Cohomology of twisted tensor products

COHOMOLOGY OF TWISTED TENSOR PR ODUCTS PETTER ANDREAS BER GH AND STEFFEN OPPERMANN Abstract. It is we ll kno wn that the cohomology of a tensor pro duct i s es- sen tially the tensor pro duct of the cohomologies. W e lo ok at t wisted tensor products, and in v estigate to which extend this is still true. W e give an explicit description of the Ext-algebra of the tensor product of tw o modules, and und er certain additional conditions, describ e an essen tial part of the Ho c hsc hild co- homology ring of a twisted tensor pr oduct. As an application, we charac terize precisely when the cohomology groups o ve r a quan tum complete i n tersection are finitely generated ov er the Hochsc hild cohomology ri ng. Moreo ver, b oth for quan tum complete intersect ions and in related cases we obtain a low er b ound for the represent ation dimension of the algebra. 1. Introduction Given a field k a nd tw o k -algebra s Λ and Γ, o ne may loo k at their tensor pro duct Λ ⊗ k Γ. This is an algebra wher e multiplication is done component wise. In other words, we use the m ultiplications in Λ a nd Γ, and define element s of Λ and Γ to commute with one another. Given a Λ-mo dule M and a Γ-mo dule N , it is well known that Ext ∗ Λ ⊗ k Γ ( M ⊗ k N ) = Ext ∗ Λ ( M ) ⊗ k Ext ∗ Γ ( N ) , HH ∗ (Λ ⊗ k Γ) = HH ∗ (Λ) ⊗ k HH ∗ (Γ) , where ⊗ is the usual tensor pro duct, but with e lemen ts of o dd degree anticomm uting (and where HH ∗ denotes the Ho chsc hild cohomolog y r ing). In this pap er we sha ll study gr aded algebra s and twisted tensor pro ducts. T ha t is, for tw o g raded algebra s Λ and Γ, w e give their tensor pro duct an a lgebra structure by defining elements fr om Λ a nd Γ to c o mm ute up to certa in scalar s, dep ending on the degr ees of the elemen ts. W e denote these twisted tensor pro ducts by Λ ⊗ t k Γ. Examples of alg ebras obtained in this way are quantum exterior algebra s (see [4], [5], [8]), a nd, mo re g enerally , quantum complete intersections (see [3], [6], [7]). The first main re sult of this pap er (Theore m 3.7) shows that the first formula ab o ve holds for t wisted tens or pro ducts. More pr e cisely , we may make the identifi- cation Ext ∗ Λ ⊗ t k Γ ( M ⊗ t k N ) = E xt ∗ Λ ( M ) ⊗ ˜ t k Ext ∗ Γ ( N ) , where the twist o n the rig h t-hand side is the co m bination of the twist we sta rted with and the sign which alrea dy o ccurred in the class ic a l case. This formula allows us to give an explicit description o f the Ext-algebr a of the s imple mo dule over a quantum complete intersection in Theo rem 5.3. As fo r the second formula above, we shall see (Remark 5.4) that in gener a l it do es not carry over to twisted tensor pro ducts. How ever, in Theorem 4.7 we show that the Ho chsc hild cohomolog y ring of a twisted tensor pro duct contains a subalg ebra, which is the t wisted tenso r pro duct of co r respo nding subalg ebras o f the Ho c hsch ild cohomo logy r ings of the factors. Under certain additional co nditions, we show that these subalge br as are big enough 2000 Mathematics Subje ct Classific ati on. 16E40, 16S80, 16U80, 81R50. Key wor ds and phr ases. Twisted tensor products, Ho c hsc hild cohomology , quan tum complete int ersections. 