The Avrunin-Scott theorem for quantum complete intersections

THE A VR UNIN-SCOTT THEOREM F OR QUANTUM COMPLETE INTERSECTIONS PETTER ANDREAS BERGH & KARIN ERDMANN Abstract. W e prov e the Avrunin-Scott th eorem f or quan tu m co mplete inter- sections; the rank v ariet y of a mo dule is isomorphi c to its supp ort v ariety . 1. Introduction Inspired by the impact of the theories of v arie ties for mo dules ov er group alge- bras, similar theories hav e been studied for other classes of algebr as. F or example, using Ho chsch ild cohomolo g y , Snashall and Solb erg developed a theory of supp or t v a rieties for finite dimensional a lgebras in [SnS]. As shown in [E HSST], this theory is very p ow erful when the co homology o f the algebra satisfies sufficient finite g en- eration, a nd for selfinjective such algebras the theor y shares many of the pr o p erties of that for g r oup alg ebras. Ho w ever, support v arieties are difficult to compute. In [Car], Ca rlson in tro duced r ank v ar ieties for mo dules o ver gr o up algebras of elemen- tary ab elia n groups, v a rieties defined without using coho mology . Given a module ov er such an algebra, its r ank v ariety is very explicit and easy to compute. Mor e - ov er, Avrunin and Scott pr oved in [AvS] that the suppo rt v ariety of a module is in fact isomorphic to its rank v ariet y . Needless to say , this result has ha d imp orta nt consequences (see, for example, the introduction in [AvS] or [Er H]). Motiv a ted by this, t he sec o nd author and Hollow a y intro duced in [ErH] rank v a rieties for truncated poly nomial algebras in whic h the generators square to zero. Such alg ebras also hav e s upp o r t v arieties, and it was sho wn that these t wo v arieties are isomor phic. In this pap er, we study rank v arieties for quantum co mplete intersections, a class of algebra s origina ting fro m w ork by Manin and Avramov, Gasharov and Peev a (cf. [Man] and [A GP]). When the defining parameters are ro o ts of unit y , then these alge br as hav e supp ort v arieties; it was shown in [BeO] that finite generation of cohomology holds. How ev er, certa in quantum complete int ersectio ns also have rank v arieties, and one w ould therefore like to k now if and ho w the supp ort and rank v arieties are related. W e show that they ar e indeed isomorphic. 2. V arie ties Throughout this section, let k b e an algebra ic ally closed field. All mo dules considered are assumed to b e le ft mo dules and finitely gener a ted. W e start by recalling the definitions and some results on suppor t v arieties; details ca n b e found in [EHSST] a nd [SnS]. Let Λ be a finite dimensiona l k -algebra with Jaco bson ra dical r . W e de no te by Λ e the env eloping algebra Λ ⊗ k Λ op of Λ. The n th Ho chschild c ohomolo gy gro up 2000 Mathematics Subje ct Classific ation. 16E30, 16E40, 16S80, 16U80, 81R50. Key wor ds and phr ases. Quantum complete in tersections, rank v arieties, supp ort v arieties, Avrunin-Scott theorem. The first author was supp orted b y NFR Storforsk grant no. 167 130. 