Ext-symmetry over quantum complete intersections

Ext -SYMMETR Y O VER QUANTUM COMPLETE INTERSECTIONS PETTER ANDREAS BERGH Abstract. W e sho w that symmetry in the v anishing of cohomology holds for graded modules o ver quan tum complete inte rsections. Moreov er , symmetry holds f or all modules if the algebra is symmetric. 1. Introduction F or which algebra s do symmetry in the v anishing of cohomolo gy hold? That is, given an alg ebra Λ, do es the implication Ext i Λ ( M , N ) = 0 for i ≫ 0 ⇒ Ext i Λ ( N , M ) = 0 for i ≫ 0 hold for finitely gene r ated mo dules M and N ? As shown in [AvB], this implication holds for finitely generated mo dules over commutativ e lo cal complete in tersec tions. The pro o f involv es the theory of certain supp or t v arieties attached to each pair of finitely genera ted modules over s uch a ring A . Namely , denote b y c the co dimension of A a nd by K the algebra ic clo sure of its residue field. A cone V( M , N ) in K c is asso ciated to every pa ir ( M , N ) of finitely generated A -mo dules, with the following prop erties: V( M , N ) = { 0 } ⇔ Ext i A ( M , N ) = 0 for i ≫ 0 , V( M , N ) = V ( M , M ) ∩ V( N , N ) . The symmetry in the v anis hing of cohomology follo ws immediately from these prop- erties. As sho wn in [Mo r, Corolla ry 4 .8], group algebras of finite groups provide another class of ex a mples wher e Ex t-symmetry holds. W e sho w in this pap er that Ext-symmetry holds for a ll gr ade d mo dules over quantum complete int erse c tions, provided all the defining co mmu tator s ar e ro o ts of unit y . W e a lso s how that, if such an alg ebra is symmetric, that is, if it is is omorphic as a bimo dule to its own dual, then symmetry holds fo r a ll mo dules. 2. Quantum complete intersections Quantum complete in tersec tio ns are nonco mmut ative analogues of trunca ted po lynomial rings. Included in this clas s of algebr a s are e xterior a lgebras and finite dimensional c o mplete intersections o f the form k [ X 1 , . . . , X c ] / ( X a 1 1 , . . . , X a c c ). Fix a field k , let c ≥ 1 b e a n integer, and let q = ( q ij ) b e a c × c commutation matrix with entries in k . That is , the diag onal ent ries q ii are all 1, and q ij q j i = 1 for all i, j . F ur thermore, let a c = ( a 1 , . . . , a c ) b e an ordered sequence o f c integers with a i ≥ 2. The qu antum c omplete 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 ) . 2000 Mathematics Subject Classific ation. 16E30, 16S80, 16U80, 81R50. Key wor ds and phr ases. Quantum complete intersect ions, v anishing of cohomology , symmetry . The author wa s supp orted by NFR Storforsk gran t no. 167130. 1 2 PETTER ANDREAS BERGH This is a finite dimensional selfinjective a lgebra of dimension Q c i =1 a i . The imag e of X i in this quotient will b e denoted by x i . W e shall consider A a c q as a Z c -graded algebra, in which the deg r ee of the gener a tor x i is the i th unit vector (0 , . . . , 1 , . . . 