1 2 PETTER ANDREAS BERGH AND STEFFEN OPPERMANN to contain all the infor mation o n finite gener ation and complexity (Corolla ry 4.8). When finite generation holds, we ma y use these subalgebra s and a r esult from [4] to find a low er b ound for the representation dimension of the twisted tensor pro duct. In the final se ction we apply these r esults to quantum complete intersections. In particular, we s ho w that the cohomology groups of such an algebra a re all finitely generated over the Hochsc hild cohomology ring if and o nly if all the commutator parameters are r oots of unity (Theorem 5.5). This allows us to give a low er b ound for the repr esen tation dimension of these a lgebras (Corollary 5.6), thus generalizing the result of [7]. 2. N ot a tion Throughout this pap er, we fix a field k . All algebras consider ed ar e a s sumed to be asso ciative k -algebra s. 2.1. Definition. Let A b e a n ab elian gro up. An A -gr ade d algebra is an algebra Λ together with a decomp osition Λ = ⊕ a ∈ A Λ a as k -vector spaces, such that Λ a · Λ a ′ ⊆ Λ a + a ′ . A module M ov er suc h a graded algebra Λ is a gr ade d mo dule if it ha s a decomp osition M = ⊕ a ∈ A M a as k -vector spaces, such that Λ a · M a ′ ⊆ M a + a ′ . W e denote the categ ory of finitely gener ated gra de d Λ-mo dules by Λ -mo d gr . Let Λ, A a nd M b e as above. W e denote the degree o f homogeneous elements λ ∈ Λ and m ∈ M by | λ | and | m | , resp ectiv ely . F or an ele ment a ∈ A we denote by M h a i the shift of M having the same Λ-mo dule structure a s M , but with M h a i a ′ = M a ′ − a . No w let M ′ be another gra ded Λ-mo dule. T o distinguis h betw een graded and ungraded mor phisms, we denote the set of all Λ -morphisms fro m M to M ′ by Hom Λ ( M , M ′ ), and the set of degre e preser ving morphisms b y gr Hom Λ ( M , M ′ ). With this notatio n we obtain a deco mp osition Hom Λ ( M , M ′ ) = ⊕ a ∈ A grHom Λ ( M , M ′ h a i ) . Setting Ho m Λ ( M , M ′ ) a = gr Ho m Λ ( M , M ′ h a i ) turns End Λ ( M ) and End Λ ( M ′ ) into A -graded alg ebras, and Hom( M , M ′ ) into a g raded End Λ ( M )-End Λ ( M ′ ) bimo dule. Since M has a g r aded pro jective reso lution P , we can also define Ext i,a Λ ( M , M ′ ) def = H i (grHom( P , M ′ h a i )). It follows that E xt ∗ Λ ( M , M ) a nd Ext ∗ Λ ( M ′ , M ′ ) ar e ( Z ⊕ A )- graded algebras , and that Ext ∗ Λ ( M , M ′ ) is a gra ded Ext ∗ Λ ( M , M )-Ext ∗ Λ ( M ′ , M ′ ) bimo dule. Our main ob jects of study in this pap er are t wisted tensor products of tw o graded algebras , a concept we no w define. 2.2. Defini ti on/Construction. Let A and B be ab elian g roups, let Λ b e an A - graded algebra and Γ a B -gr a ded algebra . Let t : A ⊗ Z B ✲ k × be a homo mor- phism of ab elian gr oups, where k × denotes the multiplicativ e gr oup of nonzero ele- men ts in k . W e write t h a | b i = t ( a ⊗ b ), and, by abuse of notation, also t h λ | γ i = t h| λ ||| γ |i for homogeneous elements λ ∈ Λ and γ ∈ Γ. The ( t -) twiste d tensor pr o duct of Λ and Γ is the algebra Λ ⊗ t k Γ defined by Λ ⊗ t k Γ = Λ ⊗ k Γ as k -vector spaces, ( λ ⊗ γ ) · t ( λ ′ ⊗ γ ′ ) def = t h λ ′ | γ i λλ ′ ⊗ γ γ ′ , where λ, λ ′ ∈ Λ a nd γ , γ ′ ∈ Γ a r e homogene o us elements. A straightforw ard calculation shows that this is indeed a well defined algebra . By defining (Λ ⊗ t k Γ) a,b to b e Λ a ⊗ k Γ b , this a lgebra b ecomes ( A ⊕ B )-gr aded. W e no w define what it means for an algebr a to hav e finitely generated co homolgy . COHOMOLOGY OF TWISTED TENSOR P R ODUCTS 3 2.3. Definition. Let Λ b e an alge bra. A co mm utativ e ring of c ohomolgy op er ators is a commutativ e Z -graded k -algebra H together with