1 2 PETTER ANDREAS BERGH & KARIN ERDMANN of Λ, denoted HH n (Λ), is the v ector s pace Ext n Λ e (Λ , Λ) of n -fold bimo dule exten- sions of Λ with itself. The Y oneda pro duct tur ns HH ∗ (Λ) = ⊕ ∞ n =0 HH n (Λ) in to a g raded k - algebra , the Ho chsc hild coho mology ring o f Λ. This algebra is graded commutativ e, that is, given tw o homogeneo us elements η, θ ∈ HH ∗ (Λ), the e quality η θ = ( − 1) | η | | θ | θη ho lds. In par ticular, the even pa rt HH 2 ∗ (Λ) = ⊕ ∞ n =0 HH 2 n (Λ) of HH ∗ (Λ) is a commutativ e ring. Every ho mo geneous element η ∈ HH ∗ (Λ) can b e repr esented by a bimo dule extension η : 0 → Λ → B | η | → · · · → B 1 → Λ → 0 . Given a Λ-mo dule M , the complex η ⊗ Λ M : 0 → M → B | η | ⊗ Λ M → · · · → B | η | ⊗ Λ M → M → 0 is exact, hence the tensor pro duct − ⊗ Λ M induces a ho momorphism HH ∗ (Λ) ϕ M − − → Ext ∗ Λ ( M , M ) of gr aded k -algebr a s. Thus, given t w o Λ-mo dules M and N , the gr aded v ector space E xt ∗ Λ ( M , N ) b ecomes a g raded module ov er HH ∗ (Λ) in tw o ways; either via ϕ M or via ϕ N . These t wo mo dule structures are equa l up to a graded sig n, that is, given homog eneous elements η ∈ HH ∗ (Λ) and θ ∈ Ext ∗ Λ ( M , N ), the equality ϕ N ( η ) ◦ θ = ( − 1) | η | | θ | θ ◦ ϕ M ( η ) ho lds, where “ ◦ ” denotes the Y oneda pro duct. Let H b e a commu tative graded subalgebra of HH ∗ (Λ). The supp ort variety of an order ed pa ir ( M , N ) of Λ-mo dules (with resp ect to H ), deno ted V H ( M , N ), is defined as V H ( M , N ) def = { m ∈ MaxSp ec H | Ann H Ext ∗ Λ ( M , N ) ⊆ m } , where MaxSp ec H is the set of ma ximal ideals of H . There ar e equalities V H ( M , Λ / r ) = V H ( M , M ) = V H (Λ / r , M ) , and this set is defined to b e the suppo rt v a riety V H ( M ) o f M . In general, suppor t v ar ieties do not cont ain any homolo gical information on the mo dules in v olved. F or exa mple, if HH ∗ (Λ) is finite dimensional, tha t is, if HH n (Λ) = 0 for n ≫ 0 , then the suppor t v ariety of any pa ir o f mo dules is trivia l. Howev er, under certain finiteness conditions, the situation is quite different. Suppo se H is No etherian and Ext ∗ Λ ( M , N ) is a finitely gener ated H -mo dule for a ll Λ-mo dules M and N (this is equiv alent to Ext ∗ Λ (Λ / r , Λ / r ) b eing finitely generated ov er H ). In this ca se, the dimensio n of V H ( M , N ) equa ls the p olyno mia l rate of g rowth of Ex t ∗ Λ ( M , N ). In par ticula r, the dimensio n of the supp or t v ariety o f a module equals its complexity , hence a mo dule has finite pro jective dimension if and only if its supp or t v a riety is trivia l. Moreover, the dimension o f the supp or t v a riety of a mo dule is one if and o nly if its minimal pr o jective resolution is b ounded. Under the finite gener ation hypothesis given, this happens precisely when the mo dule is even tually p erio dic, that is, w hen its minimal pro jectiv e r esolution beco mes per io dic from some step o n. When the a lgebra Λ, in addition to satisfying the finite gener a tion h ypo thesis, is also selfinjectiv e, then the pro jective suppo rt v arie ty o f an indecomposa ble mo dule is connected. Namely , let M be a Λ-mo dule who s e supp ort v ariety decompos es as V H ( M ) = V 1 ∪ V 2 , where V 1 and V 2 are clos ed homo g eneous v arieties such that V 1 ∩ V 2 is tr ivial. Then there are submo dules M 1 and M 2 of M , with the property that M = M 1 ⊕ M 2 and V H ( M i ) = V i . In particular, this implies that the suppor t v a riety of an