0). The categ ory of finitely g enerated left A a c q -mo dules (r esp ectively , graded mo dules) is denoted b y mo d A a c q (resp ectively , grmo d A a c q ); all mo dules are assumed to be finitely generated. A quantum complete intersection is built from truncated p olynomial rings using certain tensor pr o ducts. W e rec all here the basic s, details ca n b e found in [BeO]. Let A and B b e ab elian gro ups, and let Λ a nd Γ b e an A -g r aded and a B -gra ded k -alge br a, resp ectively . F urther mo re, let t : A ⊗ Z B → k \ { 0 } b e a homomor phism of groups, where k \ { 0 } is multiplicativ e. If a a nd b ar e elemen ts of A and B , resp ectively , then we write t ( a | b ) instea d o f t ( a ⊗ b ). Moreov er, given homog eneous elements λ ∈ Λ and γ ∈ Γ o f degrees d a nd d ′ , resp ec tively , w e write t ( λ | γ ) instead of t ( d | d ′ ). With these data, we can now define a new algebra Λ ⊗ t k Γ, the t wiste d tensor pr o duct of Λ and Γ with r esp ect to the homomor phism t . The underlying k -vector space of Λ ⊗ t k Γ is Λ ⊗ k Γ, and multiplication is given by ( λ 1 ⊗ γ 1 ) · ( λ 2 ⊗ γ 2 ) def = t ( λ 2 | γ 1 ) λ 1 λ 2 ⊗ γ 1 γ 2 for homog eneous element s λ 1 , λ 2 ∈ Λ and γ 1 , γ 2 ∈ Γ. This alg ebra is A ⊕ B - g raded; if a ∈ A and b ∈ B , then (Λ ⊗ t k Γ) ( a,b ) = Λ a ⊗ k Γ b . Given a graded Λ-mo dule M and a graded Γ-mo dule N , their tenso r product M ⊗ k N beco mes a graded Λ ⊗ t k Γ-mo dule by defining ( λ ⊗ γ ) · ( m ⊗ n ) def = t ( m | γ ) λm ⊗ γ n for homo geneous elements λ ∈ Λ , γ ∈ Γ , m ∈ M , n ∈ N . This mo dule is denoted M ⊗ t k N . As a b ove, this mo dule is A ⊕ B -gra ded; if a ∈ A a nd b ∈ B , then ( M ⊗ t k N ) ( a,b ) = M a ⊗ k N b . The purp ose of this pap er is to study symmetry in the v anishing o f cohomolog y ov er quantum complete intersections. W e ther efore end this section with the fol- lowing tw o results, the first of which shows that quantum c omplete intersections are made up of twisted tensor pro ducts. Lemma 2.1 . Le t A a c q b e a quantu m c omplete int erse ction with c ≥ 2 , let I b e a pr op er nonempty subset of { 1 , . . . , c } of or der c 1 , and let c 2 = c − c 1 . F urthermor e, let A a c 1 q 1 and A a c 2 q 2 b e the qu antum c omplete int erse ctions gener ate d by { x i } i ∈ I and { x i } i ∈{ 1 ,...,c }\ I , r esp e ctively. The n ther e is an isomorphism A a c q ≃ A a c 1 q 1 ⊗ t k A a c 2 q 2 for some homomorphism Z c 1 ⊗ Z Z c 2 → k \ { 0 } . Pr o of. By re-indexing the g enerator s , we may a s sume that I = { 1 , . . . , c 1 } . Thus A a c 1 q 1 is the subalgebra of A a c q generated by x 1 , . . . , x c 1 , whe r eas A a c 2 q 2 is the subal- gebra g enerated by the c 2 elements x c 1 +1 , . . . , x c . Consider the map Z c 1 × Z c 2 t ′ − → k \ { 0 } defined by (( d 1 , . . . , d c 1 ) , ( d c 1 +1 , . . . , d c )) 7→ c Y j = c 1 +1 j − 1 Y i =1 q d i d j j i . Given s e quences d 1 , d ′ 1 ∈ Z c 1 and d 2 , d ′ 2 ∈ Z c 2 , the e q ualities t ′ ( d 1 + d ′ 1 , d 2 ) = t ′ ( d 1 , d 2 ) t ′ ( d ′ 1 , d 2 ) t ′ ( d 1 , d 2 + d ′ 2 ) = t ′ ( d 1 , d 2 ) t ′ ( d 1 , d ′ 2 ) hold, hence t ′ induces a homomorphism Z c 1 ⊗ Z Z c 2 t − → k \ { 0 } of ab elian gro ups. Now let i and j b e elements of I , a nd consider the elements x i ⊗ 1 and x j ⊗ 1 in Ext-SYMMETR Y OVER QUANTUM COMPLETE INTERSECTIONS 3 the twisted tenso r pro duct A a c 1 q 1 ⊗ t k A a c 2 q 2 . The degrees of x i ∈ A a c 1 q 1 and x j ∈ A a c 1 q 1 are the unit vectors e i and e j in Z c 1 , r esp ectively , whereas the degree of 1 ∈ A a c 2 q 2 is the zero vector in Z c 2 . Ther efore ( x i ⊗ 1)( x j ⊗ 1) = t ( e j | 0) x i x j ⊗ 1 = x i x j ⊗ 1 = q ij x j x i ⊗ 1 = q ij t ( e i | 0) x j x i ⊗ 1 = q ij ( x j ⊗ 1)( x i ⊗ 1) , and similar ly (1 ⊗ x i )(1 ⊗ x j ) = q ij (1 ⊗ x j )(1 ⊗ x i ) whenever i a nd j ar e b oth in { c 1 + 1 , . . . , c } . If i ∈ I and j ∈ { c 1 + 1 , . . . , c } , then the degr ee of x j in A a c 2 q 2 is the unit vector e j in Z c 2 . In this ca se we obtain the equalities ( x i ⊗ 1)(1 ⊗ x j ) = t (0 | 0 ) x i ⊗ x j = x i ⊗ x j and (1 ⊗ x j )( x i ⊗ 1) = t ( e i | e j ) x i ⊗ x j = q j i x i ⊗ x j , and cons equently A a c q is iso morphic to A a c 1 q 1 ⊗ t k A a c 2 q 2 .  