Noncommutative tensor triangular geometry and the tensor product property for support maps
aa r X i v : . [ m a t h . C T ] A ug NONCOMMUTATIVE TENSOR TRIANGULAR GEOMETRY ANDTHE TENSOR PRODUCT PROPERTY FOR SUPPORT MAPS
DANIEL K. NAKANO, KENT B. VASHAW, AND MILEN T. YAKIMOV
Abstract.
The problem of whether the cohomological support map of a finitedimensional Hopf algebra has the tensor product property has attracted a lotof attention following the earlier developments on representations of finite groupschemes. Many authors have focussed on concrete situations where positive andnegative results have been obtained by direct arguments.In this paper we demonstrate that it is natural to study questions involvingthe tensor product property in the broader setting of a monoidal triangulatedcategory. We give an intrinsic characterization by proving that the tensor productproperty for the universal support datum is equivalent to complete primeness ofthe categorical spectrum. From these results one obtains information for othersupport data, including the cohomological one. Two theorems are proved givingcompete primeness and non-complete primeness in certain general settings.As an illustration of the methods, we give a proof of a recent conjecture ofNegron and Pevtsova on the tensor product property for the cohomological supportmaps for the small quantum Borel algebras for all complex simple Lie algebras. Introduction
Monoidal Triangular Geometry.
Tensor triangular geometry as introducedby Balmer has played a unifying role in understanding the interrelationships betweenrepresentation theory, homological algebra and commutative ring theory/algebraicgeometry. In [19], the authors developed a noncommutative version of Balmer’s tensor triangular geometry [2]. Our new theory has the advantage that it can beapplied to a wider variety of categories such as the stable module category for anyfinite-dimensional Hopf algebra. Given a monoidal triangulated category K , weassociated • a topological space Spc K of (thick) prime ideals and • a support datum map V : K → X sp (Spc K ),and we proved that this support datum is a universal final object in the category ofall support data, see Theorem 2.3.2 below.As in the case for non-commutative rings, for monoidal tensor categories, wedemonstrated that it was important to distinguish various types of prime ideals.The definition of a prime ideal in this setting involves considering products of idealswhereas the definition of a completely prime ideal entails considering products of Research of D.K.N. was supported in part by NSF grant DMS-1701768.Research of K.B.V. was supported by a Board of Regents LSU fellowship and in part by NSFgrant DMS-1901830.Research of M.T.Y. was supported in part by NSF grant DMS-1901830 and a Bulgarian ScienceFund grant DN02/05. objects in the category. The notion of semiprime ideal is also a key concept in thisnew theory.1.2.
Support Theory.
The precursor to support data, namely support varieties ,were first developed in the context of modular representations of finite groups by thepioneering work of Alperin and Carlson. Since that time, in representation theory(and in the more general setting of monoidal triangulated categories) there has beena plethora of contexts where support theory has been studied which includes(i) the cohomological support via group, Hopf algebra and Hochschild cohomol-ogy [9, 23],(ii) the rank variety and Π-support via embedded subobjects [9, 13],(iii) support via actions of commutative algebras [4, 5],(iv) support via actions of the extended endomorphism ring of the identity object[8],(v) support via tensor triangular geometry [2],and other approaches. Many fundamental connections between these support theo-ries have been established.In the aforementioned cases, a support datum map is a map σ from the objects of amonoidal triangulated category K to the set of specialization closed subsets X sp ( X )of a topological space X . The following problem has attracted a lot of attention andhas been at the heart of applications of support maps: Problem.
When does a support datum σ : K → X sp ( X ) possesses the tensorproduct property σ ( A ⊗ B ) = σ ( A ) ∩ σ ( B ) , ∀ A, B ∈ K ?For the cohomological support for modular representations of finite groups thiswas proved in [9] and for finite group schemes in [13]. In the support setting in (iii),a positive answer was obtained in [4, 5] under a stratification assumption. Therehas been a great deal of research on this problem for the cohomological support forthe stable module category StMod ( H ) of a finite dimensional Hopf algebra H . Inconcrete situation positive and negative answers were obtained in [6, 12, 21, 22].1.3. Main Results.
The main goal of this paper is to illustrate how the tensorproduct property can be characterized in terms of the intrinsic structure of theunderlying monoidal triangulated category. More specifically, the main results ofthis paper are as follows:(i) Given a monoidal triangulated category, we prove that the universal supportdatum V : K → X sp (Spc K ) has the tensor product property if and only ifall prime ideals of K are completely prime (Theorem 3.1.1).(ii) We prove that if all thick right ideals of a monoidal triangulated category K are two-sided, then the property in (i) holds for K (Theorem 3.2.1).(iii) We show that if all objects of a monoidal triangulated category are rigid andthe category has a nilpotent object, then the property in (i) does not holdfor K (Theorem 4.2.1).The power of Theorem 2.3.2 is that the verification of the support property forindividual objects of the category K is shown to be equivalent to an intrinsic globalproperty of the Balmer spectrum Spc K of the category. In noncommutative ring ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 3 theory, the question of whether all prime deals of a noncommutative ring are com-pletely prime is a much studied one. Dixmier proved that the universal envelopingalgebra U ( g ) of a finite dimensional Lie algebra algebra has this property if and onlyif the Lie algebra g is solvable [10, Theorem 3.7.2]. For general noncommutativerings there are no if and only if theorems of this sort, but positive results for quan-tum function algebras and Cauchon–Goodearl–Letzter extensions were obtained in[17, 18, 15]. Theorem 2.3.2 establishes a bridge between the tensor product propertyfor support data and the categorical versions of these questions in ring theory.Theorems 3.2.1 and 2.3.2 allow for a fast checking of the tensor product property inmany interesting situations. This can be combined with Theorems 6.2.1 and 7.3.1 in[19] where we proved that support data satisfying natural assumptions coincide withthe universal support map V : K → X sp (Spc K ). One can use this to apply Theorems3.2.1 and 2.3.2 to verify whether other support maps for a monoidal triangulatedcategory K , for instance the cohomological support map, posses the tensor productproperty. Along this path we obtain the last main result in the paper, proving theNegron and Pevtsova conjecture [21] that(iv) the cohomological support maps for all small quantum Borel algebras as-sociated to arbitrary complex simple Lie algebras and arbitrary choices ofgroup-like elements posses the tensor product property.1.4. Acknowledgements.
We thank Cris Negron for useful discussions along withcomments on an earlier version of our manuscript.2.
Preliminaries on noncommutative tensor triangular geometry
Monoidal Triangulated Categories.
We follow the conventions in [19]. A monoidal triangulated category (M∆C for short) is a monoidal category K in thesense of Definition 2.2.1 of [11] which is triangulated and for which the monoidalstructure ⊗ : K × K → K is an exact bifunctor.Recall that a thick subcategory of a triangulated category K , is a full triangulatedsubcategory of K that contains all direct summands of its objects. A thick right (resp. two-sided ) ideal of an M∆C, K , is a thick subcategory of K that is closedunder right tensoring (resp. right and left tensoring) with arbitrary objects of K . Foreach object M ∈ K there exist unique minimal right and two-sided ideals containing M , which will be denoted by h M i r and h M i , respectively.2.2. Prime Ideals and the Balmer Spectrum.