graded k -alg ebra morphisms φ M : H ✲ Ext ∗ Λ ( M , M ), for every M ∈ Λ -mo d, such that for ev ery pair M , M ′ ∈ Λ -mo d the induced H - module structure s on E xt ∗ Λ ( M , M ′ ) via φ M and φ M ′ coincide. If A is an ab elian group and Λ is A -graded, then we r equire that H be a ( Z ⊕ A )- graded algebra , a nd that the ma ps φ M are morphisms of ( Z ⊕ A )-gra ded algebra s. The main example of such a ring H is the even Ho chsc hild cohomology ring of an algebra. Namely , by [15] the Hochsc hild cohomo lo gy r ing is graded commutativ e, so its even part is c o mm utativ e. Note that whenever an alg ebra is graded, then so is its Hochschild cohomology ring, a nd its even part is a comm utative ring of graded cohomolo gy op erators . 2.4. Definiti on. An a lgebra Λ satisfies the finite gener ation hyp othesis Fg if it has a commutativ e ring of op erators H which is No etherian and of finite type (i.e. dim k H i < ∞ for all i ), and such that for any M , M ′ ∈ Λ -mo d the H -module Ext ∗ Λ ( M , M ′ ) is finitely genera ted. W e end this section with some remar ks conce r ning finite gener ation of cohomol- ogy . 2.5. Remarks. (i) Ass ume that Λ is a finite dimens io nal alg ebra, a nd let H be a commutativ e No etherian ring of coho mology op erators. Then all Ext ∗ Λ ( M , M ′ ) are finitely genera ted ov er H if a nd only if E x t ∗ Λ (Λ / Rad Λ , Λ / Rad Λ) is. This follows from an induction a rgumen t on the length of M a nd M ′ . (ii) By [13, P ropo sition 5.7] the following are eq uiv alent for a n algebra Λ. (1) Λ satisfies Fg with resp ect to its even Ho c hschild cohomology ring HH 2 ∗ (Λ), (2) Λ satisfies Fg with resp ect to some s ubalgebra of its even Ho chsc hild co - homology ring. 3. Tensor products of graded modul es Throughout this section, we fix tw o ab elian groups A a nd B , toge ther with an A -graded algebr a Λ and a B -gr aded algebr a Γ. Mor eo ver, we fix a homomorphism t : A ⊗ Z B ✲ k × of ab elian groups. Given a gr a ded Λ-mo dule and a gr aded Γ- mo dule, we construct a Λ ⊗ t k Γ-mo dule, and study homomor phisms a nd extensio ns betw een such mo dules. 3.1. Definitio n/Construction. Given modules M ∈ Λ -mo d gr and N ∈ Γ -mo d gr , the tensor pr oduct M ⊗ k N b ecomes a gr aded Λ ⊗ t k Γ-mo dule by defining ( λ ⊗ γ ) · ( m ⊗ n ) def = t h m | γ i λm ⊗ γ n. W e denote this mo dule b y M ⊗ t k N , its grading is given by ( M ⊗ t k N ) a,b = M a ⊗ k N b . W e now prov e some element ary results o n these tensor pr o ducts, the first of which shows tha t the tensor pro duct of tw o shifted mo dules is the shifted tenso r pro duct. 3.2. Lemma. Given mo dules M ∈ Λ -mo d gr and N ∈ Γ -mo d gr , the gr ade d Λ ⊗ t k Γ - mo dules M h a i ⊗ t k N h b i and ( M ⊗ t k N ) h a, b i ar e isomorphic via the map M h a i ⊗ t k N h b i ✲ ( M ⊗ t k N ) h a, b i m ⊗ n ✲ t h a | n i m ⊗ n Pr o of. The given map is clea rly bijective, and it is straightforward to verify that it is a ho mo morphism.  4 PETTER ANDREAS BERGH AND STEFFEN OPPERMANN The follo wing lemma shows that the tensor pro duct of pro jective mo dules is again pro jectiv e. Giv en a gr a ded algebra ∆, we denote by ∆ -pro j the ca tegory of finitely generated pro jective ∆-mo dules, and by ∆ -pr o j gr the category of finitely generated grade d pr o jective ∆-mo dules. 3.3. Lemm a. Given mo dules P ∈ Λ -pro j gr and Q ∈ Γ -pr o j gr , the tensor pr o duct P ⊗ t k Q is a gr ade d pr oje ctive Λ ⊗ t k Γ -mo dule. Pr o of. By Lemma 3.2 we only hav e to co nsider the case P = Λ and Q = Γ. In this case P ⊗ t k Q = Λ ⊗ t k Γ, so the lemma holds.  