Torsion theories induced from commutative subalgebras
aa r X i v : . [ m a t h . R T ] M a r TORSION THEORIES INDUCED FROMCOMMUTATIVE SUBALGEBRAS
VYACHESLAV FUTORNY, SERGE OVSIENKO, AND MANUEL SAORIN
Abstract.
We begin a study of torsion theories for representa-tions of an important class of associative algebras over a field whichincludes all finite W -algebras of type A , in particular the universalenveloping algebra of gl n (or sl n ) for all n . If U is such and alge-bra which contains a finitely generated commutative subalgebra Γ,then we show that any Γ-torsion theory defined by the coheight ofprime ideals is liftable to U. Moreover, for any simple U -module M , all associated prime ideals of M in Spec Γ have the same co-height. Hence, the coheight of the associated prime ideals of Γ isan invariant of a given simple U -module. This implies a strati-fication of the category of U -modules controlled by the coheightof associated prime ideals of Γ. Our approach can be viewed asa generalization of the classical paper by R.Block [Bl], it allowsin particular to study representations of gl n beyond the classicalcategory of weight or generalized weight modules. Introduction
A classical, very difficult and intriguing problem in representationtheory of Lie algebras is the classification of simple modules over com-plex simple finite dimensional Lie algebras. Such a classification is onlyknown for the Lie algebra sl due to results of R.Block [Bl]. It remainsan open problem in general, even in the subcategory of weight mod-ules with respect to a fixed Cartan subalgebra. On the other hand, aclassification of simple weight modules with finite dimensional weightspaces is well known for any simple finite dimensional Lie algebra, dueto Fernando [Fe] and Mathieu [Ma].The basic idea, proposed in [Bl] in the case of sl can be explainedas follows. First, we consider a maximal commutative subalgebra Γ ⊂ U (sl ) (in our terms, Gelfand-Tsetlin subalgebra), which is generatedby a Cartan subalgebra and the center of U (sl ). Then one fixes acentral character χ of U (sl ). After that all simples modules withcentral character χ are divided into torsion (or generalized weight)and torsionfree modules with respect to Γ / (Ker χ ). Thereafter theinvestigation of both classes of modules is reduced to the investigationof simples over a (skew) group algebra of the group Z . An analogous Mathematics Subject Classification.
Primary: 16D60, 16D90, 16D70,17B65. dea works in the more general context of generalized Weyl algebras ofrank 1 ([Ba], [BavO]), which allow a complete classification of simplemodules.A similar approach applied in the case of a Lie algebra gl( n ) (or sl n )allows to go beyond the category of weight modules with finite dimen-sional spaces. Namely, one considers the full subcategory of weightGelfand-Tsetlin gl n -modules with respect to the Gelfand-Tsetlin sub-algebra (certain maximal commutative subalgebras of U (gl n )) [DFO1],[FO2], that is, those modules V that have a decomposition V = ⊕ m ∈ Specm Γ V ( m ) , where V ( m ) = { v ∈ V |∃ N, m N v = 0 } as Γ-modules. This class isbased on natural properties of a Gelfand-Tsetlin basis for finite dimen-sional representations of simple classical Lie algebras [GTs], [Zh], [M].Gelfand-Tsetlin subalgebras were considered in various connections in[FM], [Vi], [KW1], [KW2], [Gr]. The theory developed in [FO1] and[FO2] was an attempt to unify the representation theories of the uni-versal enveloping algebra of gl n and of the generalized Weyl algebras.We underline that Gelfand-Tsetlin modules over gl n are weight mod-ules with respect to some Cartan subalgebra of gl n but they are allowedto have infinite dimensional weight spaces.In this paper we begin a study of general torsion theories for repre-sentations of a certain class of associative algebras which includes allfinite W -algebras of type A . In particular, the universal envelopingalgebra of gl n (or sl n ) is an example of such algebra for all n , where Γis a Gelfand-Tsetlin subalgebra.In the rest of the paper we shall work over an algebraically closedfield K of characteristic zero and consider the following situation. Setup 1.1. U will be a K -algebra having a commutative (not neces-sarily central) subalgebra Γ , fixed from now on, satisfying the followingproperties: (1) Γ is finitely generated as a K -algebra (2) There is a finite subset { u , . . . , u n } ⊂ U such that U is gener-ated as a K -algebra by Γ ∪ { u , . . . , u n } (3) Γ is a Harish-Chandra subalgebra , i.e., for each u ∈ U the Γ -bimodule Γ u Γ is a finitely generated Γ -module both on the leftand on the right. If M is a Gelfand-Tsetlin U -module with respect to Γ then the asso-ciated prime ideals of V in Spec Γ which form the assassin Ass( M ) aremaximal. Our goal is to understand torsion categories of modules over U more general than Gelfand-Tsetlin categories. Such modules haveassociated primes in Spec Γ which are not maximal.Our main result is the following theorem. We refer to Section 2 fordefinitions. heorem 1.2. Let Γ be a finitely generated subalgebra and U ⊃ Γ asabove. Then (1) The Γ -torsion theory associated to the subset Z i ⊂ Spec (Γ) ofprime ideals of coheihgt ≤ i is liftable to U. (2) For any simple U -module M all associated prime ideals of M in Spec Γ have the same height.
