Relative support varieties

RELA TIVE SUPPOR T V ARIETIES PETTER ANDREAS BER GH & ØYVIND SOLBERG Abstract. W e define relative support v arieties with resp ect to some fixed module o v er a finite dimensional algebra. These v arieties share man y of the standard prop erties of classical supp ort v ari eties. Moreov er, when introduc- ing finite generation conditions on cohomology , we sho w that relativ e supp ort v arieties contain homological information on the mo dules i n v olve d. A s an ap- plication, we pro vide a new criterion for a selfinjective algebra to be of wil d represen tation t yp e. 1. Introduction Suppo rt v arieties for mo dules over a given alg ebra ar e defined in terms of the maximal ideal spectrum o f some commutativ e gra de d ring of cohomo logy op erator s, op erators which act cent rally on the c ohomology groups of the alg ebra. F or gro up algebras o f finite gr o ups, or, more generally , for finite dimensio nal co c ommu tative Hopf algebra s, this role is play ed by the cohomology ring of the alg ebra (cf. [Ben], [Car], [E ve ], [F rS]). F or commutativ e lo c al co mplete intersections, one uses the po lynomial ring of Eisenbud op erato rs (cf. [Avr], [AvB]). In a ll these c a ses, the ring of cohomolog y o per ators is No etheria n, and all the cohomology groups of the algebra are finitely g e nerated as mo dules. Consequently , the theory of suppo rt v arieties ov er these rings is v ery pow erful, in that the v a riety of a module cont ains a lot of homologica l infor mation on the mo dule itself. As shown in [SnS], for a finite dimensiona l algebra , the Ho chsc hild cohomolo gy ring, with its maxima l ideal sp ectrum, is a natural ca ndidate as a ring of c e n tral cohomolgy op erators . How ever, in gener a l this ring is not No etheria n, and the cohomolog y g roups of the algebra are not alwa ys finitely g enerated mo dules. But, as sho wn in [EHSST], when the Ho chsc hild co homology ring is No etherian and all the c o homology groups are finitely genrated, then o ne obtains a supp ort v ariety theory very muc h like in the c lassical cases. It is therefo re impo r tant to establish which finite dimens ional algebra s hav e “nice” Ho chschild coho mology r ings. F or quantum complete intersections, this has b een solved (cf. [E rS], [BeO]). In this pap e r, we define r elative supp ort v a rieties with resp ect to a fix e d mo dule. These are defined in terms of the maxima l ide a l sp ectrum of some commutativ e graded s ubalgebra of the E xt-algebra of the mo dule. As one would exp ect, these v arieties share many of the sa me pro p er ties of “or dinary” supp ort v arieties, s uch as the standa rd behavior on exact s equences etc. Moreov er, when we in tr o duce finite g eneration conditions, then the relative supp ort v arieties contain homolo gical information on the mo dules inv o lved, just as in the classic al ca se. As an applicatio n, we provide a new criterio n for a finite dimensional selfinjective algebra to be of wild r epresentation type. Namely , we show that if there exis ts a mo dule whose Ext-algebr a is “lar g e” enoug h, Noetheria n and finitely generated as a mo dule ov er its center, then the algebra is wild. This genera liz e s F ar nsteiner’s 2000 Mathematics Subje ct Classific ation. 16E05, 16E30, 16G60. Key wor ds and phr ases. Relativ e supp ort v arieties, complexit y , wi ld algebras. The fir s t author was supp orted by NFR Storforsk gran t no. 167130. 1 2 PETTER ANDREAS BERGH & ØYVIND S OLBER G theorem, which sta tes that the complexity of every mo dule of a tame blo ck of a finite group scheme is a t most t wo (cf. [F a r]). 