We call a proper two-sided ideal P of K prime if(2.2.1) I ⊗ J ⊆ P ⇒ I ⊆ P or J ⊆ P for all thick two-sided ideals I and J of K . This property is equivalent to saying that(2.2.1) holds for all pairs of thick right ideals I and J of K . It is also equivalent tothe condition that for all A, B ∈ K , A ⊗ C ⊗ B ∈ P , ∀ C ∈ K ⇒ A ∈ P or B ∈ P , see [19, Theorem 3.2.2].One can define a notion of primeness on objects of K as follows. An ideal P is completely prime if and only if A ⊗ B ∈ P ⇒ A ∈ P or B ∈ P DANIEL K. NAKANO, KENT B. VASHAW, AND MILEN T. YAKIMOV for all objects A and B in K .With these definitions of primeness, one can define a topological space that isanalogous to the spectrum of a non-commutative ring. Definition 2.2.1. (a)
The noncommutative Balmer spectrum
Spc K of an M ∆ C, K , is the set of its prime ideals with the topology generated by the closed sets V ( M ) = { P ∈ Spc K | M P } for M ∈ K . (b) Let
CP-Spc K be the topological subspace consisting of all completely primeideals of K . Its topology is generated by the sets V CP ( M ) = { P ∈ CP-Spc K | M P } for M ∈ K . From the definitions, one can easily verify that every completely prime ideal in anM∆C is prime. Therefore, one has V CP ( M ) = V ( M ) ∩ CP-Spc K . It is clear that an intersection of prime ideals need not be a prime ideal.
Definition 2.2.2.
A semiprime ideal of an M ∆ C, K , is an intersection of primeideals of K . The following characterization of semiprime ideals was proved in [19, Theorem3.4.2]:
Theorem 2.2.3.
The following are equivalent for a proper thick ideal Q of andM ∆ C, K : (a) Q is a semiprime ideal; (b) For all A ∈ K , if A ⊗ C ⊗ A ∈ Q , ∀ C ∈ K , then A ∈ Q ;(c) If I is any thick two-sided ideal of K such that I ⊗ I ⊆ Q , then I ⊆ Q ; (d) If I is any thick right ideal of K such that I ⊗ I ⊆ Q , then I ⊆ Q . Support data maps, universality of
Spc K. One of the important featuresabout monoidal triangulated categories is the use of maps that take objects of K tosubsets of a topological space. For a given topological space Y , we will denote by X ( Y ), X cl ( Y ) and X sp ( Y ) the collections of its subsets, closed subsets and specializa-tion closed subsets, respectively. Given a map σ : K → X ( Y ), denote its extensionto the set of thick subcategories of K given by(2.3.1) Φ σ ( I ) = [ A ∈ I σ ( A ) . Definition 2.3.1.
A support datum for an M ∆ C, K , is a map σ : K → X ( Y ) for a topological space Y such that (i) σ (0) = ∅ and σ (1) = Y ; (ii) σ ( L i ∈ I A i ) = S i ∈ I σ ( A i ) , ∀ A i ∈ K ; (iii) σ (Σ A ) = σ ( A ) , ∀ A ∈ K ; (iv) If A → B → C → Σ A is a distinguished triangle, then σ ( A ) ⊆ σ ( B ) ∪ σ ( C ) : ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 5 (v) S C ∈ K σ ( A ⊗ C ⊗ B ) = σ ( A ) ∩ σ ( B ) , ∀ A, B ∈ K .A weak support datum is a map σ : K → X ( Y ) which satisfies conditions (i-iv) and the condition (v’) Φ σ ( I ⊗ J ) = Φ σ ( I ) ∩ Φ σ ( J ) for all thick two-sided ideals I and J of K . Each support datum is a weak support datum [19, Lemma 4.3.1 and 4.5.1]. Forevery M∆C K , the map V : K → X cl (Spc K ) given by V ( A ) = { P ∈ Spc K : M / ∈ P } is a support datum. It is universal as proved in [19, Theorems 4.2.2 and 4.5.1]: Theorem 2.3.2.
Let K be an M ∆ C. (a) The support V is the final object in the collection of support data σ for K such that σ ( A ) is closed for each A ∈ K : for any such σ : K → X ( Y ), thereis a unique continuous map f σ : Y → Spc K satisfying σ ( A ) = f − σ ( V ( A )) for A ∈ K . (b) The support V is the final object in the collection of weak support data σ for K such that Φ σ ( h A i ) is closed for each A ∈ K : for any such σ : K → X ( Y ),there is a unique continuous map f σ : Y → Spc K satisfyingΦ σ ( h A i ) = f − σ ( V ( A )) for A ∈ K . The tensor product property for the universal support datum of amonoidal triangulated category
Complete Primeness of
Spc and the Tensor Product Property.
We be-gin by proving a theorem that indicates how the structural properties of a monoidaltriangulated category are captured by characterizations involving the universal sup-port datum.
Theorem 3.1.1.
For every monoidal triangulated category K , the following areequivalent: (a) The universal support datum V : K → X (Spc K ) has the tensor productproperty V ( A ⊗ B ) = V ( A ) ∩ V ( B ) , ∀ A, B ∈ K . (b) Every prime ideal of K is completely prime.Proof. (a ⇒ b) Let P ∈ Spc K and A, B ∈ K be such that A ⊗ B ∈ P . Then P / ∈ V ( A ⊗ B ) = V ( A ) ∩ V ( B ) . Hence, either P / ∈ V ( A ) or P / ∈ V ( B ), and thus, either A ∈ P or B ∈ P .(b ⇒ a) For A, B ∈ K , we haveSpc K \ V ( A ⊗ B ) = { P ∈ Spc K | A ⊗ B ∈ P } = { P ∈ Spc K | A ∈ P } ∪ { P ∈ Spc K | B ∈ P } = (Spc K \ V ( A )) ∪ (Spc K \ V ( B )) . Thus V ( A ⊗ B ) = V ( A ) ∩ V ( B ). (cid:3) DANIEL K. NAKANO, KENT B. VASHAW, AND MILEN T. YAKIMOV
The proof of Theorem 3.1.1 immediately gives the following fact.
Corollary 3.1.2.
For every monoidal triangulated category, K , the map V CP : K → CP-Spc K given by V CP ( A ) = V ( A ) ∩ CP-Spc K has the tensor product property. In many cases for monoidal triangulated categories, K , the space CP-Spc K canbe much smaller than Spc K . So in general, the support datum V CP captures muchless information than the universal support datum V .3.2. A Criterion for Complete Primeness of
Spc K. In this section we inves-tigate monoidal tensor categories where the right ideals coincide with the two-sidedideals. In this situation, every prime ideal is completely prime and the tensor productproperty holds. This key observation will be applied in Section 5.7.
