v erlaps of [27, 38]; the definitions here use the notation of [32]. 10 SNASH ALL Definition 3.4. A path q in K Q o v erlaps a path p in K Q with o ve r lap pu if there are paths u and v suc h that pu = v q and 1 ≤ ℓ ( u ) < ℓ ( q ), where ℓ ( x ) denotes the length of the path x ∈ K Q . W e illustrate the definition with the follo wing diagram. (Note that we allo w ℓ ( v ) = 0 here.)  p  o o v / /  q  o o u / / A path q properly ov erlaps a path p with ov erlap pu if q ov erlaps p and ℓ ( v ) ≥ 1. Let Λ = K Q /I b e a finite-dimensional monomial algebra where I has a minimal set o f generators ρ of paths of length at least 2. W e fix this set ρ and now recursiv ely define sets R n (con tained in K Q ). Let R 0 = the set of vertice s of Q , R 1 = the set of ar ro ws of Q , R 2 = ρ . F or n ≥ 3 , w e sa y R 2 ∈ R 2 maximally ov erlaps R n − 1 ∈ R n − 1 with o ve rlap R n = R n − 1 u if (1) R n − 1 = R n − 2 p for some path p ; (2) R 2 o ve rlaps p with o ve rlap pu ; (3) there is no elemen t of R 2 whic h ov erlaps p with o ve rlap b eing a prop er prefix of pu . The set R n is defined to be the set of all o ve r laps R n formed in this w ay . Eac h of the elemen ts in R n is a path in the quiv er Q . W e follow the con v en t ion that paths are written from left to righ t. F or an arrow α in the quiv er Q , w e write o ( α ) for the idemp oten t corresp onding to the origin of α and t ( α ) for the idemp oten t corresponding to the tail of α so that α = o ( α ) α t ( α ). F or a path p = α 1 α 2 · · · α m , w e write o ( p ) = o ( α 1 ) and t ( p ) = t ( α m ). The imp ortance of these sets R n lies in the fact that, f or a finite- dimensional monomial algebra Λ = K Q /I , [2] uses them to giv e an ex- plicit construction of a minimal pro j ectiv e bimo dule resolution ( P ∗ , δ ∗ ) of Λ, show ing that P n = ⊕ R n ∈R n Λ o ( R n ) ⊗ K t ( R n )Λ . W e no w use these same sets R n to define a ( D , A )-stac ked monomial algebra. SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 11 Definition 3.5. ([32, D efinition 3.1]) Let Λ = K Q /I b e a finite- dimensional monomial algebra, where I is an admissible ideal with minimal set o f generators ρ . Then Λ is said to b e a ( D , A )-stack ed monomial algebra if there is some D ≥ 2 and A ≥ 1 suc h that, for all n ≥ 2 and R n ∈ R n , ℓ ( R n ) =    n 2 D if n eve n, ( n − 1) 2 D + A if n o dd. In particular a ll relations in ρ are of length D . The class of ( D , A )-stac ked mono mial algebras includes the Koszul monomial algebras (equiv alen tly , the quadratic monomial algebras) and the D -Koszul monomial a lgebras of Berger ([4]). Recall that the Ex t algebra E ( Λ) of Λ is defined by E (Λ) = Ext ∗ Λ (Λ / r , Λ / r ) . It is we ll- kno wn that the Ext alg ebra of a Koszul algebra is generated in degrees 0 and 1; moreov er the Ext algebra of a D -Koszul algebra is generated in degrees 0, 1 and 2 ([2 8]). It w as show n b y Green and Snashall in [33, Theorem 3.6] that, for algebras of infinite global dimension, the class of ( D , A )-stac ke d mono mial alg ebras is precise ly the class of monomial algebras Λ where eac h pro jectiv e mo dule in the minimal pro jectiv e resolution of Λ / r as a righ t Λ-mo dule is generated in a single degree and whe re the Ext algebra of Λ is finitely generated as a K -algebra. It w as also sho wn, for a ( D, A )- stac k ed monomial algebra of infinite global dimension, that the Ext algebra is generated in degrees 0, 1, 2 and 3. Theorem 