Theorem A provides a stratification of the module category with re-spect to the coheight of the associated primes. In classical cases asfinite W -algebras it happens that the endomorphism algebra of anysimple U -module is one dimensional and the center Z = Z ( U ) of U isan integral domain (polynomial ring) contained in Γ, which is in turn isalso an integral domain (polynomial ring) and flat over Z . Under thesecircumstances (see Proposition 5.1, all simple objects in the modulecategory U − M od are exhausted by simple U -modules whose asso-ciated primes have a fixed coheight 0 ≤ i ≤ Kdim (Γ) − Kdim ( Z ),where Kdim denotes the Krull dimension. The case i = 0 corre-sponds to Gelfand-Tsetlin modules (with respect to Γ) and the case i = Kdim (Γ) − Kdim ( Z ) corresponds to the simple U -modules whichare torsionfree with respect to some central character χ : Z −→ K .Our second main result provides information about the assassin of asimple U -module. Theorem 1.3.
Let U , Γ , u , . . . , u n be as in Setup 1.1, M = U x acyclic U -module generated by an element x such that ann Γ ( x ) = p is aprime ideal of Γ and suppose that all ideals in Ass( M ) have the samecoheight. If q ∈ Ass( M ) then there is a sequence q = q , q , . . . , q s = p of prime ideals with coheight equal to the coheight of p and a sequenceof indices k , . . . , k s ∈ { , . . . , n } such that Γ u ki Γ q i − u ki Γ+Γ u ki q i = 0 ,for all i = 1 , . . . , s . All these results can be applied to the class of Galois orders overfinitely generated Noetherian domains [FO1]. In particular, the resultsare valid for all finite W algebras of type A , e.g. U (gl n ) for all n .2. Torsion theories over a commutative Noetherian ring
In this section we collect some facts concerning torsion theories overcommutative Noetherian rings. Recall that, given a not necessarilycommutative ring R , a torsion theory over R is a pair ( T , F ) of fullsubcategories of R − Mod satisfying the following two conditions:(1) T = ⊥ F consists of those R -module T such that Hom R ( T, F ) =0, for all F ∈ F (2) F = T ⊥ consists of those R -module F such that Hom R ( T, F ) =0, for all T ∈ T ote that any of the component class of a torsion theory determinesthe other. In the above situation, for every R -module M there existsa (unique up to isomorphism) exact sequence0 → T −→ M −→ F → T ∈ T and F ∈ F . Then the assignments M t ( M ) := T and M F =: M/t ( M ) are functorial and yield a right adjoint and aleft adjoint, respectively, to the inclusion functors T ֒ → R − Mod and F ֒ → R − Mod. The functor t : R − Mod −→ T is called the torsionradical associated to T . The torsion theory is called hereditary when T is closed under taking submodules, which is equivalent to say that F is closed under taking injective envelopes (see chapter VI of [St] forall details and terminology concerning torsion theories).In this paper we are mainly interested in torsion theories over com-mutative Noetherian rings. In this section, unless otherwise stated, Γ will be a commutative Noetherian ring , We shall denote by Spec Γ (resp.Specm Γ) the prime (resp. maximal) spectrum of Γ. Given a Γ-module M and a prime ideal p ∈ Spec Γ, we shall denote by M p the localisationof M at p . We shall consider two important subsets of Spec Γ associatedto M . Namely the support of M , Supp( M ) = { p ∈ Spec Γ | M p = 0 } ,and the so-called assassin of M , Ass( M ), which consists of those p ∈ Spec(Γ) such that p = ann Γ ( x ) := { g ∈ Γ : gx = 0 } , for some x ∈ M .We now recall some properties of these sets. In the statement andin the sequel, we denote by Min X (resp. Max X ) the set of minimal(resp. maximal) elements of X , for every subset X ⊂ Spec Γ.