Theorem 3.2.1.
Let K be a monoidal triangulated category in which every thickright ideal is two-sided. Then every prime ideal of K is completely prime, and as aconsequence, the universal support datum V : K → X (Spc K ) has the tensor productproperty V ( A ⊗ B ) = V ( A ) ∩ V ( B ) , ∀ A, B ∈ K . Proof.
First we claim that(3.2.1) h M i r = h M i , ∀ M ∈ K . The inclusion h M i r ⊆ h M i is obvious. The reverse inclusion is proved as follows.The hypothesis states that h M i r is a a two-sided thick ideal and, in particular, itcontains h N i for all N ∈ h M i r . Applying this for N = M yields h M i r ⊇ h M i .Let P ∈ Spc K and A, B ∈ K be such that A ⊗ B ∈ P . Therefore A ⊗ h B i r ⊆ P and, by (3.2.1), A ⊗ h B i ⊆ P . This implies that A ⊗ C ⊗ B ∈ P for all C ∈ K and,by the primeness of P , A ∈ P or B ∈ P . Therefore, the thick ideal P is completelyprime. The second statement follows from the first and Theorem 3.1.1. (cid:3) If a monoidal triangulated category K has the property that A ⊗ B ∼ = B ⊗ A forall A, B ∈ K , then K satisfies the assumption of Theorem 3.2.1. This in particularholds for all braided monoidal triangulated categories. The next section containsmuch more nontrivial applications of this theorem.4. A criterion for non-complete primeness of
Spc K Rigidity and Semi-Primeness.
Recall that an object A of a monoidal cat-egory K is rigid if it has a dual object A ∗ : by definition, this means there areevaluation and coevaluation mapsev : A ∗ ⊗ A → → A ⊗ A ∗ , such that the compositions(4.1.1) A coev ⊗ id −−−−−→ A ⊗ A ∗ ⊗ A id ⊗ ev −−−−→ A and A ∗ id ⊗ coev −−−−−→ A ∗ ⊗ A ⊗ A ∗ ev ⊗ id −−−−→ A ∗ are the identity maps on A and A ∗ , respectively. Proposition 4.1.1. If K is a monoidal triangulated category in which every objectis rigid, then every thick ideal of K is semiprime. ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 7
Proof.
Fix a thick two-sided ideal I of K . Let A ∈ K be such that A ⊗ B ⊗ A ∈ I forall B ∈ K . In particular, A ⊗ A ∗ ⊗ A ∈ I . It follows from (4.1.1) that A is a directsummand of A ⊗ A ∗ ⊗ A . Since I is a thick subcategory of K , A ∈ I . Theorem 2.2.3now implies that I is a semiprime ideal of K . (cid:3) Existence of Nilpotent Elements.
Given a monoidal tensor category whereall objects are rigid, one can now show that the existence of a nilpotent element in-sures that the universal support datum does not satisfy the tensor product property.
Theorem 4.2.1.
Let K be a monoidal triangulated category in which every object isrigid. If K has a non-zero nilpotent object M (i.e., M = 0 but M ⊗ n := M ⊗· · ·⊗ M ∼ =0 , for some n > ) then not all prime ideals of K are completely prime. As aconsequence, the universal support datum V : K → X (Spc K ) does not have thetensor product property.Proof. By Proposition 4.1.1, h i is a semiprime ideal of K . Hence, the prime radicalof K equals h i .On the other hand M lies in all completely prime ideals P of K because M ⊗ n ∼ =0 ∈ P . If all prime ideals of K are completely prime, this would imply that M belongs to the prime radical of K (i.e. M ∈ h i ), which is a contradiction. (cid:3) The following corollary follows from Theorem 4.2.1, because all objects of stmod ( H )are rigid for finite dimensional Hopf algebras H . Corollary 4.2.2.
Assume that H is a finite dimensional Hopf algebra which admitsa non-projective finite dimensional module M such that M ⊗ n is projective. Then notall prime ideals of the stable module category stmod ( H ) are completely prime, i.e.,the universal support datum V : K → X (Spc( stmod ( H ))) does not have the tensorproduct property. Remarks on the Work of Benson-Witherspoon.
In [6] Benson and With-erspoon considered the stable module categories of Hopf algebras of the form H G,L := ( k [ G ] k L ) ∗ , where G and L are finite groups with L acting on G by group automorphisms, k is afield of positive characteristic dividing the order of G , k L is the group algebra of L , k [ G ] is the dual of the group algebra of G , and p be a prime number and n be a positive integer. In [6, Example 3.3] Ben-son and Witherspoon proved that for G := ( Z /p Z ) n , L := Z /n Z (with L cyclicallypermuting the factors of G ) and k a field of characteristic p , H G,L admits a non-projective finite dimensional module M such that M ⊗ M is projective. By Corol-lary 4.2.2, the universal support data does not satisfy the tensor product property.If K is a monoidal triangulated category in which every object is rigid and K hasobjects A and B , such that A ⊗ B ∼ = 0 but B ⊗ A = 0, then not all prime ideals of K are completely prime, i.e., the universal support datum V : K → X (Spc K ) doesnot have the tensor product property. This follows from Theorem 4.2.1, because for M := B ⊗ A is not the zero object in K , but M ⊗ M ∼ = B ⊗ ( A ⊗ B ) ⊗ A ∼ = 0.Benson and Witherspoon constructed [6, Example 3.2] a Hopf algebra of the form H G,L such that stmod ( H G,L ) has a pair of objects
A, B with this property. The
DANIEL K. NAKANO, KENT B. VASHAW, AND MILEN T. YAKIMOV group G is chosen to be the Klein 4-group, L is the cyclic group of order 3 whosegenerator cyclically permutes the non-identity elements of G , and the field k hascharacteristic 2.5. The tensor product property for the cohomological support forsmall quantum Borels
Preliminaries.