2. Rela tive suppor t v arieties Throughout this pap er , we let k b e a field and Λ a finite dimensional k -algebra . W e denote by mo d Λ the categor y of finitely genera ted left Λ-mo dules, and we fix a mo dule M ∈ mod Λ whose higher self-extensions do not all v anish. Whenever we deal with Λ- mo dules, we assume they b elong to mod Λ. Finally , for tw o Λ-mo dules X and Y , we denote by Ext ∗ Λ ( X, Y ) the direct sum ⊕ ∞ i =0 Ext i Λ ( X, Y ). Consider the Ext-a lgebra Ext ∗ Λ ( M , M ) of M , in which m ultiplication is given by the Y o neda pro duct. Then for a ny Λ-mo dule N , the graded k -vector space Ext ∗ Λ ( M , N ) is a graded rig h t Ext ∗ Λ ( M , M )-mo dule, wherea s Ext ∗ Λ ( N , M ) is a graded left Ext ∗ Λ ( M , M )-mo dule. Mor eov er , a Λ-homomor phism N 1 f − → N 2 induces homomorphisms Ext ∗ Λ ( M , N 1 ) f ∗ − → Ext ∗ Λ ( M , N 2 ) Ext ∗ Λ ( N 2 , M ) f ∗ − → Ext ∗ Λ ( N 1 , M ) of right and left Ex t ∗ Λ ( M , M )-mo dules. The homo morphism f ∗ is g iven as follows: given a homog eneous elemen t η : 0 → N 1 → X n → · · · → X 1 → M → 0 in Ext ∗ Λ ( M , N 1 ), the element f ∗ ( η ) is the low er exact sequence in the diagra m 0 / / N 1 f   / / X n   / / · · · / / X 1   / / M   / / 0 0 / / N 2 / / K / / · · · / / X 1 / / M / / 0 in whic h the mo dule K is a pushout. Similarly , the homomo rphism f ∗ is induced by pullback along f . It follows immediately that f ∗ and f ∗ are w ell defined homo- morphisms of right and left Ext ∗ Λ ( M , M )-mo dules, resp ectively . The rela tive supp or t v arieties are defined with re spe c t to so me commutativ e graded subalgebra of Ext ∗ Λ ( M , M ), and therefore we now fix such a subalgebr a. Assumption. Fix a commutativ e gra ded subalgebr a H ⊆ Ext ∗ Λ ( M , M ) such that H 0 is a lo cal ring. As men tio ned, the relative supp or t v arieties to be defined are defined with respect to this gra ded subalgebra H . The assumption that H 0 is lo cal is ma de in order to get a nice characterizatio n of the trivial v arieties. This assumption is not very restrictive. F o r example, when we introduce finiteness as s umptions later , then we may a ctually take H to be a po lynomial ring ov er k , so that H 0 is just k itself. Moreov er, the following result shows that w he n M is a n indecomp osable mo dule, then H 0 is automatically a lo cal ring. Lemma 2.1. If M is inde c omp osable, then H 0 is a lo c al ring. Pr o of. Since H 0 is a finite dimensional commutativ e k -algebra , the facto r algebra H 0 / r ad H 0 is a pro duct K 1 × · · · × K t of fields. If t ≥ 2, then this factor algebr a contains non trivial idemp otents, and these lift to H 0 . But H 0 , b eing a subalgebr a of End Λ ( M ), cannot co n tain a ny nontrivial idemp otent, hence t = 1. Therefore the radical of H 0 is a maximal ideal.  RELA TIVE SUPPOR T V ARIETIES 3 Since we have assumed that H 0 is a lo cal ring, the graded ideal rad H 0 ⊕ H 1 ⊕ · · · is maximal in H (and it is the only maximal gr aded ideal). W e denote this ideal by m gr ( H ). W e now define a r elative supp or t v arie t y theory for Λ-mo dules, in which the commutativ e g raded ring H is the co or dinate ring. Given a Λ- mo dule N , denote by Ann i H N the annihilator of Ext ∗ Λ ( M , N ) in H , and by Ann p H N the annihilator of Ext ∗ Λ ( N , M ). As the annihilator o f any gra ded module ov er any gr aded ring is graded, the ideals Ann i H N and Ann p H N are graded ideals of H . W e define the injectiv e and pro jective su pp ort varieties of N with r esp e ct to H as V i H ( N ) def = { m ∈ Ma xSp e c H | Ann i H N ⊆ m } , V p H ( N ) def = { m ∈ Ma xSp e c H | Ann p H N ⊆ m } , resp ectively , where Max Spec H denotes the set