As the follo wing result shows, the tensor product of morphism spaces is the morphism space o f tensor pr oducts. 3.4. Lemma. Given m o dules M , M ′ ∈ Λ -mo d gr and N , N ′ ∈ Γ -mo d gr , the natur al map grHom Λ ( M , M ′ ) ⊗ k grHom Γ ( N , N ′ ) ✲ grHom Λ ⊗ t k Γ ( M ⊗ t k N , M ′ ⊗ t k N ′ ) is an isomorphism. Pr o of. If M = Λ h a i and N = Γ h b i for some a ∈ A and b ∈ B , then grHom Λ (Λ h a i , M ′ ) ⊗ k grHom Γ (Γ h b i , N ′ ) = M ′ − a ⊗ k N ′ − b = ( M ′ ⊗ t k N ′ ) − a, − b = gr Hom ((Λ ⊗ t k Γ) h a, b i , M ′ ⊗ t k N ′ ) = gr Hom (Λ h a i ⊗ t k Γ h b i ) , M ′ ⊗ t k N ′ ) . Now note that both sides co mm ute with cokernels in the M and N p osition.  Note that given degree a and b mor phisms ϕ : M ✲ M ′ and ψ : N → N ′ , we obtain a degr e e ( a, b )-morphism ϕ ⊗ ψ : M ⊗ N ✲ M ′ ⊗ N ′ by comp osing the maps fr om Lemmas 3.4 and 3 .2. Explicitly , the map is given by ( m ⊗ n ) · ( ϕ ⊗ ψ ) = t h ϕ | n i m · ϕ ⊗ n · ψ (w e think o f a mo dule as a right mo dule over its endomorphism ring ). By apply ing this to the situation M = M ′ and N = N ′ , w e obta in the following re sult, show- ing that the endomorphism ring o f a tenso r pr o duct is the tensor pro duct of the endomorphism ring s. 3.5. Lemma. L et M ∈ Λ -mo d gr and N ∈ Γ -mo d gr . Then End Λ ⊗ t k Γ ( M ⊗ t k N ) = End Λ ( M ) ⊗ t k End Γ ( N ) . As for pro jectiv e resolutions, the b eha vior is also as exp ected. Namely , the following result shows that the tensor pro duct of t wo pro jectiv e reso lutio ns is a gain a pro jectiv e r esolution. 3.6. Lemm a. Given mo dules M ∈ Λ -mo d gr and N ∈ Γ -mo d gr with gr ade d pr oje c- tive r esolutions P : · · · ✲ P i ✲ · · · ✲ P 1 ✲ P 0 ✲ M ✲ 0 , Q : · · · ✲ Q i ✲ · · · ✲ Q 1 ✲ Q 0 ✲ N ✲ 0 , the total c omplex of P ⊗ t k Q is a gr ade d pr oje ctive r esol ution of M ⊗ t k N . Pr o of. By Le mma 3.3 all the terms o f the total complex T ot( P ⊗ t k Q ) of P ⊗ t k Q ar e pro jectiv e. Moreover, since k is a field T ot( P ⊗ t k Q ) is exa ct.  COHOMOLOGY OF TWISTED TENSOR P R ODUCTS 5 W e are now ready to prov e the main result of this section. It shows that the Ext-algebr a of a tenso r pr oduct is the tensor pro duct of the Ex t-algebras. 3.7. Theorem. If M , M ′ ∈ Λ - mod gr and N , N ′ ∈ Γ -mo d gr ar e mo dules, then Ext ∗ Λ ⊗ t k Γ ( M ⊗ t k N , M ⊗ t k N ) = E xt ∗ Λ ( M , M ) ⊗ ˜ t k Ext ∗ Γ ( N , N ) , with ˜ t (( i, a ) , ( j, b )) = ( − 1) ij t h a | b i . Mor e ove r Ext ∗ Λ ⊗ t k Γ ( M ⊗ t k N , M ′ ⊗ t k N ′ ) = E xt ∗ Λ ( M , M ′ ) ⊗ ˜ t k Ext ∗ Γ ( N , N ′ ) as Ext ∗ Λ ( M , M ) ⊗ t k Ext ∗ Γ ( N , N ) − Ext ∗ Λ ( M ′ , M ′ ) ⊗ t k Ext ∗ Γ ( N ′ , N ′ ) bimo dule. Pr o of. Let P and Q b e gra ded pro jective resolutions of M a nd N receptively . Then by Lemma 3.6 T ot( P ⊗ t k Q ) is a pr o jective resolution o f M ⊗ t k N , and therefor e Ext ∗ Λ ⊗ t k Γ ( M ⊗ t k N , M ′ ⊗ t k N ′ ) = H ∗ (Hom Λ ⊗ t k Γ (T o t( P ⊗ t k Q ) , M ′ ⊗ t k N ′ ) = H ∗ (T o t(Hom Λ ⊗ t k Γ ( P ⊗ t k Q ) , M ′ ⊗ t k N ′ ) = H ∗ (T o t(Hom Λ ( P , M ′ ) ⊗ t k Hom Γ ( Q , N ′ ))) = Ext ∗ Λ ( M , M ′ ) ⊗ ˜ t k Ext ∗ Γ ( N , N ′ ) . Here the third eq ualit y holds b y Lemma 3.4, whereas the final o ne holds since k is a field. The m ultiplication is induced b y the multiplication o f mor phisms in Lemma 3.5, with the additional signs needed b e c ause of the signs added when passing from the double complex to its total complex .  