Let ∆ be an irreducible root system of rank n . Let ℓ be apositive integer and ζ be a primitive ℓ th root of unity.We begin by introducing a general construction of the small quantum group for aBorel algebra that generalizes the well-known construction using group like elementsarising from the root lattice. For a given ∆, let X be the corresponding weightlattice and ∆ + be a set of positive roots. Denote by { α , . . . , α n } the base of simpleroots for ∆ corresponding to ∆ + and by { d , . . . , d n } the collection of relativelyprime positive integers that symmetrizes the corresponding Cartan matrix. Denoteby h− , −i the Weyl group invariant nondegenerate symmetric inner product on theEuclidean space t ∗ R spanned by ∆, normalized by h β, β i = 2 for short roots β . Interms of this form, the integers d i are given by d i = h α i , α i i /
2. Let { α ∨ , . . . , α ∨ n } bethe corresponding coroots thought of as elements of t ∗ R by setting α ∨ i = 2 α i h α i , α i i = α i d i · Choose a Z -lattice, Γ, with Z ∆ ⊆ Γ ⊆ X . Such a lattice Γ has rank n . Let { µ , . . . , µ n } be a Z -basis for Γ.Let u ζ ( b ) be the small quantum group as described in [3, Section 2.2]. Then u ζ ( b ) = u ζ ( u ) u ζ ( t ) where u ζ ( u ) is generated by the root vectors { E β | β ∈ ∆ + } satisfying E ℓβ = 0 and u ζ ( t ) is a Hopf algebra isomorphic to the group algebra of Z ∆ / ( ℓ Z ∆) over C , realized as u ζ ( t ) = C [ K ± α , . . . , K ± α n ] / ( K ℓα i − , ≤ i ≤ n )where K α i are group like elements. The relations in u ζ ( b ) defining the smash productare(5.1.1) K α i E β K − α i = ζ h β,α i i E β for β ∈ ∆ + .We can consider the following generalization of the small quantum group for theBorel subalgebra. Given a lattice Γ with Z ∆ ⊆ Γ ⊆ X as above, define its sublatticeΓ ′ := { ν ∈ Γ | h ν, ∆ i ⊆ ℓ Z } . Obviously, Γ ′ ⊇ ℓ Γ, so Γ / Γ ′ is a factor group of Γ /ℓ Γ ∼ = ( Z /ℓ Z ) n . Denote thecanonical projection(5.1.2) Γ ։ Γ / Γ ′ by µ µ. Let(5.1.3) u ζ, Γ ( t ) denote the group algebra of Γ / Γ ′ over C . For µ ∈ Γ / Γ ′ denote by K µ the element of u ζ, Γ ( t ) corresponding to µ . Consider theHopf algebra u ζ, Γ ( b ) = u ζ ( u ) u ζ, Γ ( t ) ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 9 with relations(5.1.4) K µ E α K − µ = ζ h α,µ i E α for µ ∈ Γ / Γ ′ , α ∈ ∆ + , where µ ∈ Γ is a preimage of µ . By the definition of the lattice Γ ′ , the right handside does not depend on the choice of preimage. The coproduct of the generators E α i is given by(5.1.5) ∆( E α i ) = E α i ⊗ K α i ⊗ E α i for 1 ≤ i ≤ n . The antipode is given by S ( E α i ) = − K − α i E α i .In all of the above definitions, the lattice Γ ′ can be replaced with any sublattice ofΓ ′ . The motivation for the use of the full lattice Γ ′ is that this makes u ζ, Γ ( b ) smallin the sense that the only group-like central elements of u ζ, Γ ( b ) are the scalars. Remark 5.1.1.
Consider two lattices Γ and Γ such that Z ∆ ⊆ Γ ⊆ Γ ⊆ X .Then Γ ′ = Γ ∩ Γ ′ . Hence, we have a Hopf algebra embedding u ζ, Γ ( b ) ֒ → u ζ, Γ ( b ) given by K µ +Γ ′ K µ +Γ ′ , E α E α for µ ∈ Γ , α ∈ ∆ + .5.2. Assumptions on ℓ . For the remainder of this section we will employ one ofthe following assumptions in the statements of our results where ζ is an ℓ th root ofunity. Assumption 5.2.1.
Let ℓ be a positive integer such that (a) ℓ is odd; (b) If ∆ is of type G then ∤ ℓ ; (c) If ∆ is of type A then ℓ ≥ , otherwise ℓ > . Conditions (a)-(b) in Assumption 5.2.1 are equivalent to saying that ℓ is an oddpositive integer which is coprime to { d , . . . , d n } . Assumption 5.2.2.
Let ℓ be a positive integer such that (a) ℓ is odd; (b) If ∆ is of type G then ∤ ℓ ; (c) ℓ > h where h is the Coxeter number for ∆ . Note that if ℓ satisfies Assumption 5.2.2 then ℓ satisfies Assumption 5.2.1.The group of group-like elements of u ζ, Γ ( t ) is isomorphic to Γ / Γ ′ . Next we explic-itly describe this finite abelian group. Proposition 5.2.3. (a) If ℓ is coprime to { d , . . . , d n } , then Γ ′ = Γ ∩ ℓX. That is, Γ / Γ ′ ∼ = Γ / (Γ ∩ ℓX ) . (b) If ℓ is coprime to { d , . . . , d n } and | X/ Γ | , then Γ ′ = ℓ Γ . That is, Γ / Γ ′ ∼ = Γ / ( ℓ Γ) ∼ = ( Z /ℓ Z ) n . Proof. (a) Let ν = P m i ω i ∈ Γ ⊆ X for some m i ∈ Z . Then ν ∈ Γ ′ ⇔h ν, α i i ∈ ℓ Z , ∀ ≤ i ≤ n ⇔ m i d i ∈ ℓ Z , ∀ ≤ i ≤ n ⇔ m i ∈ ℓ Z , ∀ ≤ i ≤ n ⇔ ν ∈ Γ ∩ ℓX. (b) In view of part (a), we have to prove that under the assumptions in part (b),Γ ∩ ℓX = ℓ Γ. Clearly, Γ ∩ ℓX ⊇ ℓ Γ . For the opposite inclusion, take ν ∈ Γ ∩ ℓX . Then the order of ν/ℓ + Γ in X/ Γdivides ℓ . Since ℓ is coprime to the order of the group X/ Γ, the order of ν/ℓ + Γequals 1. Therefore ν/ℓ ∈ Γ, and thus, ν ∈ ℓ Γ. Hence, Γ ∩ ℓX = ℓ Γ. (cid:3) Example 5.2.4.
The standard notion of a small quantum Borel subalgebra u ζ ( b ) isrecovered from the above one as follows. Proposition 5.2.3(b), applied for the rootlattice Γ = Z ∆, implies that, if ℓ is coprime to { d , . . . , d n } and | X/ Z ∆ | , then u ζ, Z ∆ ( b ) ∼ = u ζ ( b ) . Note that both aforementioned algebras are defined for general values of ℓ , butbecome isomorphic under the coprimeness conditions.5.3. Automorphisms, Representations and Cohomology.
In this section wewill generalize many of the properties presented in [19, Section 8.3] for u ζ ( b ) to u ζ, Γ ( b ). For the readers convenience, we will use the same notational conventions.Denote the character group of Γ / Γ ′ by [ Γ / Γ ′ . By abuse of notation, for λ ∈ [ Γ / Γ ′ we denote by the same symbol the one dimensionalrepresentation of u ζ, Γ ( b ) given by K µ λ ( µ ) , E α , ∀ µ ∈ Γ / Γ ′ , α ∈ ∆ + . For each λ ∈ [ Γ / Γ ′ , one can define an automorphism, γ λ of u ζ, Γ ( b ) as follows: γ λ ( E α ) = λ ( α ) E α , γ λ ( K µ ) = K µ , ∀ µ ∈ Γ / Γ ′ , α ∈ ∆ + . Denote the subgroup Π = { γ λ : λ ∈ [ Γ / Γ ′ } ⊆ Aut( u ζ, Γ ( b )). For any u ζ, Γ ( b )-module, Q , the automorphism γ λ can be used to define a new module structure on it calledthe twist: Q γ λ . The underlying vector space of Q γ λ is still Q with the action givenby x.m = γ λ ( x ) m for all x ∈ u ζ, Γ ( b ) and m ∈ Q γ λ .Let R = H • ( u ζ, Γ ( b ) , C ) be the cohomology ring of u ζ, Γ ( b ). An automorphism in Πacts on the cohomology ring by taking an n -fold extension of C with C and twistingeach module in the n -fold extension to produce a new n -fold extension. This providesan action of the group Π on the ring R . The following proposition summarizesproperties of the automorphisms in Π and how they interact with representationsand the cohomology. Proposition 5.3.1.