The final r esult of this section shows that the cohomo logy of a twisted tensor pro duct o f g raded mo dules is the tensor pro duct of the resp ective cohomolo gies. Theorem 2. 2. [BeO, Theorem 3.7] L et A and B b e ab elian gr oups, and let Λ and Γ b e an A - gr ade d and a B -gr ade d k -algebr a, r esp e ctively. F urthermor e, let t : A ⊗ Z B → k \ { 0 } b e a homomorphism, and let M 1 , M 2 ∈ gr mo d Λ and N 1 , N 2 ∈ grmo d Γ b e gr ade d mo dules. Then ther e is an isomorphism Ext ∗ Λ ⊗ t k Γ ( M 1 ⊗ t k N 1 , M 2 ⊗ t k N 2 ) ≃ E xt ∗ Λ ( M 1 , M 2 ) ⊗ k Ext ∗ Γ ( N 1 , N 2 ) of gr ade d ve ctor sp ac es. 3. E xt -symmetr y In this section, we prove that symmetry ho lds in the v a nishing of coho mology for graded mo dules ov er a quantum co mplete intersection, provided the commutators are a ll ro o ts of unity . The idea of the pro o f is to use twisted tensor pro ducts to pass to a bigger quantum complete intersection where E xt-symmetry ho lds, namely a symmetric one. In order to do this, we must determine prec is ely when a quantum complete intersection is symmetric. Thr o ughout this s ection, we fix a field k . Recall that a finite dimensional k - a lgebra Λ is F ro b enius if Λ Λ and D (Λ Λ ) are isomorphic as left Λ-mo dules, where D denotes the usua l k - dua l Hom k ( − , k ). If Λ and D (Λ) a re isomo r phic as bimo dules, then Λ is symmetric . Now supp ose Λ is F r ob enius, and le t φ : Λ Λ → D (Λ Λ ) b e an isomor phism. Let y ∈ Λ b e any element, and consider the linear functional φ (1) · y ∈ D (Λ), i.e. the k -linear map Λ → k defined by λ 7→ φ (1)( y λ ). Since φ is surjective, there is an element x ∈ Λ having the pr op erty tha t φ ( x ) = φ (1) · y , giving x · φ (1) = φ (1) · y since φ is a ma p of left Λ- mo dules. It is no t difficult to show that the map y 7→ x defines a k -a lgebra automorphism on Λ, a nd its inv erse ν is called the Nakayama automorphism of Λ (with resp ect to φ ). Thu s ν is defined by φ (1)( λx ) = φ (1)( ν ( x ) λ ) for all λ ∈ Λ. This automorphism is unique up to a n inner automor phism; if φ ′ : Λ Λ → D (Λ Λ ) is another isomo rphism of left mo dules yielding a Nak ayama automorphis m ν ′ , then there exists a n inv ertible element z ∈ Λ such that ν = z ν ′ z − 1 . Note that φ is a n isomorphism b etw een the bimo dules 1 Λ ν − 1 and D (Λ). Moreov er, no te that Λ is symmetric if a nd o nly if the Nak ay ama automor phism is the identit y . As D (Λ Λ ) is a n injective left Λ-mo dule, we see that a F rob enuis algebr a is al- wa ys left selfinjectiv e, but in fact the definition is left-right sy mmetric. F o r if 4 PETTER ANDREAS BERGH φ : Λ Λ → D (Λ Λ ) is an isomo rphism of left Λ-mo dules, we can dualize and get an isomorphism D ( φ ) : D 2 (Λ Λ ) → D ( Λ Λ) of rig ht mo dules. Comp osing with the nat- ural is omorphism Λ Λ ≃ D 2 (Λ Λ ), we obtain an isomorphis m Λ Λ → D ( Λ Λ) of rig ht Λ-mo dules. A