W e end this section with the following re s ult, which w as s ho wn in [14] for un- t wisted tensor pro ducts (in whic h case we may forget about the g rading). It will help us find upp er b ounds fo r the representation dimensio n of twisted tensor pro d- ucts. Given an alge br a ∆, we denote by gld ∆ its globa l dimension. 3.8. Prop ositi o n. L et M ∈ Λ -mo d gr and N ∈ Γ - mo d gr b e gr ade d mo dules, su ch that M gener ates and c o gener ates Λ -mo d , and such that N gener ates and c o gen- er ates Γ -mo d . Th en M ⊗ t k N is a gener ator-c o gener ator of Λ ⊗ t k Γ -mo d , and gld End Λ ⊗ t k Γ ( M ⊗ t k N ) = g ld End Λ ( M ) + gld End Γ ( N ) . 4. Tensor products of bimodules Throughout this se c tion, we k eep the notation from the last section. That is, we fix tw o ab elian groups A and B , tog ether with an A -g raded algebra Λ and a B - graded alg ebra Γ. Mo reo ver, w e fix a homomorphism t : A ⊗ Z B ✲ k × of ab elian groups. Giv en an a lg ebra ∆, we denote by ∆ e its env eloping algebra ∆ ⊗ k ∆ op . Note that if ∆ is G -graded, where G is some ab elian group, then so is ∆ e , and ∆ is a g raded ∆ e -mo dule. 4.1. Definition/Co n s truct ion. Given mo dules X ∈ Λ e -mo d gr and Y ∈ Γ e -mo d gr , the tensor pr oduct X ⊗ k Y b ecomes a graded (Λ ⊗ t k Γ) e -mo dule by defining ( λ ⊗ γ )( x ⊗ y )( λ ′ ⊗ γ ′ ) def = t h x | γ i t h λ ′ | y i t h λ ′ | γ i λxλ ′ ⊗ γ y γ ′ . W e denote this bimo dule by X ⊗ t k Y . 4.2. Re m ark . In gener al the graded (Λ ⊗ t k Γ) e -mo dules X h a i ⊗ t k Y h b i a nd ( X ⊗ t k Y ) h a, b i are not isomorphic. The fo llo wing res ults are analog ue s of Le mma s 3.3, 3.4 and 3.6. W e pr ove only the fir st result, as the pro ofs of the other tw o results are mo r e or le ss the same as those of Lemma s 3.4 and 3.6. 6 PETTER ANDREAS BERGH AND STEFFEN OPPERMANN 4.3. Lem m a. Given mo dules X ∈ Λ e -pro j gr and Y ∈ Γ e -pro j gr , the tensor pr o duct X ⊗ t k Y is a gr ade d pr oje ctive (Λ ⊗ t k Γ) e -mo dule. Pr o of. It suffices to show that Λ e h a i ⊗ t k Γ e h b i is g raded pro jective for any a ∈ A and b ∈ B . This can b e seen by noting that the map (Λ ⊗ t k Γ) e h a, b i ✲ Λ e h a i ⊗ t k Γ e h b i ( l ⊗ g ) ⊗ ( l ′ ⊗ g ′ ) ✲ t h l ′ | g i t h a | g i t h l ′ | b i ( l ⊗ l ′ ) ⊗ ( g ⊗ g ′ ) is an iso morphism of graded (Λ ⊗ t k Γ) e -mo dules.  4.4. Lemma. Gi ven mo dules X , X ′ ∈ Λ e -mo d gr and Y , Y ′ ∈ Γ e -mo d gr , the natur al map grHom Λ e ( X, X ′ ) ⊗ k grHom Γ e ( Y , Y ′ ) ✲ grHom (Λ ⊗ t k Γ) e ( X ⊗ t k Y , X ′ ⊗ t k Y ′ ) is an isomophism. 4.5. Lemm a. Given mo dules X ∈ Λ e -mo d gr and Y ∈ Γ e -mo d gr with gr ade d pr o- je ct ive bimo dule r esolutions P : · · · ✲ P i ✲ · · · ✲ P 1 ✲ P 0 ✲ X ✲ 0 , Q : · · · ✲ Q i ✲ · · · ✲ Q 1 ✲ Q 0 ✲ Y ✲ 0 , the total c omplex of P ⊗ t k Q is a gr ade d pr oje ctive bimo dule r esolution of X ⊗ t k Y . Now note that for a fixed b ∈ B the map t induces a morphism t h−| b i : A ✲ k × (and similarly for a fixed a ∈ A ). With this notation, w e make the following observ ation. 4.6. Lemma. L et a ′ ∈ ∩ b ∈ B Ker t h−| b i ≤ A and b ′ ∈ ∩ a ∈ A Ker t h a |−i ≤ B . Then the map Λ h a ′ i ⊗ t k Γ h b ′ i ✲ (Λ ⊗ t k Γ) h a, b i λ ⊗ γ ✲ λ ⊗ γ is an isomorphism of gr ade d (Λ ⊗ t k Γ) e -mo dules. Using the a bov e notation, we now pr o ve the main re sult of this s ection. It shows that Ho chsc hild co homology commutes with twisted tensor pro ducts, provided we only consider the g raded parts cor respo nding to the subgroups ∩ b ∈ B Ker t h−| b i ≤ A and ∩ a ∈ A Ker t h a |−i ≤ B . 