of max imal ideals of H . Note that Ann i H N and Ann p H N are contained in m gr ( H ), hence m gr ( H ) is tr ivially a po in t in both V i H ( N ) and V p H ( N ). W e call a v ariety trivial if it o nly co nt ains this p oint. In the following r esult we r e cord some elementary facts on r elative v arieties. Whenever w e write V ∗ H ( N ) or Ann ∗ H N and make a statement , it is to be understo o d that the statement holds in both the injective and pro jective cases. F ur thermore, denote by M ⊥ the categ ory of all Λ-mo dules X such that E xt n Λ ( M , X ) = 0 for n ≫ 0, and by ⊥ M the categ o ry of all Λ-mo dules Y such that E x t n Λ ( Y , M ) = 0 for n ≫ 0. Prop ositio n 2.2. F or Λ -mo dules M , N , N 1 , N 2 , N 3 , t he fol lowing hold: (i) V ∗ H ( M ) = Ma xSp e c H . (ii) If N ∈ M ⊥ , then V i H ( N ) is trivial. In p articular, this hold s if the inje ctive dimension of N is finite. (iii) If N ∈ ⊥ M , then V p H ( N ) is trivial. In p articular, this holds if the pr oje ctive dimension of N is finite. (iv) F or any exact se quenc e 0 → N 1 → N 2 → N 3 → 0 , the inclusion V ∗ H ( N u ) ⊆ V ∗ H ( N v ) ∪ V ∗ H ( N w ) holds whenever { u, v , w } = { 1 , 2 , 3 } . (v) If N = N 1 ⊕ N 2 , t hen V ∗ H ( N ) = V ∗ H ( N 1 ) ∪ V ∗ H ( N 2 ) . Pr o of. Since H is a subalgebr a o f Ext ∗ Λ ( M , M ), no nonzer o e le ment of H can an- nihilate Ext ∗ Λ ( M , M ). Therefo r e Ann ∗ H M = 0, and this shows (i). T o pr ov e (ii), no te that if Ext n Λ ( M , N ) = 0 for n ≫ 0 and η is a homogeneous element in H o f p o sitive degree, then some p ower of η b elongs to Ann i H N . More- ov er, if θ is any element o f rad H 0 , then it is nilp otent, and therefore s ome power of θ also b elong s to Ann i H N . Co nsequently V i H ( N ) = { m gr ( H ) } . This pr ov es (ii), and the pro of of (iii) is similar . As for (iv), we pr ove only the inclusion V i H ( N 3 ) ⊆ V i H ( N 1 ) ∪ V i H ( N 2 ); the other inclusions are pr ov ed a nalogously . The given s hort ex act seq ue nc e induces an exact sequence Ext ∗ Λ ( M , N 2 ) → Ext ∗ Λ ( M , N 3 ) → Ext ∗ +1 Λ ( M , N 1 ) of right Ext ∗ Λ ( M , M )-mo dules, fr o m which we o btain Ann i H N 2 · Ann i H N 1 ⊆ Ann i H N 3 . The inclus ion V i H ( N 3 ) ⊆ V i H ( N 1 ) ∪ V i H ( N 2 ) now follows. The pro of of (v) is straightforward.  By com bining prop erties (ii), (iii) and (iv) in Prop osition 2.2, w e see that injectiv e v arieties a re in v ariant under cosyzyg ies, whereas pr o jective v ar ieties are inv ariant 4 PETTER ANDREAS BERGH & ØYVIND S OLBER G under syzygies. W e reco rd these facts in the following s lig ht ly more ge neral result, which concludes this sectio n. Corollary 2.3. L et N b e a Λ -mo dule, and let 0 → N 1 → N 2 → N 3 → 0 b e an ex act se qu en c e in mo d Λ . (i) If N 2 ∈ M ⊥ , then V i M ( N 1 ) = V i M ( N 3 ) . In p articular, the inje ctive variety of M e quals that of Ω − 1 Λ ( M ) . (ii) If N 2 ∈ ⊥ M , then V p M ( N 1 ) = V p M ( N 3 ) . In p articular, the pr oje ctive variety of M e quals that of Ω 1 Λ ( M ) . 3. Finite genera tion The reason why the theorie s of support v ar ieties for g roup rings, co co mmutative Hopf algebras a nd complete in ter sections are a ll very p owerful, is the existence of a central comm utative No ether ian ring over which all the cohomolog y g r oups are finitely generated (cf. [Avr], [AvB], [Ben], [Ca r], [Eve], [F rS]). As shown in [E HSST], a similar theory is obtained for supp ort v ar ieties defined in terms o f the Hochschild cohomolog y r ing , when one assumes the existence of such a commutativ e ring. Motiv ated by this, we now make the follo wing assumption on the fixed subalg ebra H of E xt ∗ Λ ( M , M ). Assumption. The ring H is Noe ther ian. A priori, the algebra H is just so me unknown gr aded subalgebr a of Ext ∗ Λ ( M , M ), and this is of co urse not satisfactor y if we want to do r eal computations. How ever, the following r e sult shows that when H is a subalgebra of the c e n ter Z( M ) of Ext ∗ Λ ( M , M ), a nd we require Ext ∗ Λ ( M , M ) to b e a finitely genera ted H -mo dule, then we may take H to be Z( M ) itself. Note tha t Z( M ) is a graded alg ebra. Indeed, suppos e η is an element of Z( M ), a nd write η = η 0 + · · · + η n , where η i is an element of Ext i Λ ( M , M ) for each i . Le t θ b e any homog eneous elemen t of Ext ∗ Λ ( M , M ). Then since η θ = θη , we see that each η i m ust commute with θ . Therefore eac h η i belo ngs to Z( M ), and this shows that Z( M ) is a graded algebra. Prop ositio n 3.1. The fol lowing ar e e quivalent. (i) Ther e ex ists a c ommutative N o etherian gr ade d sub algebr a R ⊆ Z( M ) over which E xt ∗ Λ ( M , M ) is a finitely gener ate d mo dule. (ii) The ring Z( M ) is No etherian, and Ext ∗ Λ ( M , M ) is a finitely gener ate d Z( M ) -mo dule. (iii) The ring Ext ∗ Λ ( M , M ) is No etherian and a finitely gener ate d Z( M ) -mo dule. Pr o of. The implication (ii) ⇒ (i) is obvious. Suppo se (i) holds, and let G b e an algebra “lying b etw een” R a nd Ext ∗ Λ ( M , M ), i.e. R ⊆ G ⊆ Ext ∗ Λ ( M , M ). Then Ext ∗ Λ ( M , M ) must b e a finitely generated G -mo dule. Mo reov er, s ince R is Noe ther- ian and E x t ∗ Λ ( M , M ) is a finitely generated R - mo dule, we see that Ext ∗ Λ ( M , M ) is a No etherian r ing. This shows the implication (i) ⇒ (iii). Finally , the implication (iii) ⇒ (ii) is [ArT, Theorem 1].  As men tio ned in the previo us se ction, the a ssumption that H 0 be a lo ca l ring is sup e rfluous once we hav e in tro duced finiteness conditions. Namely , the following result s hows tha t we may take H to be a p olyno mia l ring ov er k , so that H 0 is just k itself. Recall first tha t if V is a graded k -vector spa ce of finite type (i.e. dim k V i < ∞ for all i ), then the r ate of gr owth of V , denoted γ ( V ), is defined as γ ( V ) def = inf { t ∈ N ∪ { 0 } | ∃ a ∈ R suc h that dim k V n ≤ an t − 1 for n ≫ 0 } . RELA TIVE SUPPOR T V ARIETIES 5 Prop ositio n 3.2. L et N b e a Λ -mo dule, and supp ose E xt ∗ Λ ( M , N ) (r esp e ctively, Ext ∗ Λ ( N , M ) ) is a finitely gener ate d H -mo dule. Then ther e exists a p olynomia l ring k [ x 1 , . . . , x c ] ⊆ H , with c = γ ( H ) , such that H and Ext ∗ Λ ( M , N ) (r esp e ctively, Ext ∗ Λ ( N , M ) ) ar e fin itely gener ate d k [ x 1 , . . . , x c ] -mo dules. Pr o of. F ollows from the No ether normalizatio n lemma.  In the fo llowing result w e characterize pr ecisely when al l the cohomology modules are finitely gener ated over H . Prop ositio n 3.3. Conside r the fol lowing c onditions. (i) F or al l N ∈ mod Λ , t he H -mo dule E xt ∗ Λ ( M , N ) is finitely gener ate d. (ii) The H - mo dule Ext ∗ Λ ( M , Λ / rad Λ) is finitely gener ate d. (iii) F or al l N ∈ mo d Λ , the H -mo dule Ext ∗ Λ ( N , M ) is finitely gener ate d. (iv) The H - mo dule Ext ∗ Λ (Λ / r ad Λ , M ) is fi nitely gener ate d. Then t he implic ations (i) ⇔ (ii) and (iii) ⇔ (iv) hold. Pr o of. W e pro ve only the implication (ii) ⇒ (i); the implication (iv) ⇒ (iii) is prov ed analogo us ly . The pro of is by induction o n the length ℓ ( N ) o f a mo dule N . Since the H -mo dule Ext ∗ Λ ( M , Λ / rad Λ) is finitely gener ated, so is Ex t ∗ Λ ( M , S ) for an y simple Λ-mo dule S . No w supp os e ℓ ( N ) > 1, and choo s e a nonzero prop er submo dule L of N . The exac t sequence 0 → L → N → N /L → 0 induces an exact sequence Ext ∗ Λ ( M , L ) → Ext ∗ Λ ( M , N ) → Ext ∗ Λ ( M , N /L ) of H -mo dules. By a ssumption, b oth the end ter ms are finitely generated H - mo dules, hence so is the middle term since H is No etherian.  