Proposition 2.1.
Let X ⊆ Spec Γ be any nonempty subset and M bea Γ -module. The following assertions hold: (1) Every element of X contains a minimal element of X (2) Ass( M ) ⊆ Supp( M ) and Min Ass( M ) = Min Supp( M ) .Proof. The set Spec Γ satisfies DCC with respect to inclusion. Indeedif p = p ⊇ p ⊇ ... is a descending chain of prime ideals, then thenumber of nonzero terms in it is bounded above by the height of p ,which is always finite (cf. [Mat][Theorem 13.5]).If X ⊆ Spec Γ is any nonempty subset and p ∈ X , then, by the DCCproperty, the set { q ∈ X : q ⊆ p } has a minimal element which is turna minimal element of X .Let now take p ∈ Ass( M ), so that p = ann Γ (Γ x ), for some x ∈ M .Then p ∈ Ass(Γ x ) ⊆ Supp(Γ x ) (see [Mat][Theorem 6.5]). Putting N = Γ x , we get that N p = 0, which implies that M p = 0 due to theexactness of localization. Then Ass( M ) ⊆ Supp( M ).Since M is the directed union of its finitely generated submodulesand localization is exact and preserves direct unions it follows thatSupp( M ) = S N Supp( N ), where the union is taken over all finitelygenerated submodules N of M . In particular, if p ∈ Min Supp( M )hen p ∈ Min Supp( N ), for some N < M finitely generated. But then p ∈ Ass( N ) (cf. [Mat][Theorem 6.5]), and so p ∈ Ass( M ). From theinclusion Ass( M ) ⊆ Supp( M ) we conclude that p ∈ Min Ass( M ), thusproving that Min Supp( M ) ⊆ Min Ass( M ).Conversely, if p ∈ Min Ass( M ) then we fix a cyclic submodule N =Γ x such that p = ann Γ ( N ). Then we have p ∈ Ass( N ) ⊂ Supp( N ) ⊂ Supp( M ). By assertion 1, there exists q ∈ Min Supp( M ) such that q ⊆ p . But equality must hold since we already know that Min Supp( M ) ⊆ Min Ass( M ) and p is minimal in Ass( M ). Therefore p ∈ Min Supp( M )and we get that Min Ass( M ) = Min Supp( M ). (cid:3) Definition 1. A subset Z ⊆ Spec Γ is called closed under specializa-tion when the following property holds:(*) If p ⊆ q are prime ideals with p ∈ Z , then q belongs to Z . The prototypical examples of closed under specialization subsets ofSpec Γ are the Zariski-closed subsets and those of the form Supp( M ),where M is a Γ-module. The following is a crucial result from [Ga]. Theorem 2.2. Let Γ be a commutative Noetherian ring. The assign-ments Z ( T Z , T ⊥ Z ) , where T Z = { T ∈ Γ − Mod : Supp( T ) ⊆ Z } , and ( T , F ) Z ( T , F ) = { p ∈ Spec Γ : Γ / p ∈ T } define mutually inverseorder-preserving one-to-one correspondences between the closed underspecilization subsets of Spec Γ and the hereditary torsion theories in Γ − Mod . For our purposes it is convenient to identify for a given module M the torsion submodule t Z ( M ) with respect to the torsion theory ( T Z , T ⊥ Z ). Proposition 2.3. Let Z ⊆ Spec Γ be a closed under specializationsubset and M be a Γ -module. For an element x ∈ M , the followingassertions are equivalent: (1) x belongs to t Z ( M )(2) Ass(Γ x ) ⊆ Z (resp. Min Ass(Γ x ) ⊆ Z ) (3) If p is a prime ideal such that ann Γ ( x ) ⊆ p , then p ∈ Z (4) There are prime ideals p , . . . , p r ∈ Z (resp. p , . . . , p r ∈ Min Z ) and integers n , . . . , n r > such that p n · · · · · p n r r x = 0 Proof. ⇐⇒ ⇐⇒ 3) Due to the fact that T Z is closed under takingsubmodules, assertion 1) is equivalent to say that Γ x ∈ T Z , i.e., to saythat Supp(Γ x ) ⊆ Z . But Supp(Γ x ) is precisely the set of prime idealscontaining ann Γ ( x ) (cf. Proposition III.4.6 in [Ku]). Moreover, being Z closed under specialization, Proposition 2.1 implies that Supp(Γ x ) ⊆ Z holds exactly when (Min) Ass(Γ x ) ⊆ Z .3) = ⇒ 4) Let { p , . . . , p r } be the (finite) set of prime ideals of Γwhich are minimal among those containing ann Γ ( x ). In particular, theybelong to Z . Then we have p · . . . · p r ⊆ p ∩ . . . ∩ p r = p ann Γ ( x ),here √ I denotes the radical of I , for every ideal I of Γ. It followsthe existence of a positive integer n > p n · . . . · p nr =( p · . . . · p r ) n ⊆ ann Γ ( x ). By Proposition 2.1(1), replacing each p i by minimal element of Z contained in it if necessary, we can find theneeded p i in Min Z .4) = ⇒ 3) Let p , . . . , p r ∈ Z and n , . . . , n r > p n · . . . · p n r r ⊆ ann Γ ( x ). If p is a prime ideal suchthat ann Γ ( x ) ⊆ p then there is some j = 1 , . . . , r such that p j ⊆ p . Itfollows that p ∈ Z since Z is closed under specialization. (cid:3) The following example of closed under specialization subsets of Spec Γwill be the most interesting for us. Example 2.4. One defines a transfinite ascending chain of subsets ( Z i ) i ordinal as follows. We put Z = Specm Γ . If i > is any ordinaland Z j has been defined for all j < i , then Z i = S j
6∈ T j , for all j < i . We also have t i ( M ) ⊆ t j ( M ), for all i ≤ j , where t i denotes the torsion radical associated to T i . Corollary 2.5. Let Γ be a commutative Noetherian ring, M be anonzero Γ -module and i be a nonlimit ordinal. The following asser-tions are equivalent: (1) t i ( M ) = M but t i − ( M ) = 0(2) The next two conditions hold: (a) For every x ∈ M there are prime ideals p , . . . , p r of co-height exactly i and positive integers n , . . . , n r > suchthat p n · . . . · p n r r x = 0(b) If p is a prime ideal of coheight < i and x ∈ M is anelement such that p x = 0 , then x = 0 . (3) The prime ideals in Ass( M ) have coheight exactly i .Proof. ⇐⇒ 3) By Proposition 2.3 and the fact that the mentionedtorsion theories are hereditary, we have that t i ( M ) = M iff Ass( M ) ⊆ Z i and t i − ( M ) = 0 iff Ass( M ) ∩ Z i − = ∅ . Therefore assertion 1 holdsif and only if Ass( M ) ⊆ Z i \ Z i − , which is equivalent to assertion 3.) = ⇒ 1) From Proposition 2.3 and condition 2.a we get that t i ( M ) = M . On the other hand, if we had 0 = x ∈ t i − ( M ) that same propo-sition would give that ∅ 6 = Ass(Γ x ) ⊆ Z i − . We then get g ∈ Γ suchthat gx = 0 and ann Γ ( gx ) = p is a prime ideal in Z i − . That wouldcontradict condition 2.b).1) , 3) = ⇒ 2) Let’s prove condition 2.b by way of contradiction. Sup-pose that there are 0 = x ∈ M and p ∈ Z i − such that p x = 0. Takinga maximal element in the set { ann Γ ( gx ) : g ∈ G and gx = 0 } , we ob-tain a q ∈ Ass(Γ x ) ⊆ Ass( M ) (cf. [Mat][Teorem 6.1]) such that p ⊆ q .Since Z i − is closed under specialization we get that q ∈ Z i − , againstassertion 3).We next prove condition 2.a. Let us take 0 = x ∈ M . Then, byProposition 2.3, we have prime ideals p , . . . , p r ∈ Z i (hence of coheight ≤ i ) and positive integers n , . . . , n r > p n · . . . · p n r r x = 0.It is not restrictive to choose the p i and the n i in such a way that thelatter ones are minimal, i.e., that p n · . . . · p n k − k · . . . · p n r r x = 0 forall k = 1 , . . . , r . That immediately implies the existence of elements g k ∈ Γ such that g k x = 0 and p k ⊆ ann Γ ( g k x ), for all k = 1 , . . . , r .By [Mat][Theorem 