3.6. [32] L et Λ = K Q /I b e a finite-di m ensional ( D , A ) - stacke d monomial algebr a, wher e I is an admis s i b le ide al with minimal set o f gener ators ρ . Supp ose char K 6 = 2 and gldimΛ ≥ 4 . Then ther e is some inte ger r ≥ 0 such that HH ∗ (Λ) / N ∼ = K [ x 1 , . . . , x r ] / h x i x j for i 6 = j i . Mor e over the de gr e es of the x i and the value of the p ar ameter r may b e explicitly and e asily c alculate d. W e do not give the full details o f t he x i and t he parameter r here; they ma y b e fo und in [32]. Ho we ver, it is w orth remarking that, giv en an y integer r ≥ 0 and eve n in tegers n 1 , . . . , n r , t here is a finite- dimensional ( D , A )-stac ke d monomial algebra Λ with HH ∗ (Λ) / N ∼ = K [ x 1 , . . . , x r ] / h x i x j for i 6 = j i where the degree of x i is n i , for all i = 1 , . . . , r . 12 SNASH ALL In [25], nec essary and sufficien t conditions are giv en for a simple mo dule ov er a ( D , A )-stack ed monomial algebra to hav e trivial v a riet y . Referring bac k to Theorem 1.1(4), this g o es part wa y to determining necessary and sufficien t conditions on an y finitely generated mo dule for it to hav e trivial v ariet y for this class of algebras. W e end this section with a class of selfinjectiv e special biserial alge- bras Λ N , for N ≥ 1, studied b y Snashall and T aillefer in [51]. The study of these alg ebras w as motiv ated b y the results of [14] where the alge- bras Λ 1 arose in the presen tation by quiv er and relations o f the Drinfeld double D (Λ n,d ) of the Hopf a lgebra Λ n,d where d | n . The a lgebra Λ n,d is giv en b y an orien ted cycle with n v ertices suc h tha t a ll paths of length d are zero. These algebras also o ccur in the study of the represen tatio n theory of U q ( sl 2 ); see w o rk of P atra ([46]), Suter ([53 ]), Xiao ([56]), and also of Chin and Krop ([11]). The more general alg ebras Λ N o ccur in w ork of F arnsteine r and Sko wro ´ nski [2 2, 2 3], where they determine the Hopf algebras asso ciated to infinitesim al groups whose principal blo c ks are tame when K is an a lgebraically closed field with c har K ≥ 3. Our class of selfinjectiv e sp ecial biserial algebras Λ N is describ ed as follo ws. Firstly , for m ≥ 1, let Q b e the quiv er with m v ertices, lab elled 0 , 1 . . . , m − 1, a nd 2 m arrows as follows : · a " " ¯ a m m · a - - ¯ a r r · a   ¯ a b b · a 2 2 · ¯ a W W · Let a i denote the arr o w that go es fro m verte x i to v ertex i + 1, and let ¯ a i denote t he arro w t hat go es from v ertex i + 1 to v ertex i , for eac h i = 0 , . . . , m − 1 (with the ob vious con v entions mo dulo m ). Then, for N ≥ 1, we define Λ N to b e the algebra give n by Λ N = K Q /I N where I N is the ideal of K Q generated by a i a i +1 , ¯ a i − 1 ¯ a i − 2 , ( a i ¯ a i ) N − (¯ a i − 1 a i − 1 ) N , for i = 0 , 1 , . . . , m − 1 , SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 13 and where t he subscripts are ta k en modulo m . W e note tha t, if N = 1, then the algebra Λ 1 is a Koszul algebra. (W e con tinue to write pa ths from left to righ t.) Theorem 3.7. ( [51, Theorem 8.1] ) F or m ≥ 1 and N ≥ 1 , let Λ N b e as define d ab ove. Then HH ∗ (Λ N ) is a fini tely gener ate d K -algebr a. Mor e over HH ∗ (Λ N ) / N is a c om mutative finitely g ener ate d K - a l g ebr a of Krul l dime nsion two. F urthermore, if N = 1 then [51] also sho wed that the conditions (Fg1) and (Fg2) hold with H = HH ev (Λ 1 ). 