There are situations when finiteness alwa ys o ccurs, regar dless of the mo dule M we s ta rt with. Namely , when all the cohomology groups of the algebra are finitely generated over a cent ral r ing o f cohomo logy ope r ators, as in the following definition. Definition. The algebra Λ satisfies Fg if there exists a commutative No etherian graded k -algebr a R = L ∞ i =0 R i of finite type (i.e. dim k R i < ∞ for all i ) satisfying the following: (i) F or every X ∈ mo d Λ there is a gr aded r ing homomorphism φ X : R → Ext ∗ Λ ( X, X ) . (ii) F o r each pair ( X, Y ) of finitely g enerated Λ- mo dules , the sca lar actions from R on Ex t ∗ Λ ( X, Y ) via φ X and φ Y coincide, and E x t ∗ Λ ( X, Y ) is a finitely generated R -mo dule. As ment ioned, this holds if Λ is the group algebra o f a finite g roup, a co com- m utative Hopf algebra , a finite dimensional commutativ e c o mplete in ter section, or if the Ho chsc hild cohomology ring of Λ is suitably “nice” (cf. [Avr], [AvB], [Ben], [Car], [Eve], [F rS], [EHSST], [Er S], [BeO ]). Now supp ose Λ satisfies Fg with resp ect to a gra ded ring R as in the definition, and let X be a Λ-mo dule. Then φ X ( R ) is a co mm utative No etherian graded subalg e bra of the c e nter o f E xt ∗ Λ ( X, X ). Mor e- ov er, for any Y ∈ mo d Λ b oth Ext ∗ Λ ( X, Y ) and Ext ∗ Λ ( Y , X ) are finitely generated φ X ( R )-mo dules. When Λ satisfies Fg , we may a lso define supp ort v a r ieties with res p ect to the r ing of co homology op era tors. Namely , let R b e as in the definition. Given Λ-mo dules X and Y , we define V R ( X, Y ) def = { m ∈ Ma xSpe c R | Ann R Ext ∗ Λ ( X, Y ) ⊆ m } . 6 PETTER ANDREAS BERGH & ØYVIND S OLBER G Is this v ariety comparable to V i φ X ( R ) ( Y ) and V p φ Y ( R ) ( X )? The following r esult shows that the three v arieties V i φ X ( R ) ( Y ) , V p φ Y ( R ) ( X ) and V R ( X, Y ) are in fact isomorphic. Prop ositio n 3.4. Su pp ose Λ satisfies Fg with r esp e ct to a gr ade d ring R as in the definition ab ove, and let X and Y b e Λ -mo dules. Then the varieties V i φ X ( R ) ( Y ) , V p φ Y ( R ) ( X ) and V R ( X, Y ) ar e isomorphic. Pr o of. Let m b e a ma ximal ideal in R . Since φ X (Ann R Ext ∗ Λ ( X, Y )) e q uals Ann i φ X ( R ) Y , we s ee that Ann R Ext ∗ Λ ( X, Y ) ⊆ m if and o nly if Ann i φ X ( R ) Y ⊆ φ X ( m ). Therefore m b elongs to V R ( X, Y ) if a nd only if φ X ( m ) belo ngs to V i φ X ( R ) ( Y ), and this shows that the v ar ieties V R ( X, Y ) and V i φ X ( R ) ( Y ) are is o - morphic. Similarly the v a rieties V R ( X, Y ) and V p φ Y ( R ) ( X ) are isomorphic.  As we saw ab ov e, when Λ satisfies Fg then for every Λ - mo dule M there ex- ists a co mm utative No etherian g r aded subalgebra H ⊆ Ext ∗ Λ ( M , M ) ov er which Ext ∗ Λ (Λ / r ad Λ , M ) and Ext ∗ Λ ( M , Λ / rad Λ) are finitely generated. How ever, the fol- lowing example shows that this may very well hold for a mo dule even if the alg ebra do es not satisfy Fg . Example. Supp ose Λ is s elfinjectiv e, a nd let M b e a nonzero per io dic Λ-mo dule, i.e. Ω p Λ ( M ) ≃ M for s ome p ≥ 1. Then the first part of the minimal pro jective resolution of M is a p -fold extension 0 → M → P p − 1 → · · · → P 0 → M → 0 . Denote this extensio n by µ , and consider the s uba lgebra k [ µ ] of Ex t ∗ Λ ( M , M ). This subalgebra is a No etherian ring ov er which Ext ∗ Λ ( M , M ) is finitely generated as a mo dule. In fact, g iven any Λ-mo