Let u ζ, Γ ( b ) be the small quantum group for the Borel subalgebraand R = H • ( u ζ, Γ ( b ) , C ) be the cohomology ring. ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 11 (a)
The irreducible representations for u ζ, Γ ( b ) are one-dimensional and are pre-cisely the representations λ for λ ∈ d Γ / Γ ′ . (b) For any u ζ, Γ ( b ) -module, Q , and λ ∈ d Γ / Γ ′ one has λ ⊗ Q ⊗ λ − ∼ = Q γ λ . (c) The action of Π on R is trivial. (d) The action of Π on Proj( R ) is trivial.Proof. (a) The relations E ℓα = 0 for α ∈ ∆ + imply that all root vectors E β arein the radical of the finite dimensional algebra u ζ, Γ ( b ) and so they act by 0 onevery irreducible representation of u ζ, Γ ( b ). Hence, every irreducible representationsof u ζ, Γ ( b ) is an irreducible representation of u ζ, Γ ( t ), which is the group algebra ofΓ / Γ ′ , so the irreducible representation of u ζ, Γ ( t ) are precisely the representations λ for λ ∈ [ Γ / Γ ′ .(b) The isomorphism follows from coproduct formula (5.1.5) and the fact that theset { K µ , E α i | µ ∈ Γ , i = 1 , . . . , n } generates the algebra u ζ, Γ ( b ).(c and d) Note that (d) follows immediately from (c). So to finish the proof weshow that the action of Π on the cohomology ring R is trivial.By using the Lyndon-Hochschild-Serre (LHS) spectral sequence and the fact thatthe representations for u ζ, Γ ( t ) are completely reducible (because u ζ, Γ ( t ) is isomorphicto the group algebra over C of a finite group), it follows that R = H • ( u ζ ( u ) , C ) u ζ, Γ ( t ) with respect to the action (5.1.4) (cf. [14, Theorem 2.5]). Consequently, for everyweight ν ∈ Z ∆ of R h ν, Γ i ⊆ ℓ Z ⇒ h ν, ∆ i ⊆ ℓ Z ⇒ ν ∈ Z ∆ ∩ Γ ′ ⇒ ν = 0 . Let f ∈ R be of weight ν . The automorphism γ λ ∈ Π acts on f by γ λ ( f ) = λ ( ν ) f = f, which proves the triviality of the Π-action on R . (cid:3) Finite Generation.
In order to verify the finite generation conditions on thecohomology, we state the following result from [3, Proposition 5.6.3] on the coho-mology for u ζ ( u ). Theorem 5.4.1.
Let ℓ satisfy Assumption 5.2.1, and ζ be an ℓ th root of unity.There exists a polynomial ring S • ( u ∗ ) such that the following holds: (a) H • ( u ζ ( u ) , C ) is finitely generated over S • ( u ∗ ) ; (b) H • ( u ζ ( u ) , C ) is a finitely generated C -algebra. Theorem 5.4.1 allows us to consider the issue of finite generation of cohomologyfor u ζ, Γ ( b ). The filtration in [3, Section 2.9] on u ζ ( u ) that induces the grading asin [3, Lemma 5.6.1] is stable under the action of K µ i , i = 1 , . . . , n . Consequently,there exists a spectral sequence(5.4.1) E i,j = H i + j (gr u ζ ( u ) , C ) ( i ) ⇒ H i + j ( u ζ ( u ) , C )such that H n (gr u ζ ( u ) , C ) ∼ = M a + b = n S a ( u ∗ ) [1] ⊗ Λ bζ . Here S • ( u ∗ ) [1] is the symmetric algebra on u ∗ (the dual of u ) and Λ bζ is a deformationof the exterior algebra on u ∗ with generators and relations defined in [3, Section2.9]. In the proof of Theorem 5.4.1 (given in [3, Proposition 5.6.3]), it is shown thatunder the assumptions on ℓ , d r ( S • ( u ∗ ) [1] ) = 0 for r ≥ d r is the differentialon the E r -page of the spectral sequence (5.4.1). One can then conclude part (a) ofTheorem 5.4.1.Since u ζ ( u ) is normal in u ζ, Γ ( b ) (cf. [3, Section 2.8]) with quotient u ζ, Γ ( t ), and thefiltration is stable under u ζ, Γ ( t ), it follows that u ζ, Γ ( t ) acts on the spectral sequence(5.4.1). Furthermore, one can verify that u ζ, Γ ( t ) acts trivially on S • ( u ∗ ) [1] .Since finite-dimensional representations for u ζ, Γ ( t ) are completely reducible, thefixed point functor ( − ) u ζ, Γ ( t ) is exact. By using the LHS spectral sequence and theexactness, one shows thatH • ( u ζ, Γ ( b ) , C ) ∼ = H • ( u ζ ( u ) , C ) u ζ, Γ ( t ) . Moreover, the fixed point functor can be applied to get a spectral sequence:(5.4.2) E i,j = [H i + j (gr u ζ ( u ) , C ) ( i ) ] u ζ, Γ ( t ) ⇒ H i + j ( u ζ ( b ) , C ) . We can now verify the requisite finite generation assumptions on the cohomology for u ζ, Γ ( b ). Theorem 5.4.2.
Let ℓ satisfy Assumption 5.2.1, ζ be an ℓ th root of unity, and u ζ, Γ ( b ) be a small quantum group for a Borel subalgebra. Then (a) H • ( u ζ, Γ ( b ) , C ) is a finitely generated C -algebra; (b) For any finite-dimensional u ζ, Γ ( b ) -module, M , H • ( u ζ, Γ ( b ) , M ) is finitely gen-erated over H • ( u ζ, Γ ( b ) , C ) .Proof. (a) Let R := H • ( u ζ, Γ ( b ) , C ). From Theorem 5.4.1(a), and the spectral se-quence (5.4.2), we have polynomial ring S := S • ( u ∗ ) [1] with d r ( S ) = 0 for r ≥ R finitely generated over S . This shows (a).(b) By using induction on the composition length of M and the long exact se-quence in cohomology one can reduce the statement to showing that H • ( u ζ, Γ ( b ) , M )is finitely generated over R for M a simple u ζ, Γ ( b )-module.The simple u ζ, Γ ( b )-modules are one-dimensional and indexed by λ ∈ [ Γ / Γ ′ . Byusing the LHS spectral sequence, one hasH • ( u ζ, Γ ( b ) , λ ) ∼ = Hom u ζ, Γ ( t ) ( − λ, H • ( u ζ ( u ) , C )) = A λ . Now S acts on H • ( u ζ, Γ ( b ) , λ ) and thus acts on A λ . This action is compatible withthe action on T = H • ( u ζ ( u ) , C ). We have T ∼ = ⊕ λ ∈ [ Γ / Γ ′ A λ , and by Theorem 5.4.1, T is finitely generated over S . Consequently, A λ is finitely generated over S , thusfinitely generated over R . (cid:3) Calculation of the Cohomology Ring.