finite dimensional lo cal algebr a is F ro b enius if and only if it is selfinjective. In particular, a qua nt um complete intersection A a c q is F rob enius, and the fo llowing result shows tha t ther e is a particula rly nice Nak ayama a utomorphism. Lemma 3.1. A quantum c omplete int erse ction A a c q is F r ob enius, with a Nakayama automorphism A a c q ν − → A a c q given by x w 7→ c Y i =1 q a i − 1 iw ! x w for 1 ≤ w ≤ c . Pr o of. Consider the map A a c q φ − → D ( A a c q ) defined by φ (1) : X i 1 ,...,i c α i 1 ,...,i c x i c c · · · x i 1 1 7→ α a 1 − 1 ,...,a c − 1 . That is, the element φ (1) maps a n e le ment y ∈ A a c q to the co efficient o f the so cle element x a c − 1 c · · · x a 1 − 1 1 in y . This is an isomo r phism of left A a c q -mo dules. By definition, a Nak ay ama a utomorphism A a c q ν − → A a c q has the pr op erty that y · φ (1) = φ (1) · ν ( y ) for all y ∈ A a c q . The g iven map satisfies this pro pe r ty .  Thu s qua nt um complete intersections ar e not symmetric in general. How ever, the following r esult shows that fo r every such algebr a, there exists a s y mmetric quantum co mplete int erse c tion “ex tending” the given one. Prop ositi on 3.2. Gi ven any quantu m c omplete interse ction A a c q , the r e exists a symmetric qu antum c omplete interse ction A a 2 c q ′ with the fol lowing pr op erties: (i) The sub algebr a of A a 2 c q ′ gener ate d by x 1 , . . . , x c is isomorphic to A a c q . (ii) The c ommutators of A a 2 c q ′ ar e the c ommutators of A a c q . Pr o of. Supp ose A a c q is g iven b y the sequence a c = ( a 1 , . . . , a c ) and the commutation matrix q =    q 11 · · · q 1 c . . . . . . . . . q c 1 · · · q cc    in which q ij q j i = 1 and q ii = 1 for all i , j . Define a sequence a 2 c and a 2 c × 2 c commutation matrix q ′ by a 2 c def = ( a 1 , . . . , a c , a 1 , . . . , a c ) q ′ def =  q q T q T q  where Q T denotes the transp o se of a matr ix. Then A a c q is isomorphic to the sub- algebra of the quantum complete int erse ction A a 2 c q ′ generated by x 1 , . . . , x c . The commutators q ′ uv in A a 2 c q ′ satisfy q ′ uv =        q uv if u ≤ c, v ≤ c q vi if u = c + i, v ≤ c q j u if u ≤ c, v = c + j q ij if u = c + i, v = c + j, Ext-SYMMETR Y OVER QUANTUM COMPLETE INTERSECTIONS 5 and so if i and w are integers with 1 ≤ i ≤ c and 1 ≤ w ≤ 2 c , we see that q ′ iw q ′ ( c + i ) w = 1. This g ives 2 c Y i =1 ( q ′ iw ) a i − 1 = c Y i =1 ( q ′ iw ) a i − 1 ! c Y i =1 ( q ′ ( c + i ) w ) a i − 1 ! = 1 for 1 ≤ w ≤ 2 c , and ther efore A a 2 c q ′ is symmetr ic b y Lemma 3 .1.  As ment ioned, the cr ucial step when proving Ext-symmetry for gr aded mo dules ov er a quantum complete intersection is the passing to a big g er symmetric alge - bra, as in the previo us r esult. Namely , the next result s hows that Ext-s ymmetry holds for al l mo dules over a symmetric quantum co mplete intersection whose com- m utator s are all ro ots of unity . Recall first the following; details can b e found in [SnS] and [Sol]. Let Λ a finite dimensional k -algebra, a nd denote the env eloping algebra Λ ⊗ k Λ op of Λ by Λ e . F o r n ≥ 0, the n th Ho chschild c ohomolo gy group of Λ, denoted HH n (Λ), is the vector spa ce Ext n Λ e (Λ , Λ). The gr aded vector space HH ∗ (Λ) = E xt ∗ Λ e (Λ , Λ) is a graded-co mmu tative ring with Y oneda pro duct, a nd