dule N , the k [ µ ]-mo dules Ext ∗ Λ ( M , N ) and Ext ∗ Λ ( N , M ) are finitely genera ted (cf. [Sc1] a nd [Sc2 ] for a discussion o f these phenomena). As an example, consider the quant um exterior algebr a k h x, y i / ( x 2 , xy − q y x, y 2 ) , where the element q is a nonzero non-r o ot of unit y in k . Let M be a t wo dim ensional vector space with bas is { u, v } , sa y . By defining xu = 0 , xv = 0 , y u = v , yv = 0 , this vector spac e becomes a module o ver the quantum exterior algebra. Mor eov er, it is not difficult to see that this module is p er io dic of per io d one (cf. [Ber , Example 4.5]). How ever, b y [E rS] a nd [BeO, Theorem 5.5] the a lgebra do es not satisfy Fg , since q is not a ro ot of unity . W e now return to the general theory . A natural question to ask is how big the relative suppo rt v a riety o f a mo dule is. F or an arbitrary mo dule N , this cannot be answered unless we introduce finiteness co nditions, since a priori there is no rela- tionship betw een H and Ext ∗ Λ ( M , N ) or Ext ∗ Λ ( N , M ). How ever, when we intro duce finite generatio n, the situation b ecomes muc h more manageable. Let X b e a Λ-mo dule with minimal pro jective and injective r esolutions · · · → P 2 → P 1 → P 0 → X → 0 , 0 → X → I 0 → I 1 → I 2 → · · · , say . Then we define the c omplexity and plexity of X , denoted cx X a nd p x X , resp ectively , as cx X def = inf { t ∈ N ∪ { 0 } | ∃ a ∈ R suc h that dim k P n ≤ an t − 1 for n ≫ 0 } , px X def = inf { t ∈ N ∪ { 0 } | ∃ a ∈ R suc h that dim k I n ≤ an t − 1 for n ≫ 0 } . RELA TIVE SUPPOR T V ARIETIES 7 The complexity and the plexit y of a mo dule a re no t ne c e ssarily finite. Also, from the definition we see that cx X = 0 (respectively , p x X = 0) if and only if X has finite pro jective dimension (resp ectively , finite injective dimensio n). It is well known that the complexity of X equals γ (Ext ∗ Λ ( X, Λ / rad Λ)), whereas its plexity equals γ (Ext ∗ Λ (Λ / r ad Λ , X )). Generalizing this, w e define the c omplexity of the p air ( X , Y ) of Λ- mo dules to b e γ (Ext ∗ Λ ( X, Y )), and denote it b y cx( X, Y ). Th us cx X is the c o mplexity of the pair ( X , Λ / rad Λ), wher eas p x X is the complexity of the pair (Λ / rad Λ , X ). Note that cx( X , Y ) 6 = cx( Y , X ) in general, that is, the order matters . Also, it follows from the discussion prior to [Ben, Prop osition 5.3 .5] that cx( M , N ) ≤ cx M , a nd similar ly cx( M , N ) ≤ px N . In particular cx( M , M ) is at most cx M and px M , and the following res ult sho ws that equality o c c ur s when finite generatio n holds. Prop ositio n 3.5. [B e n, Prop osition 5.3.5] If the H -m o dule Ext ∗ Λ ( M , Λ / rad Λ) is finitely gener ate d, t hen cx M = c x( M , M ) = γ ( H ) . Similarly, if the H -mo dule Ext ∗ Λ (Λ / r ad Λ , M ) is fi nitely gener ate d , t hen px M = c x( M , M ) = γ ( H ) . In p articular, if b oth Ext ∗ Λ ( M , Λ / rad Λ) and Ext ∗ Λ (Λ / r ad Λ , M ) ar e finitely gener- ate d over H , then cx M = px M . As for the “size ” of the r elative supp ort v arieties, the following result shows that it is given in terms o f the complexit y , provided finite generation holds. Prop ositio n 3.6. If the H -m o dule Ext ∗ Λ ( M , N ) is fi nitely gener ate d, then dim V i H ( N ) = cx( M , N ) . S imilarly, if the H -mo dule Ext ∗ Λ ( N , M ) is finitely gener- ate d , then dim V p H ( N ) = cx ( N , M ) . Pr o of. If Ext ∗ Λ ( M , N ) is finitely genera ted over H , then γ  H/ Ann i H N  = γ (Ext ∗ Λ ( M , N )), and so b y definition dim V i H ( N ) = cx( M , N ). The other equa lity is prov ed similarly .  