6.1], we find q k ∈ Ass(Γ x ) ⊆ Ass( M ) such that p k ⊆ q k , for all k = 1 , . . . , r . But then, by assertion 3), we have i =cht( q k ) ≤ cht( p k ) ≤ i for k = 1 , . . . , r . Therefore we have cht( p k ) = i ,for k = 1 , . . . , n . (cid:3) Our next goal is to give the precise structure of the Γ-modules in T , which is actually given by a more general result, Proposition 2.7below, which will follow from the following strengthened version of thechinese reminder’s theorem: Lemma 2.6. Let I , . . . , I r ( r > ) be pairwise distinct ideals of Γ .The following assertions are equivalent: (1) I i and I j are coprime, for all i = j (2) The canonical ring homomorphism Γ −→ Q ≤ i ≤ r Γ /I i is sur-jective.In such case T ≤ i ≤ j I i = I · . . . · I r .Proof. See [AM], Proposition 1.10, i). (cid:3) In the rest of the paper, if p ∈ Spec Γ and M is a Γ-module, we shalldenote by M ( p ) the submodule consisting of those x ∈ M such that p n x = 0, for some n ≥ 0. Note that, in such case, if p ∈ Ass ( M ( p ))then M inAss ( M ( p )) = { p } . Proposition 2.7. Let M be a Γ module such that Min Ass( M ) con-sists of pairwise coprime ideals (e.g. if Ass( M ) ⊆ Specm Γ ). Then Min Ass( M ) = Ass( M ) and M = ⊕ p ∈ Ass( M ) M ( p ) .roof. We shall prove that M = ⊕ p ∈ Min Ass( M ) M ( p ). It will follow thatAss( M ) = S p ∈ Min Ass( M ) Ass( M ( p )) = S p ∈ Min Ass( M ) { p } = Min Ass( M )and the result will follow.Let us fix p ∈ Min Ass( M ) and take x ∈ M ( p ) ∩ ( ⊕ q ∈ Min Ass( M ) , q = p M ( q )) . Then we have inclusionsAss(Γ x ) ⊆ Ass( M ( p )) ∩ Ass( ⊕ q ∈ Min Ass( M ) , q = p M ( q )) ⊆{ p } ∩ (Min Ass( M ) \ { p } ) = ∅ .It follows that x = 0 and, hence, the sum of the M ( q ), with q ∈ Min Ass( M ), is direct.Let us consider now Z := Supp( M ), which is a subset of Spec Γclosed under specialization. Then, by Theorem 2.2, M belongs to T Z and hence t Z ( M ) = M . If now x ∈ M then Proposition 2.3 guaranteesthe existence of distinct prime ideals p , . . . , p r ∈ Min Supp( M ) andpositive integer n , . . . , n r > p n · . . . · p n r r x = 0. The p i arepairwise coprime since Min Supp( M ) = Min Ass( M ) (see Proposition2.1). But then it follows easily that the ideals p n i i are also pairwisecoprime. Then Γ x is a module over the factor ring Γ / p n · . . . p n r r . But,by Lemma 2.6, we know that p n · . . . · p n r r = T ≤ i ≤ r p n i i , and then thecanonical map Γ / p n · . . . · p n r r −→ Q ≤ i ≤ r Γ / p n i i is a ring isomorphism. It follows that in the ring Γ / p n · . . . · p n r r we candecompose ¯1 = ¯ g + . . . + ¯ g r , where g i ∈ p n · . . . p n i − i − · p n i +1 i +1 . . . · p n r r .Then x = P ≤ i ≤ r g i x and p i n i g i x = 0, for i = 1 , . . . , r . It followsthat x ∈ ⊕ p ∈ Min Ass( M ) M ( p ), and we get the desired equality M = ⊕ p ∈ Min Ass( M ) M ( p ). (cid:3) Proposition 2.8. Let M and N be Γ -modules such that p and q are co-prime whenever p ∈ Ass( M ) and q ∈ Ass( N ) (resp. p ∈ Min Ass( M ) and q ∈ Min Ass( N ) ). The equality Ext i Γ ( M, N ) = 0 = Ext i Γ ( N, M ) holds for all i ≥ .Proof. Since we have Min Ass( M ) = Min Supp( M ) and similarly for N it follows that p and q are coprime whenever p ∈ Supp( M ) and q ∈ Supp( N ). If 0 → M −→ I −→ I −→ . . . is the minimal injective resolution of M in Γ − Mod and E (Γ / p ) isan injective indecomposable Γ-module appearing as direct summandof some I i , then p ∈ Supp( M ) (cf. [Mat][Theorem 18.7]). It followsthat Hom Γ ( N, I i ) = 0, and hence Ext i Γ ( N, M ) = 0, for all i ≥ 0. ThatExt i Γ ( M, N ) = 0 for all i ≥ (cid:3) . Algebras with a commutative Harish-Chandrasubalgebra and lifting of torsion theories Throughout the rest of the paper U and Γ satisfy the Setup 1.1. Wedenote by j : Γ ֒ → U the canonical inclusion and by j ∗ : U − Mod −→ Γ − Mod the restriction of scalar functor. It is clear that if T is a(hereditary) torsion class in Γ − Mod, then ˆ T = j − ∗ ( T ) := { T ∈ U − Mod : j ∗ ( T ) ∈ T } is a (hereditary) torsion class in U − Mod.However, if M is an U -module, then its torsion Γ-submodule t ( M )and its torsion U -submodule ˆ t ( M ) satisfy an inclusion ˆ t ( M ) ⊆ t ( M )that might be strict. Equality happens exactly when t ( M ) is an U -submodule of M . That justifies the following. Definition 2. A torsion theory ( T , F ) in Γ − Mod is called liftable to U − Mod in case t ( M ) is a U -submodule of M , for every U -module M . The following is a general criterion for the lifting of a torsion theory. Proposition 3.1. Let Z ⊆ Spec Γ be a closed under specializationsubset and ( T Z , F Z ) be its associated torsion theory in Γ − Mod . Thefollowing assertions are equivalent: (1) ( T Z , F Z ) is liftable to U − Mod(2) For each prime ideal p (minimal) in Z , the U -module U/U p belongs to T Z when looked at as Γ -module.Proof. (1) = ⇒ (2) Let us take p ∈ Z . Then the canonical gener-ator x = 1 + U p of U/U p belongs to t Z ( U/U p ) (see Proposition2.3). Since t Z ( U/U p ) is a U -submodule of U/U p we conclude that U/U p = t Z ( U/ p ) and condition (2) holds.(2) = ⇒ (1) Let M = 0 be an arbitrary nonzero U -module. If 0 = x ∈ t Z ( M ) then, by Proposition 2.3, there are p , . . . , p r ∈ Min Z andpositive integers n , . . . , n r > p n · . . . · p n r r x = 0. We shallprove that U x ⊆ t Z ( M ) by induction on k = n + . . . + n r . If k = 1then we have a minimal p ∈ Z such that p x = 0. Then we get anepimorphism of U -modules U/U p ։ U x (¯ u = u + U p ux ) whosedomain belongs to T Z when viewed as a Γ-module. Then U x belongsto T Z when viewed as a Γ-module, so that U x ⊆ t Z ( M ).Suppose now that k > 1. If p r x = 0 then we are done. So we canassume that p r x = 0. The induction hypothesis says that U p r x ⊆ t Z ( M ), from which it follows that the assignment ¯ u = u + U p r ux = ux + t Z ( M ) gives a well-defined map f : U/U p r −→ M/t Z ( M ),which is clearly a homomorphism of Γ-modules. Then we have thatIm( f ) = ( U x + t Z ( M )) /t Z ( M ) ∈ T Z since U/U p r belongs to T Z . Butwe also have that Im( f ) ∈ F Z because Im( f ) is a Γ-submodule of M/t Z ( M ). It follows that Im( f ) = 0, so that U x ⊆ t Z ( M ). (cid:3) Note that in our setting the commutative algebra Γ always has finiteKrull dimension, so that the (co)height of any of its prime ideal is aatural number. We are now in the position to prove our main result,which implies Theorem A. Theorem 3.2. (1) Let i be any natural number. The torsion the-ory ( T i , F i ) is liftable to U − Mod . (2) Let M be a simple U -module. There exists a (unique) naturalnumber i such that t i ( M ) = M and t i − ( M ) = 0 . In that case,all prime ideals in Ass( M ) have coheight exactly i .Proof. We prove the first statement by induction on i . If i = 0 we take m ∈ Min Z = Z = Specm Γ. In order to prove that U/U m ∈ T ,thus ending the