In this section we calculate the co-homology ring R := H • ( u ζ, Γ ( b ) , C ) for ℓ > h . We will need the following fact provedby Andersen and Jantzen [1, § Lemma 5.5.1. [1]
Let ∆ be an irreducible root system. For every weight λ of Λ • ( u ∗ ) and simple root α i , |h λ, α ∨ i i + 1 | ≤ h − , where h is the Coxeter number for ∆ . ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 13
The following theorem that provides a natural generalization to the fundamentalresult of Ginzburg and Kumar [14, Theorem 2.5].
Theorem 5.5.2.
Let ℓ satisfy Assumption 5.2.2 (in particular, ℓ > h ), ζ be an ℓ throot of unity, and u ζ, Γ ( b ) be a small quantum group for a Borel subalgebra. Then (a) H • ( u ζ, Γ ( b ) , C ) ∼ = S • ( u ∗ ) [1] ; (b) H • +1 ( u ζ, Γ ( b ) , C ) = 0 .Proof. Consider the spectral sequence (5.4.2) andH n (gr u ζ ( u ) , C ) u ζ, Γ ( t ) ∼ = M a + b = n S a ( u ∗ ) [1] ⊗ [Λ bζ ] u ζ, Γ ( t ) . The u ζ, Γ ( t )-weights of Λ bζ come from the t -weights of Λ • ( u ∗ ). If λ is a weight of Λ • ( u ∗ )corresponding to an element in [Λ bζ ] u ζ, Γ ( t ) , then h λ, Γ i ⊆ ℓ Z . Therefore h λ, α i i ∈ ℓ Z for all 1 ≤ i ≤ n . For each simple root α i of ∆ we have h λ, α ∨ i i = 1 d i h λ, α i i . Since h λ, α ∨ i i is an integer, h λ, α i i ∈ ℓ Z and gcd( ℓ, d i ) = 1, we have that that h λ, α ∨ i i is a multiple of ℓ . Lemma 5.5.1 gives that |h λ, α ∨ i i| ≤ h < ℓ. The combination of the two facts implies that h λ, α ∨ i i = 0 for all simple roots α i .Thus λ = 0 and(5.5.1) [Λ bζ ] u ζ, Γ ( t ) ∼ = ( b > C if b = 0 . Consequently, the E i,j -term of the spectral sequence only contains terms of theform S a ( u ∗ ) [1] where 2 a = i + j . From Theorem 5.4.2, d r ( S • ( u ∗ ) [1] ) = 0 for r ≥ (cid:3) Classification of Tensor Ideals.
Let stmod ( u ζ, Γ ( b )) be the stable modulecategory of finitely generated u ζ, Γ ( b )-modules. The stable module category for all u ζ, Γ ( b )-modules will be denoted by StMod ( u ζ, Γ ( b )). The category stmod ( u ζ, Γ ( b )) isa monoidal triangulated category. The goal of this section will to describe the thicktensor ideals in stmod ( u ζ, Γ ( b )) and its Balmer spectrum.Let R := H • ( u ζ, Γ ( b ) , C ) be cohomology ring for the small quantum group u ζ, Γ ( b ).In Theorem 5.4.2(a), it was shown that R is a a finitely generated C -algebra. There-fore, Y = Proj( R ), the space of (nontrivial) homogeneous prime ideals of R , is aNoetherian topological space. In fact, Y is a Zariski space.For brevity, the set of subsets, closed subsets, and specialization-closed subsets of Y will be denoted respectively by X , X cl , and X sp . The finite generation result inTheorem 5.4.2(b) can be used to define a (cohomological) support variety theory for u ζ, Γ ( b ). Let W ( − ) be the cohomological support stmod ( u ζ, Γ ( b )) → X cl , defined by W ( M ) = { p ∈ Proj R : Ext • ( M, M ) p = 0 } . This extends to a support map
StMod ( u ζ, Γ ( b )) → X sp by [4, Theorem 5.5], whichwe will also denote by W ( − ). Let Φ = Φ W : { thick right ideals of stmod ( u ζ, Γ ( b )) } → X be the map given by (2.3.1). Note that it takes values in X sp because W ( M ) ∈ X cl for all M ∈ stmod ( u ζ, Γ ( b )). On the other hand, we can define an assignmentΘ : X sp → { thick right ideals of stmod ( u ζ, Γ ( b )) } by Θ( Z ) = { M ∈ stmod ( u ζ, Γ ( b )) | W ( M ) ⊆ Z } for Z ∈ X sp . We can now state the theorem that classifies thick ideals in stmod ( u ζ, Γ ( b )). Ourresults extend the results due to the authors in [19, Theorems 8.2.1, 8.3.1]. Theorem 5.6.1.
Let u ζ, Γ ( b ) be the small quantum group for the Borel subalgebra foran arbitrary finite dimensional complex simple Lie algebra. Assume that ℓ satisfiesAssumption 5.2.2 (in particular, ℓ > h ), which implies that R ∼ = S • ( u ∗ ) . (a) The above Φ and Θ are mutually inverse bijections { thick right ideals of stmod ( u ζ, Γ ( b )) } Φ −→←− Θ { specialization closed sets of Proj( R ) } . (b) Every thick right ideal of stmod ( u ζ, Γ ( b )) is two-sided. (c) There exists a homeomorphism f : Proj( R ) → Spc( stmod ( u ζ, Γ ( b ))) . For the proof of the theorem we will need the following auxiliary lemma
Lemma 5.6.2.
In the setting of Theorem 5.6.1, for every finite dimensional u ζ, Γ ( b ) -module Q and its dual Q ∗ , W ( Q ) = W ( Q ∗ ) . Proof.
Every object of stmod ( u ζ, Γ ( b )) is rigid. The first composition in (4.1.1) givesthat if Q is a finite dimensional u ζ, Γ ( b )-module, then Q is a summand of Q ⊗ Q ∗ ⊗ Q .So, W ( Q ) ⊆ W ( Q ⊗ Q ∗ ⊗ Q ) . Since Q has a composition series by subquotients isomorphic to the one dimensionalmodules λ ∈ [ Γ / Γ ′ , W ( Q ⊗ Q ∗ ⊗ Q ) = [ λ ∈ [ Γ / Γ ′ W ( λ ⊗ Q ∗ ⊗ Q ) . The cohomological support W is automatically a quasi support datum. Applyingthis fact and Proposition 5.3.1 (b-c), we obtain that W ( λ ⊗ Q ∗ ⊗ Q ) ⊆ W (( Q ∗ ) γ λ ⊗ λ ⊗ Q ) ⊆ W (( Q ∗ ) γ λ ) = W ( Q ∗ )for all λ ∈ [ Γ / Γ ′ . Combining the above inclusions gives W ( Q ) ⊆ W ( Q ∗ ). Since thesquare of the antipode of u ζ, Γ ( b ) is an inner automorphism, Q ∗∗ ∼ = Q . Interchangingthe roles of Q and Q ∗ gives W ( Q ∗ ) ⊆ W ( Q ). Hence, W ( Q ) = W ( Q ∗ ). (cid:3) Proof of Theorem 5.6.1. (a) This statement follows by [19, Theorem 7.4.3]. The(fg) assumption is established in Theorem 5.4.2. The arguments in [7, Section 7.4],together with Lemma 5.6.2, verify [19, Assumption 7.2.1].To prove (b) and (c), we will employ [19, Theorem 6.2.1]. As noted earlier, thecohomological support W is a quasi support datum and satisfies [19, Assumption ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 15 W satisfiesthe following two properties:(Faithfulness) If Φ( h M i ) = ∅ , then M ∼ = 0.(Realization) If V is a closed set in Y , then there exists a compact object M withΦ( h M i ) = V .For any Hopf algebra, the cohomological support satisfies faithfulness automati-cally. For realization, we compute:Φ( h M i ) = [ C,D ∈ K c W ( C ⊗ M ⊗ D )= [ C ∈ K c W ( C ⊗ M )= [ λ ∈ [ Γ / Γ ′ W ( λ ⊗ M )= [ λ ∈ [ Γ / Γ ′ W ( λ ⊗ M ⊗ λ − )= [ λ ∈ [ Γ / Γ ′ W ( M γ λ )= Π · W ( M )= W ( M ) . The second and fourth equalities follow from the fact that W is a quasi supportdatum, the fourth since W ( λ ⊗ M ) ⊆ W ( λ ⊗ M ⊗ λ − ) ⊆ W ( λ ⊗ M ⊗ λ − ⊗ λ ) = W ( λ ⊗ M ) . The third, fifth, and seventh equalities follow from Proposition 5.3.1, parts (a), (b),and (d) respectively. Since Φ( h M i ) = W ( M ) and every closed set of Proj R may berealized as W ( M ) for some compact M , W satisfies the realization property.We may now apply [19, Theorem 6.2.1], which gives us both part (c) and a bijectionof the form: { thick two-sided ideals of stmod ( u ζ, Γ ( b )) } Φ −→←− Θ { specialization closed sets of Proj( R ) } . Since we already know by (a) that Φ induces a bijection between the thick right idealsof stmod ( u ζ, Γ ( b )) and specialization closed sets of Proj( R ), it follows immediatelythat every thick right ideal is two-sided. (cid:3) The Tensor Product Property for the Cohomological Support Map.
In this section we illustrate Theorem 3.2.1. We prove that the cohomological supportmaps for all small quantum Borel algebras associated to arbitrary complex simpleLie algebras and arbitrary choices of group-like elements have the tensor productproperty. This was conjectured by Negron and Pevtsova [21] and proved by them inthe type A case. Theorem 5.7.1.
Let u ζ, Γ ( b ) be the small quantum group for the Borel subalgebra ofan arbitrary finite dimensional complex simple Lie algebra and a lattice Z ∆ ⊆ Γ ⊆ X . Assume that ℓ satisfies Assumption 5.2.2 (in particular, ℓ > h ). Then the followinghold: (a) All prime ideals of stmod ( u ζ, Γ ( b )) are completely prime. (b) The cohomological support W ( − ) : stmod ( u ζ, Γ ( b )) → X cl (Proj(H • ( u ζ, Γ ( b ) , C ))) has the tensor product property W ( A ⊗ B ) = W ( A ) ∩ W ( B ) for all A, B ∈ stmod ( u ζ, Γ ( b )) .Proof. Part (a) of the theorem follows by combining Theorems 3.2.1 and 5.7.1(a).(b) Recall the universal support datum V : stmod ( u ζ, Γ ( b )) → X cp (Spc( stmod ( u ζ, Γ ( b ))))defined in Section 2.3. It follows from Theorem 3.1.1 and part (a) of this theoremthat V has the tensor product property.In the proof of Theorem 5.6.1 it was shown that W is a weak support datum. ByTheorem 2.3.2(b), there exists a homeomorphism f : Proj(H • ( u ζ, Γ ( b ) , C )) → Spc( stmod ( u ζ, Γ ( b )))satisfying Φ W ( h M i ) = f − ( V ( M )) for all M ∈ stmod ( u ζ ( b )). Applying Theorem5.6.1(b), (3.2.1) and the fact that W is a quasi support datum, we obtain W ( M ) ⊆ Φ( h M i ) = Φ( h M i r ) ⊆ W ( M )for all M ∈ stmod ( u ζ ( b )). Therefore, W ( M ) = Φ( h M i ) = f − ( V ( M )) , ∀ M ∈ stmod ( u ζ ( b )) . Now Theorem 3.2.1, the continuity of f and the fact that the universal supportdatum V has the tensor product property give W ( A ⊗ B ) = f − ( V ( A ⊗ B )) = f − ( V ( A ) ∩ V ( B ))= f − ( V ( A )) ∩ f − ( V ( B )) = W ( A ) ∩ W ( B )for all A, B ∈ stmod ( u ζ ( b )). (cid:3) Example 5.2.4 and Theorem 5.7.1 imply the following:
Corollary 5.7.2.
Let u ζ ( b ) be the standard small quantum group for the Borelsubalgebra of an arbitrary finite dimensional complex simple Lie algebra. Assumethat ℓ satisfies Assumption 5.2.2 and that ℓ is coprime to | X/ Z ∆ | . Then the followinghold: (a) All prime ideals of stmod ( u ζ ( b )) are completely prime. (b) The cohomological support W ( − ) : stmod ( u ζ ( b )) → X cl (Proj(H • ( u ζ ( b ) , C ))) has the tensor product property W ( A ⊗ B ) = W ( A ) ∩ W ( B ) for all A, B ∈ stmod ( u ζ ( b )) . Remark 5.7.3.
Assume that ℓ satisfies Assumption 5.2.2 and that ℓ is coprimeto | X/ Z ∆ | . Then by Proposition 5.2.3(b), the small quantum Borel subalgebra u ζ, Γ ( b ) is based off the group algebra of the lattice Γ /ℓ Γ, cf. (5.1.3). Therefore,the statements in parts (a) and (b) of Theorem 5.7.1 hold for the version of a smallquantum Borel subalgebra based off the group algebra of the lattice Γ /ℓ Γ. ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 17
The Negron–Pevtsova small quantum Borel algebras.