indecomposa ble per io dic mo dule is a single line. W e now tur n to the cla ss of alg ebras for which w e will prove the Avrunin-Scott theorem. These are ana logues of trunca ted p olynomia l ring s. Let c ≥ 1 be an int eger, and let q = ( q ij ) b e a c × c commutation matrix with en tries in k . That THE A VRUNIN-SCOTT THEOREM FOR QUANTUM COMPLETE INTERSECTIONS 3 is, the diagonal ent ries q ii are all 1, and q ij q j i = 1 for i 6 = j . F urthermor e, let a c = ( a 1 , . . . , a c ) b e an o rdered seq uence of c integers with a i ≥ 2. The quantum c omp lete interse ction A a c q determined by these data is the algebra A a c q def = k h x 1 , . . . , x c i / ( x a i i , x i x j − q ij x j x i ) , which is selfinjective and finite dimensional of dimensio n Q a i . It was prov ed in [Be O] that if all the commutators q ij are r o ots of unity , then HH 2 ∗ ( A a c q ) is No etherian, and E xt ∗ A a c q ( M , N ) is a finitely gener ated HH 2 ∗ ( A a c q )-mo dule for a ll A a c q -mo dules M and N . Thus, in this case, the suppo rt v arieties with res p e c t to HH 2 ∗ ( A a c q ) detec t pro jective and per io dic mo dules, as w e s aw ab ov e. The Krull dimension of HH 2 ∗ ( A a c q ) is c , that is, the num ber o f g enerator s defining the quan- tum complete intersection. Therefore, the supp ort v arieties ar e homogeneous affine subsets of k c . Now fix a n integer a ≥ 2, and define a ′ by a ′ def =  a/ gcd( a, char k ) if char k > 0 a if char k = 0 . Let q ∈ k b e a primitive a ′ th r o ot o f unity , let q be the commutation matrix with q ij = q for i < j , a nd let a c be the c -tuple ( a, . . . , a ). Then w e denote the corres p o nding quan tum complete intersection A a c q by A c q , i.e. A c q = k h x 1 , . . . , x c i / ( { x a i } , { x i x j − qx j x i } i 0 a if char k = 0 . Moreov er, we fix an in teger c ≥ 1 a nd a pr imitive a ′ th ro o t of unity q ∈ k , a nd work with the corr esp onding quantum co mplete intersection A c q given b y A c q = k h x 1 , . . . , x c i / ( { x a i } , { x i x j − qx j x i } i dim M , and so the res ult follows.  Our next aim is to determine the suppor t v ariet y V H ( Au λ ) for all nonzero points λ ∈ k c . W e start with the following gener al result. Recall that an a lgebra is F r ob enius if, as a left mo dule ov er itse lf, it is iso morphic to its vector space dual. Lemma 3.2. L et Λ and Γ b e two finite dimensiona l k - algebr as, with Λ selfinje ctive and Γ F r ob enius. F urthermor e, let B b e a Γ - Λ -bimo dule which is pr oje ctive b oth as a Γ -mo dule and as a Λ -mo dule. Then for every Λ -mo dule X and every Γ -mo dule Y , ther e is a natur al isomorphism Ext n Λ ( X, D ( B ) ⊗ Γ Y ) ≃ Ext n Γ ( B ⊗ Λ X , Y ) for e ach n ≥ 0 . Pr o of. Adjointness gives a natural isomorphism Hom Λ ( X, Hom Γ ( B , Y )) ≃ Hom Γ ( B ⊗ Λ X , Y ) for ev ery Λ-mo dule X and every Γ-module Y . The functor Hom Γ ( B , − ) is isomor- phic to Hom Γ ( B , Γ ⊗ Γ − ), which in turn is isomorphic to Ho m Γ ( B , Γ) ⊗ Γ − since B is a pr o jective Γ-module. Now Γ, b eing F rob enius, is isomorphic a s a left Γ-module to D (Γ), a nd so adjointn ess gives Hom Γ ( B , Γ) ≃ Hom Γ ( B , D (Γ)) ≃ D (Γ ⊗ Γ B ) ≃ D ( B ) . Therefore, the functor Hom Γ ( B , − ) is isomo rphic to D ( B ) ⊗ Γ − , proving the case n = 0. F or the case n > 0, note that since B is a pro jective Λ -mo dule, the Λ-mo dule D ( B ) ⊗ Γ P is pr o jective for every pro jectiv e Γ-mo dule P . How ever, since b oth Λ and Γ are s elfinjective, this means that D ( B ) ⊗ Γ I is an injective Λ-mo dule for every injectiv e Γ-mo dule I . Moreover, g iven a Γ-mo dule Y with an injective r esolution I , the complex D ( B ) ⊗ Γ I is an injectiv e resolution of the Λ-mo dule D ( B ) ⊗ Γ Y , since D ( B ) is a pro jectiv e right Γ-module. Therefo r e the isomor phisms Ext n Γ ( B ⊗ Λ X , Y ) ≃ H n (Hom Γ ( B ⊗ Λ X , I )) ≃ H n (Hom Λ ( X, D ( B ) ⊗ Γ I )) ≃ Ex t n Λ ( X, D ( B ) ⊗ Γ Y ) 6 PETTER ANDREAS BERGH & KARIN ERDMANN hold for every Λ-module X .  In or der to determine the s uppo rt v ariety of Au λ , w e exploit some nice proper ties of certain bimo dules arising from elemen ts o f the Ho chsc hild c o homolog y ring. Let ζ b e a homoge neous elemen t of HH ∗ ( A ), represented b y a map Ω | ζ | A e ( A ) f ζ − → A , say . Then ζ corresp onds to the bo ttom s ho rt exact s equence in the exact co mmu tative pushout diagr am 0 / / Ω | ζ | A e ( A ) f ζ   / / P | ζ | − 1   / / Ω | ζ | − 1 A e ( A ) / / 0 0 / / A / / K ζ / / Ω | ζ | − 1 A e ( A ) / / 0 where P | ζ | − 1 is the pro jective cov er of Ω | ζ | − 1 A e ( A ). If ζ is an element of H and M is an A -mo dule, then by [EHSST, P rop osition 4.3 ] the suppor t v a riety o f the A -mo dule K ζ ⊗ A M is given b y V H ( K ζ ⊗ A M ) = V H ( ζ ) ∩ V H ( M ) , where V H ( ζ ) = { α ∈ k c | ζ ( α ) = 0 } . In the following result, we use the pushout bimo dule K ζ to give a criterio n for when tw o lines in k c are p erp endicular . Given a nonzero p oint µ ∈ k c , we denote the set { α ∈ k c | P α i µ i = 0 } b y ℓ ⊥ µ ; this is the hyperplane p erp endicular to the line ℓ µ . Lemma 3. 3. L et M b e a p erio dic A -m o dule of p erio d 1 , and supp ose V H ( M ) is a single line ℓ α , wher e α ∈ k c is a nonzer o p oint. Then, given any nonzer o p oi nt µ ∈ k c , the implic ation Hom A ( M , K ζ ⊗ A k ) 6 = 0 ⇒ ℓ α ⊆ ℓ ⊥ µ holds, wher e ζ = P µ i η i ∈ H 2 . Pr o of. Since the bimo dule K ζ is pro jective both as a left and as a rig ht A -mo dule, we see from the previous lemma that Hom A ( M , K ζ ⊗ A k ) is naturally isomorphic to Hom A ( D ( K ζ ) ⊗ A M , k ). Th us Hom A ( M , K ζ ⊗ A k ) is nonzero if a nd only if the same holds for Hom A ( D ( K ζ ) ⊗ A M , k ), and this is equiv a lent to D ( K ζ ) ⊗ A M not being a pro jective A -module. The latter happ ens if and only if V H ( D ( K ζ ) ⊗ A M ) 6 = 0. Using the pre vious lemma onc e mor e, w e see that the H -mo dules Ext ∗ A ( D ( K ζ ) ⊗ A M , k ) and Ext ∗ A ( M , K ζ ⊗ A k ) ar e isomorphic. Therefor e V H ( D ( K ζ ) ⊗ A M ) = V H ( M , K ζ ⊗ A k ) ⊆ V H ( M ) ∩ V H ( K ζ ⊗ A k ) = V H ( M ) ∩ V H ( ζ ) = ℓ α ∩ ℓ ⊥ µ , and so if Hom A ( M , K ζ ⊗ A k ) is nonzer o then ℓ α ⊆ ℓ ⊥ µ .  Next, define a map k c F − → k c of affine spaces b y ( α 1 , . . . , α c ) 7→ ( α a 1 , . . . , α a c ) . Our aim now is to show that V H ( Au λ ) = ℓ F ( λ ) for every nonzero p o int λ ∈ k c . In order to prove this, w e need the following lemma. Lemma 3. 