4.7. Theorem. L et A ′ = ∩ b ∈ B Ker t h−| b i ≤ A and B ′ = ∩ a ∈ A Ker t h a |−i ≤ B . The n ther e is an isomorphism HH ∗ ,A ′ (Λ) ⊗ ( − 1) ∗∗ k HH ∗ ,B ′ (Γ) ✲ HH ∗ ,A ′ ⊕ B ′ (Λ ⊗ t k Γ) , wher e ( − 1) ∗∗ denotes the morphism mapping (( i, a ′ ) , ( j, b ′ )) to ( − 1) ij . Pr o of. Let P and Q be graded bimo dule pro jective r esolutions of Λ a nd Γ , resp ec- tively . Given a ∈ A and b ∈ B , the same ar gumen ts as in the pro of of Theorem 3 .7 give HH ∗ ,a,b (Λ ⊗ t k Γ) = H ∗ (grHom (Λ ⊗ t k Γ) e (T o t( P ⊗ t k Q , (Λ ⊗ t k Γ) h a, b i ))) = H ∗ (T o t(Hom (Λ ⊗ t k Γ) e ( P ⊗ t k Q , (Λ ⊗ t k Γ) h a, b i ))) and H ∗ (T o t(Hom (Λ ⊗ t k Γ) e ( P ⊗ t k Q , Λ h a i ⊗ t k Γ h b i ))) = H ∗ (T o t(Hom Λ e ( P , Λ h a i ) ⊗ t k Hom Γ e ( Q , Γ))) = HH ∗ ,a (Λ) ⊗ ˜ t k HH ∗ ,b (Γ) , COHOMOLOGY OF TWISTED TENSOR P R ODUCTS 7 where ˜ t = ( − 1) ∗∗ · t as in Theor em 3.7. Now if a ∈ A ′ and b ∈ B ′ , then from Lemma 4.6 we see that we may identif y H ∗ (T o t(Hom (Λ ⊗ t k Γ) e ( P ⊗ t k Q , (Λ ⊗ t k Γ) h a, b i ))) with H ∗ (T o t(Hom (Λ ⊗ t k Γ) e ( P ⊗ t k Q , Λ h a i ⊗ t k Γ h b i ))) . Finally , note that HH ∗ ,A ′ (Λ) ⊗ ˜ t k HH ∗ ,B ′ (Γ) = HH ∗ ,A ′ (Λ) ⊗ ( − 1) ∗∗ k HH ∗ ,B ′ (Γ), since all degrees o ccurring ar e in the kernel of t .  W e end this s ection with the following corollar y to Theorem 4.7. It shows that, given cer tain conditio ns, if Λ and Γ satisfy Fg , then so do es Λ ⊗ t k Γ. 4.8. Corollary . With the same notation as in The or em 4.7, assume Λ and Γ satisfy Fg with r esp e ct to t hei r even Ho ch schild c ohomolgy rings HH 2 ∗ (Λ) and HH 2 ∗ (Γ) . Mor e over, supp ose [ A : A ′ ] and [ B : B ′ ] ar e finite, and that Λ / Rad Λ and Γ / Rad Γ ar e sep ar able ove r k . The n Λ ⊗ t k Γ satisfies Fg with r esp e ct its even Ho chschild c oh omolgy ring HH 2 ∗ (Λ ⊗ t k Γ) . Pr o of. Since [ A : A ′ ] is finite, the alg ebra HH 2 ∗ ,A (Λ) is a finit ely g ener- ated module o ver HH 2 ∗ ,A ′ (Λ). Therefore, since Λ satisfies Fg , we see that Ext ∗ Λ (Λ / Rad Λ , Λ / Rad Λ) is finitely generated ov er HH 2 ∗ ,A ′ (Λ). The same argu- men ts a pply to Γ, hence Ext ∗ Λ (Λ / Rad Λ , Λ / Rad Λ) ⊗ t k Ext ∗ Γ (Γ / Rad Γ , Γ / Rad Γ) is finitely genera ted ov er HH 2 ∗ ,A ′ (Λ) ⊗ k HH 2 ∗ ,B ′ (Γ). Then by Theorems 3.7 and 4.7, we se e that Ext ∗ Λ ⊗ t k Γ (Λ / Rad Λ ⊗ t k Γ / Rad Γ , Λ / Rad Λ ⊗ t k Γ / Rad Γ) m ust b e a finitely generated HH 2 ∗ ,A ′ ⊕ B ′ (Λ ⊗ t k Γ)-mo dule. Finally , since Λ / Rad Λ and Γ / Rad Γ ar e separ able ov er k , the equality Λ / Rad Λ ⊗ t k Γ / Rad Γ = (Λ ⊗ t k Γ) / Rad(Λ ⊗ t k Γ) holds. The c laim now follows fro m Remark s 2.5.  5. Qua ntum complete intersections W e now apply the cohomo logy theory of twisted tenso r pr oducts to the cla ss of finite dimensional a lgebras known as quantum c omple te interse ctions . Throughout this sec tion, fix integers n ≥ 1 and a 1 , . . . , a n ≥ 2, together with a nonze r o element q ij ∈ k for every 1 ≤ i < j ≤ n . W e define the alge bra Λ by Λ def = k h x 1 , . . . , x n i / ( x a i i , x j x i − q ij x i x j ) , a co dimension n quantum complete intersection in its mos t general form. This is a selfinjective algebra o f dimension Q a i . W e shall deter mine prec is ely when such an algebra satisfies Fg , and co nsequen tly obtain a lower bo und for its representation dimension. Note that Λ is Z n graded b y | x i | def = (0 , . . . , 1 , . . . 