4. Cou nterexample to the conjecture of [50] The previous se ctio n concerned algebras where the conjecture of [50] concerning the finite generation of the Hochsc hild cohomology ring mo dulo nilp otence has b een sho wn to hold. In this section w e presen t a counterex a mple to the conjecture of [50]. In [57], Xu ga ve a coun- terexample in the case where the field K has c haracteristic 2. It can easily b e seen, fo r ch ar K = 2, tha t the category algebra he presen ted in [57] is isomorphic to the f ollo wing algebra A given as a quotien t of a path algebra. Moreo v er, we will sho w that this a lgebra A pro vides a counte rexample to the conjecture irresp ectiv e o f the c ha racteristic of the field. Example 4.1. Let K b e an y field and let A = K Q /I where Q is the quiv er 1 a   b F F c / / 2 and I = h a 2 , b 2 , ab − ba, ac i . The rest o f t his se ctio n is dev o ted to study ing this algebra A and to sho wing that HH ∗ ( A ) / N is not finitely g enerated as an algebra. W e b egin b y giving an explicit minimal pro jectiv e resolution ( P ∗ , d ∗ ) for A as an A , A -bimo dule. The des cription of the re solution g iv en here is motiv ated by [31] where the first terms o f a minimal pro jectiv e bi- mo dule resolution of a finite-dimensional quotien t of a path algebra w ere determined explicitly from the minimal pro jectiv e resolution of Λ / r as a rig h t Λ-mo dule of Green, Solb erg a nd Z ac haria in [37]. This same tec hnique for constructing a minimal pro jective bimo dule resolu- tion w as used in [26] for an y Koszul algebra, and in [5 1] for the algebras Λ N whic h w ere discussed at the end of Section 3. 14 SNASH ALL F rom Happ el [3 9], w e kno w that the multiplic it y of Λ e i ⊗ K e j Λ as a direct summand of P n is equal to dim Ext n Λ ( S i , S j ), where S i is the simple A -mo dule corres p onding to the v ertex i of Q . F ollo wing [31, 37], w e start b y defining sets g n in K Q inductiv ely , and then lab elling the summands of P n b y the elemen ts of g n . The se t g 0 is determined b y the v ertices of Q , the set g 1 b y the arrows of Q , and the set g 2 b y a minimal generating set of the ideal I . Let g 0 = { g 0 0 = e 1 , g 0 1 = e 2 } , g 1 = { g 1 0 = a, g 1 1 = − b, g 1 2 = c } , g 2 = { g 2 0 = a 2 , g 2 1 = ab − ba, g 2 2 = − b 2 , g 2 3 = ac } . F or n ≥ 3 and r = 0 , 1 , . . . , n , let g n r = X p ( − 1) s p where the sum is o ve r all paths p of length n , written p = α 1 α 2 · · · α n where the α i are arrows in Q , such that (i) p con tains n − r arr o ws equal to a and r arrows equal to b , a nd (ii) s = P α j = b j . In addition, fo r n ≥ 3, define g n n +1 = a n − 1 c. F or r = 0 , 1 , . . . , n , w e hav e that g n r = e 1 g n r e 1 so w e define o ( g n r ) = e 1 = t ( g n r ). Moreo v er o ( g n n +1 ) = e 1 and t ( g n n +1 ) = e 2 . Th us P n = ⊕ n +1 r =0 A o ( g n r ) ⊗ K t ( g n r ) A . T o desc r ib e the map d n : P n → P n − 1 , w e first need to write eac h of the elemen ts g n r in terms of t he elemen t s of the set g n − 1 , that is, in terms of g n − 1 0 , . . . , g n − 1 n . The f ollo wing result is straigh tforward to v erify . Prop osition 4.2. Supp ose n ≥ 2 . Then, ke eping the ab ove notation, g n 0 = g n − 1 0 a = ag n − 1 0 g n r = g n − 1 r a + ( − 1) n g n − 1 r − 1 b = ( − 1) r ( ag n − 1 r + bg n − 1 r − 1 ) for 1 ≤ r ≤ n − 1 g n n = ( − 1) n g n − 1 n − 1 b = ( − 1) n bg n − 1 n − 1 g n n +1 = g n − 1 0 c = ag n − 1 n . W e define the map d 0 : P 0 → A to b e the multiplic a tion map. T o define d n for n ≥ 1, w e need one f urther piece of notation. In de- scribing the image d n ( o ( g n r ) ⊗ t ( g n r )) in the pro jective mo dule P n − 1 , we SUPPOR T V ARIETIES AN D THE H OCHSCHILD COHO MOLOGY RING . . . 