4. L et µ and λ b e nonzer o p oints in k c with ℓ µ ⊆ ℓ ⊥ F ( λ ) , and denote the element P µ i η i in H by ζ . Then ther e ex ists a monomorphism Au λ → K ζ ⊗ A k , and c onse quently Hom A ( Au λ , K ζ ⊗ A k ) is nonzer o. THE A VRUNIN-SCOTT THEOREM FOR QUANTUM COMPLETE INTERSECTIONS 7 Pr o of. Note that we have implicitly assumed that c is a t least 2 . Denote the r adical of A by r . The sec ond syzygy Ω 2 A ( k ) is the kernel of the pro jective cover A c → r , which maps the i th generator ε i of the free A -mo dule A c to x i . The generato rs of Ω 2 A ( k ) are the t wo se ts { x a − 1 i ε i } c i =1 , { q x j ε i − x i ε j } i 0. This implies that V H ( Au λ , M ) is nontrivial, and since this v a riety is co ntained in V H ( Au λ ) ∩ V H ( M ), we see from Propo sition 3.5 that the line ℓ F ( λ ) m ust b e co ntained in V H ( M ). Consequently , the inclusion F (V r A ( M )) ⊆ V H ( M ) ho lds. T o prov e the reverse inclusion, we argue by induction o n the dimension of the suppo rt v ariety o f M . W e may ass ume that M is indeco mpo sable, since the v ariet y (rank or supp or t) of a direct sum is the union of the v arieties of the s ummands. Suppo se fir s t that the dimension of V H ( M ) is one, i.e. that M is p erio dic. Then V H ( M ) is a line, and w e proved ab ov e that this v ariet y contains F (V r A ( M )). If F (V r A ( M )) = 0, then V r A ( M ) = 0, which is not the case since M is not pr o jec- tive. Hence F (V r A ( M )) is nontrivial, and so F (V r A ( M )) = V H ( M ) in this cas e. Next, supp ose that dim V H ( M ) > 1, a nd let µ ∈ k c be a nonzero p oint such that dim  ℓ ⊥ µ ∩ V H ( M )  < dim V H ( M ). Denote the element µ 1 η 1 + · · · + µ c η c ∈ H by ζ , and let Ω 2 A e ( A ) f ζ − → A be a bimo dule map repre s enting this element. Since the bo ttom exact seq ue nc e in the pushout dia g ram 0 / / Ω 2 A e ( A ) f ζ   / / P 1   / / Ω 1 A e ( A ) / / 0 0 / / A / / K ζ / / Ω 1 A e ( A ) / / 0 splits as rig ht A -modules , it stays e xact after tenso ring with M . This giv es a shor t exact sequence 0 → M → K ζ ⊗ A M → Ω 1 A ( M ) ⊕ P → 0 , in which P is pro jective, and so V r A ( K ζ ⊗ A M ) ⊆ V r A ( M ) ∪ V r A (Ω 1 A ( M ) ⊕ P ) = V r A ( M ) . Now since V H ( K ζ ⊗ A M ) = V H ( ζ ) ∩ V H ( M ) = ℓ ⊥ µ ∩ V H ( M ), induction gives V H ( K ζ ⊗ A M ) ⊆ F ( V r A ( K ζ ⊗ A M )). There fo re ℓ ⊥ µ ∩ V H ( M ) ⊆ F (V r A ( M )), and since this inclusio n holds for every nonzero point µ ∈ k c such that dim  ℓ ⊥ µ ∩ V H ( M )  is strictly less tha n dim V H ( M ), we see that V H ( M ) ⊆ F (V r A ( M )).  THE A VRUNIN-SCOTT THEOREM FOR QUANTUM COMPLETE INTERSECTIONS 9 The Avrunin-Sco tt theo r em follows immediately from this theorem; the r ank v a riety of an A -module is isomorphic to its supp or t v ariet y . Corollary 3.7. F or every A -m o dule M , the varieties V r A ( M ) and V HH 2 ∗ ( A ) ( M ) ar e isomorphic. Consequently , the dimension of the r ank v a riety of an A -mo dule is the co mplexity of the mo dule. In particular , an indecomp os a ble module is p erio dic if a nd only if its rank v ariet y is of dimension one. Corollary 3.8. F or every A -mo dule M , the dimension of V r A ( M ) is the c omplexity of M . Corollary 3.9. An inde c omp osable A -mo dule M is p erio dic if and only if the di- mension of V r A ( M ) is one. 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