0 ), the i th unit vector. In particular, we use the Z -gra ding | x | = 1 for the sp ecial ca s e of a codimensio n one quantum co mplete intersection k [ x ] / ( x a ). The following observ ation allows us to study the co homology inductively , starting with the well known case k [ x ] / ( x a ). 5.1. Lemma. L et Λ ′ b e the su b algebr a of Λ gener ate d by x 1 , . . . , x n − 1 . Then Λ = Λ ′ ⊗ t k k [ x n ] / ( x a n n ) , wher e t h d 1 ,...,d n − 1 | d n i def = Q n − 1 i =1 q d i d n in . 8 PETTER ANDREAS BERGH AND STEFFEN OPPERMANN As for quantum complete in tersections of co dimension one, that is, truncated po lynomial algebra s, their c o homology is well known. W e record this in the follow- ing lemma. 5.2. Lemma. If Γ = k [ x ] / ( x a ) , then (1) HH 2 ∗ (Γ) = k [ x, z ] / ( x a , ax a − 1 z ) , (2) Ext ∗ Γ ( k , k ) =  k [ y , z ] / ( y 2 = z ) if a = 2 k [ y , z ] / ( y 2 ) if a 6 = 2 , with | x | = 0 , | y | = 1 and | z | = 2 . I n p articular, the algebr a Γ satisfies Fg with r esp e ct t o its even H o chschild c ohomol o gy ring. Pr o of. The first part is [9, Theor em 3 .2], the second part can b e r ead off directly from the pro jective reso lution.  Using this lemma a nd Theo rem 3.7, w e obtain the fo llowing result on the E xt- algebra of the s imple module of a qua n tum co mplete in tersection. 5.3. Theorem. The Ext -algebr a of k is given by Ext ∗ Λ ( k , k ) = k h y 1 , . . . y n , z 1 , . . . z n i / I , wher e I is t he ide al in k h y 1 , . . . y n , z 1 , . . . z n i define d by the r elations           y i z i − z i y i y j y i + q ij y i y j i < j y j z i − q 2 ij z i y j i < j z j y i − q 2 ij z i y j i < j z j z i − q 4 ij z i z j i < j y 2 i = z i a i = 2 y 2 i a i 6 = 2           5.4. Remark. Lemma 5.2 shows that if Γ a nd ∆ are ar bitrary algebra s, then the algebra HH ∗ (Γ) ⊗ e t k HH ∗ (∆) do es not in general embed into HH ∗ (Γ ⊗ t k ∆). Namely , the latter is always graded co mm utativ e, wherea s HH ∗ (Γ) ⊗ e t k HH ∗ (∆) need not b e. W e are now rea dy to c haracterize pr e cisely when a quantum complete intersection satisfies Fg . 5.5. Theorem. The fol lowing ar e e quivalent. (1) Λ satisfies Fg , (2) Λ satisfies Fg with r esp e ct to its even Ho chschild c ohomolo gy ring HH 2 ∗ (Λ) , (3) al l the c ommutators q ij ar e r o ots of un ity. Pr o of. The implication (2 ) ⇒ (1) is obvious, and the implication (3) ⇒ (2) follows from Coro llary 4.8. T o show (1) ⇒ (3), w e as s ume that (1) holds but not (3), so there are i and j such that q ij is not a ro ot of unity . By (1), the Ext-alg e bra of k is finitely generated as a mo dule over its cent er, hence so is ev ery quotient of this ring. By factoring out all y k , z k with k 6∈ { i , j } and { y k | k ∈ { i, j } a nd y 2 k = 0 } , we obtain a r ing o f the for m k h r, s i / ( sr − q rs ), where q is not a ro ot o f unity . The center of this ring is trivial, hence the ring cannot b e finitely gener ated ov er its center, a co n tradiction.  