fo r every M ∈ mo d Λ the tenso r pro duct − ⊗ Λ M induces a homomo rphism HH ∗ (Λ) ϕ M − − → E xt ∗ Λ ( M , M ) of g raded k -a lgebras . If N ∈ mo d Λ is another module and η ∈ HH ∗ (Λ) a nd θ ∈ Ext ∗ Λ ( M , N ) are homogeneous elements, then the relation ϕ N ( η ) ◦ θ = ( − 1) | η | | θ | θ ◦ ϕ M ( η ) ho lds, where “ ◦ ” deno tes the Y oneda pro duct. In the terminology used in [Ber] and [BIKO ], the Ho chsc hild cohomolo gy r ing HH ∗ (Λ) acts c ent ra lly on the bo unded derived categor y D b (Λ) o f mo d Λ . Theorem 3. 3. L et A a c q b e a symmetric quantum c omplete int erse ct ion whose c om- mutators q ij ar e al l ro ots of u nity. Then for al l mo dules M , N ∈ mod A a c q , the fol lowing ar e e qu ivalent: (i) Ext i A a c q ( M , N ) = 0 for i ≫ 0 . (ii) Ext i A a c q ( M , N ) = 0 for i > 0 . (iii) Ext i A a c q ( N , M ) = 0 for i ≫ 0 . (iv) Ex t i A a c q ( N , M ) = 0 for i > 0 . Pr o of. By [BeO, Theor em 5.5 ], the Ho chsc hild cohomo logy ring HH ∗ ( A a c q ) is No e - therian, and Ext i A a c q ( k , k ) is a finitely generated HH ∗ ( A a c q )-mo dule. The result now follows from [Be r, Theo rem 4 .2 ].  Examples. (i) Let A b e the exterior a lgebra on a c -dimensio nal k -vector spa ce, i.e. A = k h X 1 , . . . , X c i / ( X 2 i , { X i X j + X j X i } i 6 = j ) . F r om Le mma 3.1, we see that when a pplying the Nak ay ama auto morphism to a generator x i , then the result is the element ( − 1) c − 1 x i . Therefore A is s ymmetric precisely when c is an o dd num b er. Consequently , s ymmetry in the v anishing of cohomolog y holds for mo dules ov er exterior algebras on o dd-dimensio nal vector spaces (cf. [Mo r, Co rollar y 4 .9]). (ii) Fix integers c and a , b oth at le a st tw o, and let q b e an ele men t in k with the prop erty that q a − 1 = 1. F urthermore, let a c be the c -tuple ( a, . . . , a ), let q b e the commutation matrix        1 q · · · · · · q q − 1 1 q · · · q . . . . . . . . . . . . . . . q − 1 · · · q − 1 1 q q − 1 · · · · · · q − 1 1        6 PETTER ANDREAS BERGH and c o nsider the quantum complete int erse c tion A a c q . Explicitly , this is the alge br a k h X 1 , . . . , X c i / ( X a i , { X i X j − q X j X i } i 0 . (iii) Ext i A a c q ( N , M ) = 0 for i ≫ 0 . (iv) Ex t i A a c q ( N , M ) = 0 for i > 0 . Pr o of. Let A a 2 c q ′ be a symmetric qua nt um co mplete intersection with the prop erties given in Pro p osition 3.2. That is, the suba lgebra of A a 2 c q ′ generated b y x 1 , . . . , x c is isomor phic to A a c q , and the commutators o f A a 2 c q ′ are the commutators o f A a c q . By Lemma 2.1, there ex ists a ho momorphism Z c ⊗ Z Z c t − → k \ { 0 } and a qua nt um complete intersection A b c p , such that A a 2 c q ′ is isomor phic to the t wisted tensor pro d- uct A a c q ⊗ t k A b c p . Now for gr a ded A a c q -mo dules M and N , Theorem 2 .2 provides a n isomorphism Ext ∗ A a 2 c q ′ ( M ⊗ t k A b c p , N ⊗ t k A b c p ) ≃ E xt ∗ A a c q ( M , N ) ⊗ k Ext ∗ A b c p ( A b c p , A b c p ) of gr aded vector spaces. The result now follows from Theo r em 3.3, since a ll the commutators of A a 2 c q ′ are ro ots of unity .  References [AvB] L. Avramov, R.-O. 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Petter Andreas Bergh, Institutt for ma tema tiske f ag, NTNU, N-7491 Trondheim, Nor w a y E-mail addr ess : bergh@math.n tnu.no