4. Wild algebras and complexity In this sectio n we ass ume that our field k is algebraically closed. Recall that Λ is of finite r epr esentation typ e if there are only finitely many no n-isomorphic indecomp osable Λ-mo dules. F urther more, reca ll that Λ is of tame r epr esentation typ e if there exist infinitely many non-iso morphic indecomp osable Λ - mo dules, but they all b elong to o ne-parameter families, a nd in each dimensio n there are finitely many such families. Fina lly , the a lgebra Λ is of wild r epr esentation t yp e if it is not of finite or tame type. In [C-B], Crawley-Bo evey established a link b etw een the representation type of a s e lfinjectiv e finite dimensional alg ebra and the complexities of its mo dules . Namely , it was shown that for such a n algebra, in any dimension only finitely man y indecomp osable mo dules are not of complexity one. Using this, F a rnsteiner show ed in [F a r] that the complexity of every mo dule of a tame blo ck of a finite gro up sc heme is at most tw o. Suppo se o ur alge br a Λ is selfinjective and s a tisfies the “globa l” finite gener ation hypothesis Fg defined immediately after Pr op osition 3.3. That is, supp ose there exists a commutativ e Noetheria n g raded k -alg ebra R = L ∞ i =0 R i of finite type satisfying the following: (i) F or every X ∈ mo d Λ there is a gr aded r ing homomorphism φ X : R → Ext ∗ Λ ( X, X ) . 8 PETTER ANDREAS BERGH & ØYVIND S OLBER G (ii) F o r each pair ( X, Y ) of finitely g enerated Λ- mo dules , the sca lar actions from R on Ex t ∗ Λ ( X, Y ) via φ X and φ Y coincide, and E x t ∗ Λ ( X, Y ) is a finitely generated R -mo dule. Then F arnsteiner’s pro of still applies, hence Λ is wild if there exis ts a mo dule of complexity at least three. W e end this pap er with the following r e sult, which generalizes this. Theorem 4.1 . Supp ose Λ is selfinje ctive, and ther e exist s a Λ -mo dule M satisfying the fol lowing: (i) cx( M , M ) ≥ 3 , (ii) ther e exists a c ommut ative No etherian gr ade d sub algebr a H ⊆ Ex t ∗ Λ ( M , M ) over which Ex t ∗ Λ ( M , M ) is a finitely gener ate d mo dule. Then Λ is of wild r epr esentation typ e. Pr o of. Supp ose (i) holds. By Pro po sition 3 .2, we may assume that H is a poly no- mial r ing , say H = k [ y 1 , . . . , y n ], where n ≥ 3. Denote the ideal Ann i H N ⊆ H by a . Since Ext ∗ Λ ( M , M ) is a finitely generated H / a -mo dule, we may apply the No ether normalizatio n lemma and obta in a new p olynomial r ing R = k [ x 1 , . . . , x c ] ⊆ H / a , with c = cx( M , M ), ov er which Ex t ∗ Λ ( M , M ) is finitely generated. Mo reov e r , we may assume that the homogeneous e le ments x 1 , . . . , x c are of the same deg ree, sa y | x i | = d . The max imal ideals in R co rresp ond to p oints ( α 1 , . . . , α c ) ∈ k c . Given an ideal I ⊆ R , we denote its v ariety by V R ( I ), th us V R ( I ) = { α ∈ k c | f ( α ) = 0 for all f ∈ I } . Now for each α ∈ k , denote the elemen t x 1 + αx 2 ∈ R by x α . Lifting this element to H gives a homogeneous element η α ∈ Ext ∗ Λ ( M , M ) of degree d , fro m whic h w e obtain a short exact sequence ( † ) 0 → M → K α → Ω d − 1 Λ ( M ) → 0 . By applying the same pro of as in [EHSST, Pro po sition 4.3 ], w e see that V p R ( K α ) = V R ( x 1 + αx 2 ). Thus cx( K α , M ) = c − 1, and K α is not is omorphic to K α ′ whenever α 6 = α ′ . F or each α , let K α = K 1 α ⊕ · · · ⊕ K t α α be a decomp os ition of K α int o indecom- po sable Λ-mo dules. Mo reov er , denote the idea l Ann p R K i α ⊆ R , that is, the ideal Ann R Ext ∗ Λ ( K i α , M ), by a i . Then V p R ( K i α ) is by definition the v ar ie t y V R ( a i ), and therefore V R ( x 1 + αx 2 ) = V p R ( K α ) = t α [ i =1 V p R ( K i α ) = t α [ i =1 V R ( a i ) = V R ( t α Y i =1 a i ) , which in turn implies t α Y i =1 a i ⊆ v u u t t α Y i =1 a i = p ( x 1 + αx 2 ) . Since the ideal ( x 1 + αx 2 ) is prime, it is equal to its own radical, and contains one o f the ideals a 1 , . . . , a t α , say a 1 . How ever, the v ariety V p R ( K 1 α ) is contained in V p R ( K α ), and there fore ( x 1 + αx 2 ) = √ a 1 . Consequently √ a 1 = ( x 1 + αx 2 ), and this shows that the v a r ieties V p R ( K 1 α ) and V p R ( K α ) are equal. The indecomp osable Λ-mo dules { K 1 α } α ∈ k are pairwise no nisomorphic, a nd from the exact sequence ( † ) we see that dim k K 1 α ≤ dim k M + dim k Ω d − 1 Λ ( M ) for every α ∈ k . Moreov er, by construction w e know that cx( K 1 α , M ) = cx( K α , M ), hence 2 ≤ c − 1 = cx( K 1 α , M ) ≤ cx K 1 α . RELA TIVE SUPPOR T V ARIETIES 9 The result now follows from Crawley-Bo evey’s re sult [C-B, Theorem D].  Using Prop ositio n 3.3 and Prop osition 3.5, we o btain the following co rollarie s to Theorem 4.1. Corollary 4.2. Su pp ose Λ is selfinje ctive, and ther e exists a Λ -mo dule M satisfying the fol lowing: (i) cx M ≥ 3 , (ii) ther e exists a c ommut ative No etherian gr ade d sub algebr a H ⊆ Ex t ∗ Λ ( M , M ) over which Ex t ∗ Λ ( M , Λ / rad Λ) is a finitely gener ate d mo dule. Then Λ is of wild r epr esentation typ e. Corollary 4.3. Su pp ose Λ is selfinje ctive, and ther e exists a Λ -mo dule M satisfying the fol lowing: (i) px M ≥ 3 , (ii) ther e exists a c ommut ative No etherian gr ade d sub algebr a H ⊆ Ex t ∗ Λ ( M , M ) over which Ex t ∗ Λ (Λ / r ad Λ , M ) is a finitely gener ate d mo dule. Then Λ is of wild r epr esentation typ e. Finally , by applying Propo sition 3.1 w e o btain the following co rollary to Theorem 4.1. It shows that an algebr a is wild if it poss esses a mo dule whose Ext-algebr a is “big enough”, No etherian and finitely gener ated o ver its cent er. Corollary 4.4. Su pp ose Λ is selfinje ctive, and ther e exists a Λ -mo dule M satisfying the fol lowing: (i) cx( M , M ) ≥ 3 , (ii) Ext ∗ Λ ( M , M ) is a No etherian ring and finitely gener ate d as a mo dule over its c en ter. Then Λ is of wild r epr esentation typ e. W e end this pap er with the following ex ample illustrating Theorem 4.1, an e x - ample in which the a lgebra do es not satisfy the finite gener ation h yp othesis Fg . Example. Let q be a nonzero no n- ro ot of unity in k , and denote by Γ the quantum exterior algebra k h x, y i / ( x 2 , xy − q y x, y 2 ) . Let X b e the Γ-mo dule from the example following Pr op osition 3.4, i.e. X is a t wo dimensiona l vector space with basis { u, v } , say , and with scala r m ultiplication defined by xu = 0 , xv = 0 , y u = v , yv = 0 . This module is pe r io dic of p erio d one, and so if we deno te its pro jective cov er 0 → X → P → X → 0 by µ , the Ext-algebra Ext ∗ Γ ( X, X ) is finitely ge nerated as a mo dule over the po ly- nomial subalgebra k [ µ ]. Now let Λ b e the a lg ebra Γ ⊗ k Γ ⊗ k Γ, and let M b e the Λ-mo dule X ⊗ k X ⊗ k X . Then the E x t-algebra of M is given b y Ext ∗ Λ ( M , M ) = Ext ∗ Γ ( X, X ) ⊗ k Ext ∗ Γ ( X, X ) ⊗ k Ext ∗ Γ ( X, X ) , where ⊗ differ from the usual tensor pr o duct only in that elemen ts of o dd degree anticomm ute (cf. [CaE, Chapter XI]). Therefore Ext ∗ Λ ( M , M ) is finitely generated as a mo dule over a commutativ e No etherian gr aded s uba lgebra. Moreover, since cx Γ ( X, X ) = 1, we see that cx Λ ( M , M ) = 3 . Finally , the algebra Λ does not satisfy Fg . Namely , this a lgebra is a quantum exterior a lgebra o n six generator s, where some of the defining comm uta to rs equa l q . 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