As a corollar y , we obta in a low er b ound for the r e presen tation dimension of a quantum complete intersection. Recall that the representation dimension of a finite dimensional alge br a ∆ is defined as rep dim ∆ def = inf { g ld End ∆ ( M ) } , where the infim um is taken ov er all the finitely g enerated ∆-mo dules which genera te and cogener a te ∆ -mo d. COHOMOLOGY OF TWISTED TENSOR P R ODUCTS 9 5.6. Coroll ary . Define the inte ger c ≥ 0 by c def = max { ca rd I | I ⊆ { 1 , . . . , n } and q ij is a r o ot of unity ∀ i, j ∈ I , i < j } . Then rep dim Λ ≥ c + 1 . In p art icular, if al l the c ommutators q ij ar e r o ots of unity, then rep dim Λ ≥ n + 1 . In order to prov e this result w e need to r ecall so me no tions. Let ∆ b e an alg ebra, and let M be a finitely gener ated ∆-mo dule with minimal pro jective r esolution · · · ✲ P 2 ✲ P 1 ✲ P 0 ✲ M ✲ 0 , say . The c omplexity of M , denoted c x M , is defined as cx M def = inf { t ∈ N ∪ { 0 } | ∃ r ∈ R s uc h that dim k P n ≤ r n t − 1 for n ≫ 0 } . Now let V be a p ositively gra ded k -vector space of finite t yp e, i.e. dim k V n < ∞ for all n . The r ate of gr owth of V , denoted γ ( V ), is defined as γ ( V ) def = inf { t ∈ N ∪ { 0 } | ∃ r ∈ R s uc h that dim k V n ≤ r n t − 1 for n ≫ 0 } . It is w ell known that the complexity o f a mo dule M equa ls γ (Ext ∗ ∆ ( M , ∆ / Rad ∆)). Now supp ose ∆ is selfinjective, and denote by ∆ -mo d the stable module categ ory of ∆ -mo d, that is, the categor y obtained from ∆ -mo d by factoring o ut all mor phisms which factor thro ugh a pro jective mo dule. This is a triangulated categ o ry , and we denote by dim(∆ -mo d ) its dimensio n, as de fined in [11]. Pr o of of Cor ol lary 5.6. Cho ose a subset I of { 1 , . . . , n } r ealizing the maximum in the definition of the integer c , a nd let Λ ′ be the subalge br a of Λ generated b y the x i with i ∈ I . By Theo rem 5.5 the algebra Λ ′ satisfies Fg , and so from [4, Theorem 3.1] w e see that dim (Λ ′ -mo d ) ≥ cx Λ ′ k − 1. Moreov er, b y Theorem 5.3 the co mplexit y of k as a Λ ′ -mo dule equals ca rd I , giving dim(Λ ′ -mo d ) ≥ c a rd I − 1 . The forgetful functor Λ -mo d ✲ Λ ′ -mo d is exa ct, dense, and ma ps pro jec- tive Λ- modules to pro jective Λ ′ -mo dules. Ther efore it induces a dense triangle functor Λ -mo d ✲ Λ ′ -mo d , a nd so from [11, Lemma 3.4] we o btain the ine q ual- it y dim(Λ -mo d) ≥ dim(Λ ′ -mo d) . Finally , by [12, Pro position 3.7] the inequality rep dim Λ ≥ dim(Λ -mo d) + 2 holds , and the pro of is complete.  5.7. Rem ark. By [7, Theo rem 3 .2] the inequality re pdim Λ ≤ 2 n always holds . It was s ho wn in [10] that the representation dimension of the truncated p olyno- mial algebra k [ x, y ] / ( x 2 , y a ) is thre e . Using their constr uction and exactly the same pro of, one ca n show that the quantum complete in tersection Γ = k h x, y i / ( y x − q xy , x 2 , y a ) has a g enerator-cog enerator M which is graded with gld End Γ ( M ) = 3 . Moreov er, for a quantum e x terior algebra Γ on n v ariables (that is, a co dimen- sion n quantum complete intersection where all the defining exp onen ts ar e 2), the global dimensio n of the endomorphism ring of the g raded generator -cogenerator ⊕ Γ / (Rad Γ) i is n + 1 (cf. [1]). Using this and Pr opos itio n 3.8, we obtain the fol- lowing improvemen t of Remark 5.7. 5.8. Theorem. If h = card { i | a i = 2 } , then rep dim Λ ≤  2 n − h if h ≤ n / 2 2 n − h + 1 if h > n / 2 . Pr o of. In the first case deco mp ose the alg ebra into h parts of the form k h x, y i / ( y x − q xy , x 2 , y a ), a nd n − 2 h par ts o f the form k [ x ] / ( x a ). Adding up the g lobal dimensions of the endomorphism rings of the graded Auslander gene r ators (which we may do by Pro position 3.8), we obtain h · 3 + ( n − 2 h ) · 2 = 2 n − h . 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