In [20, 21] Negronand Pevtsova considered a different version of small quantum Borel subalgebras. Fora lattice, Γ, with Z ∆ ⊆ Γ ⊆ X , setΓ ⊥ := { ν ∈ Γ | h ν, Γ i ⊆ ℓ Z } . Denote the canonical projectionΓ ։ Γ / Γ ⊥ by µ µ. Let e u ζ, Γ ( t ) denote the group algebra of Γ / Γ ⊥ over C . For µ ∈ Γ / Γ ⊥ denote by K µ the corresponding element of u ζ, Γ ( t ). Following [20, 21],define the Hopf algebra e u ζ, Γ ( b ) = u ζ ( u ) e u ζ, Γ ( t )with relations K µ E α K − µ = ζ h α,µ i E α for µ ∈ Γ / Γ ′ , α ∈ ∆ + , where µ ∈ Γ is a preimage of µ . By the definition of the lattice Γ ⊥ , the right handside does not depend on the choice of preimage. The coproduct of the generators E α i is given by(5.8.1) ∆( E α i ) = E α i ⊗ K α i ⊗ E α i for 1 ≤ i ≤ n . The antipode is given by S ( E α i ) = − K − α i E α i .Clearly, Γ ′ ⊇ Γ ⊥ and the elements { K µ | µ ∈ Γ ′ / Γ ⊥ } are in the center of e u ζ, Γ ( b ). In other words, e u ζ, Γ ( b ) has a larger center than u ζ, Γ ( b ).By abuse of notation we will denote by µ µ the canonical projection Γ / Γ ⊥ ։ Γ / Γ ′ , recall (5.1.2). We have the surjective Hopf algebra homomorphism e u ζ, Γ ( t ) ։ u ζ, Γ ( t )given by K µ K µ for µ ∈ Γ / Γ ⊥ and E α E α for α ∈ ∆ + . Its kernel is the idealgenerated by the central elements { K µ − | µ ∈ Γ ′ / Γ ⊥ } . Let d be the minimal positive integer such that the restriction of h− , −i to Γ takesvalues in Z /d . Choose a primitive ( dℓ )th root of unity ξ such that ζ = ξ d . Considerthe symmetric (multiplicative) bicharacter χ : Γ / Γ ⊥ × Γ / Γ ⊥ → C × given by χ ( µ, ν ) := ξ h µ,ν i for µ, ν ∈ Γ / Γ ⊥ , where µ and ν are preimages of µ and ν in Γ. By the definition of Γ ⊥ , thebicharacter is well-defined and nondegenerate. It induces the isomorphism(5.8.2) ϕ : Γ / Γ ⊥ ∼ = −→ \ Γ / Γ ⊥ given by ϕ ( µ ) := χ ( µ, − ) for µ ∈ Γ / Γ ⊥ . Similarly to the discussion for u ζ, Γ ( t ), for λ ∈ \ Γ / Γ ⊥ define the one dimensionalrepresentation of e u ζ, Γ ( t ) K µ λ ( µ ) , E α , ∀ µ ∈ Γ / Γ ⊥ , α ∈ ∆ + . The irreducible representations of e u ζ, Γ ( t ) are one-dimensional and are indexed by \ Γ / Γ ⊥ . We have a much simplified version of Proposition 5.3.1 for the algebras e u ζ, Γ ( t ): Proposition 5.8.1. [20](a)
The irreducible representations for e u ζ, Γ ( b ) are one-dimensional and are pre-cisely the representations λ for λ ∈ \ Γ / Γ ⊥ . (b) For any e u ζ, Γ ( b ) -module, Q , and λ ∈ \ Γ / Γ ⊥ one has λ ⊗ Q ⊗ λ − ∼ = Q. Part (a) is proved in the same way as Proposition 5.3.1(a). Part (b) follows atonce by combining the following two facts:(1) For any e u ζ, Γ ( b )-module, Q , and λ ∈ \ Γ / Γ ⊥ , λ ⊗ Q ⊗ λ − ∼ = Q γ ′′ λ where, γ ′′ λ isthe automorphism of e u ζ, Γ ( b ) given by γ ′′ λ ( E α ) = λ ( α ) E α , γ λ ( K µ ) = K µ , ∀ µ ∈ Γ / Γ ⊥ , α ∈ ∆ + (this follows from (5.8.1));(2) γ ′′ λ equals the an inner automorphism x K ϕ − ( µ ) xK − ϕ − ( µ ) (this followsfrom (5.8.2)).From this point further the proofs of Theorems 5.5.2, 5.7.1, and 5.7.1, extendmutatis mutandis from the family of algebras u ζ, Γ ( b ) to the family of algebras e u ζ, Γ ( b ).Furthermore, there is a simplification in the proof of the analog of Theorem 5.7.1:on the third line of the long display λ ⊗ M ⊗ λ − ∼ = M and the rest of the equalitiesin the display can be omitted. This proves the following: Theorem 5.8.2.
Let e u ζ, Γ ( b ) be the version of the small quantum group for the Borelsubalgebra of an arbitrary finite dimensional complex simple Lie algebra and a lattice Z ∆ ⊆ Γ ⊆ X defined in [20] . Assume that ℓ satisfies Assumption 5.2.2. Then thefollowing hold: (a) H • +1 ( e u ζ, Γ ( b ) , C ) = 0 and R := H • ( e u ζ, Γ ( b ) , C ) ∼ = S • ( u ∗ ) [1] . (b) There exist two mutually inverse bijections { thick right ideals of stmod ( e u ζ, Γ ( b )) } Φ −→←− Θ { specialization closed sets of Proj( R ) } , where Φ and Θ are given by Φ( I ) := [ A ∈ I W ( A ) for the cohomological support W : stmod ( e u ζ, Γ ( b )) → X cl (Proj( R )) and Θ( Z ) := { M ∈ stmod ( u ζ, Γ ( b )) | W ( M ) ⊆ Z } for Z ∈ X sp (Proj( R )) . (c) Every thick right ideal of stmod ( e u ζ, Γ ( b )) is two-sided. (d) There exists a homeomorphism
Proj( R ) ∼ = Spc( stmod ( u ζ, Γ ( b ))) . (e) All prime ideals of stmod ( e u ζ, Γ ( b )) are completely prime. ENSOR PRODUCT PROPERTY FOR SUPPORT MAPS 19 (f)
The cohomological support W ( − ) : stmod ( e u ζ, Γ ( b )) → X cl (Proj R ) has the tensor product property W ( A ⊗ B ) = W ( A ) ∩ W ( B ) for all A, B ∈ stmod ( e u ζ, Γ ( b )) . There is a further simplification in the proof of part (c) of the theorem comparedto that of Theorem 5.6.1(b). Since the algebras e u ζ, Γ ( b ) satisfy the property inProposition 5.8.1(b), part (c) of the theorem also follows directly from this property. References [1] H.H. Andersen and J.C. Jantzen,
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E-mail address : [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,U.S.A.
E-mail address : [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,U.S.A.
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