15 use a subscript under ⊗ to indicate the a ppropriate summ and of the pro jectiv e mo dule P n − 1 . Sp ecifically , let − ⊗ r − denote a term in the summand of P n − 1 corresp onding to g n − 1 r . Nonetheless all tensors are o ve r K ; ho w ev er, to simplify notation, we omit the subscript K . The maps d n : P n → P n − 1 for n ≥ 1 may no w b e defined. The pro of of the follo wing r esult is omitted and is similar to those in [31, Prop osition 2.8] and [51, Theorem 1.6]. Theorem 4.3. F or the algebr a A of Exa mple 4.1, the se quenc e ( P ∗ , d ∗ ) is a min i m al pr oje ctive r esolution of A as an A , A -bimo dule, wher e, for n ≥ 0 , P n = n +1 M r =0 A o ( g n r ) ⊗ t ( g n r ) A , the map d 0 : P 0 → A is the multiplic ation map, the map d 1 : P 1 → P 0 is given by d 1 ( o ( g 1 r ) ⊗ t ( g 1 r )) =    o ( g 1 0 ) ⊗ 0 a − a ⊗ 0 t ( g 1 0 ) for r = 0; − o ( g 1 1 ) ⊗ 0 b + b ⊗ 0 t ( g 1 1 ) for r = 1; o ( g 1 2 ) ⊗ 0 c − c ⊗ 1 t ( g 1 2 ) for r = 2; and, for n ≥ 2 , the map d n : P n → P n − 1 is given by d n ( o ( g n r ) ⊗ t ( g n r )) =            o ( g n 0 ) ⊗ 0 a + ( − 1) n a ⊗ 0 t ( g n 0 ) for r = 0; o ( g n r ) ⊗ r a + ( − 1) n o ( g n r ) ⊗ r − 1 b +( − 1) r + n ( a ⊗ r t ( g n r ) + b ⊗ r − 1 t ( g n r )) for 1 ≤ r ≤ n − 1; ( − 1) n o ( g n n ) ⊗ n − 1 b + b ⊗ n − 1 t ( g n n ) for r = n ; o ( g n n +1 ) ⊗ 0 c + ( − 1) n a ⊗ n t ( g n n +1 ) for r = n + 1 . Prop osition 4.4. The algebr a A of Exam ple 4.1 is a Koszul algebr a. Pr o of. W e apply the functor A / r ⊗ A − to the resolution ( P ∗ , d ∗ ) of Theorem 4.3 to g iv e a (minimal) pro j ectiv e resolution of A / r as a rig h t A -mo dule. Thus A / r has a linear pro jectiv e resolution, and so A is a Koszul algebra.  Since, A is a Koszul algebra, it no w follows from [9] that the image of the ring homomorphism φ A / r = A / r ⊗ A − : HH ∗ ( A ) → E ( A ) is the graded cen tre Z gr ( E ( A )) o f E ( A ), where Z gr ( E ( A )) is the subalgebra generated b y all homogeneous eleme n ts z suc h that z g = ( − 1) | g || z | g z for all g ∈ E ( A ). Th us φ A / r induces an isomorphis m HH ∗ ( A ) / N ∼ = Z gr ( E ( A )) / N Z , where N Z denotes the ideal of Z gr ( E ( A )) generated by all homogeneous nilp oten t elemen ts. F rom [30, Theorem 2.2], E ( A ) is t he Koszul dual of A and is giv en explicitly by quiv er and relatio ns as E ( A ) ∼ = K Q op /I ⊥ , where Q is the 16 SNASH ALL quiv er of A a nd I ⊥ is the ideal generated b y the ort hogonal relations to those of I . Sp ecifically , for this example, E ( A ) has quiv er 1 a o   b o F F 2 c o o o and I ⊥ = h a o b o + b o a o , b o c o i , where, for an arrow α ∈ Q , w e denote by α o the corresp onding arrow in Q op . Moreo ver, t he left modules ov er E ( A ) are the right mo dules o ve r K Q / h ab + ba, bc i . It is now easy to calculate Z gr ( E ( A )) to giv e the follo wing t heorem. The structure o f HH ∗ ( A ) / N for ch ar K = 2 w a s giv en by Xu in [57 ]. Theorem 4.5. L et A b e the a lgebr a of Example 4.1. (1) Z gr ( E ( A )) ∼ =  K ⊕ K [ a, b ] b if c har K = 2 K ⊕ K [ a 2 , b 2 ] b 2 if c har K 6 = 2 , wher e b is in de gr e e 1 and ab is in de gr e e 2. 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