proof (cf. Proposition3.1), it is enough to prove that Γ u Γ+ U m U m ∼ = Γ u ΓΓ u Γ ∩ U m is a ’left’ Γ-module in T , for all u ∈ U . Indeed wehave an epimorphism in Γ − Mod Γ u ΓΓ u m ։ Γ u ΓΓ u Γ ∩ U m .But since Γ u Γ is finitely generated as right Γ-module it follows that Γ u ΓΓ u m is finite dimensional as K -vector space. In particular Γ u ΓΓ u Γ ∩ U m is a’left’ Γ-module of finite length and hence belongs to T .Suppose now that i > i < d = Kdim (Γ) (the case i ≥ d istrivial). If p ∈ Min Z i and cht( p ) < i then the induction hypothe-sis says that U/U p ∈ T i − ⊂ T i . We assume then that cht( p ) = i .According to Proposition 3.1, it will be enough to prove that U/U p belongs to T i when viewed as a Γ-module. This is turn equivalentto prove that, for each u ∈ U , all the prime ideals of Γ containingann Γ ( u + U p ) = ( U p : u ) := { g ∈ Γ : gu ∈ U p } have coheight ≤ i (cf. Proposition 2.3). Therefore our goal is to prove that the Krulldimension of the algebra Γ / ( U p : u ) is ≤ i , for all u ∈ U . For thatwe shall use the fact that the Krull dimension of this latter algebracoincides with its Gelfand-Kirillov dimension (cf [KL][Proposition 7.9])We fix an element u ∈ U , a finite set of generators { u = u , u , . . . , u n } of Γ u Γ as right Γ-module and a finite set of generators { t , . . . , t m } ofΓ as a K -algebra. We consider the filtration ( F k ) k ≥ on Γ obtained bytaking as F k the vector subspace of Γ generated by the monomials ofdegree ≤ k on the t i . The induced filtration on Γ / ( U p : u ) is givenby ( F k +( U p : u )( U p : u ) ) k ≥ . The multiplication map ¯ g gu + U p is a K -linearisomorphism F k +( U p : u )( U p : u ) ∼ = −→ F k u + U p U p , for each k ≥ t i u j = P ≤ l ≤ n u l g lij , with g lij ∈ Γ,for all i = 1 , . . . , r and j = 1 , . . . , n . There exists a minimal positiveinteger s > { g lij } ⊂ F s . An easy induction gives that F k u j ⊆ P ≤ i ≤ n u i F sk , for all k ≥ j = 1 , . . . , n . In particularwe have F k u ⊆ P ≤ i ≤ n u i F sk , and hence F k u + U p U p ⊆ P ≤ i ≤ n u i F ks + U p U p ,for all k ≥ 0. Note that we have a surjective K -linear map F sk + U p U p ։ u i F ks + U p U p ( g + U p u i g + U p ).hen, taking K -dimensions, we obtain dim ( F k u + U p U p ) ≤ s · dim ( F ks + U p U p ),and hence log ( dim ( Fku + U p U p )) log ( k ) ≤ log ( s · dim ( Fks + U p U p )) log ( k ) , (*)for all k > 0. Note that we obtain a filtration ( F ′ k ) k ≥ of the algebra Γby putting F ′ k = F sk , for all k ≥ 0. Then, by applying limit superiorto the inequality (*) and bearing in mind that the Gelfand-Kirillovdimension decreases by passing to factor algebras, we get that GKdim (Γ / ( U p : u )) ≤ GKdim (Γ / ( U p ∩ Γ)) ≤ GKdim (Γ / p ) = i .This proves the first statement of the theorem. Let us now put i = min { j ≥ M ∈ T j } . Then we have t i ( M ) = M and t i − ( M ) ( M (convening that t − ( M ) = 0). By (1), it follows that t i − ( M ) is a proper U -submodule of M . The simplicity of M gives that t i − ( M ) = 0 and,using Corollary 2.5, the proof is completed. (cid:3) Question and Remark 3.3. According to Proposition 2.7, if M is asimple U -module and the prime ideals in Ass( M ) are pairwise coprime(e.g. if M ∈ T ) then, as Γ -module, we have a decomposition M = ⊕ p ∈ Ass( M ) M ( p ) . For an arbitrary simple M , using Theorem 3.2, itis not difficult to see that the sum P p ∈ Ass( M ) M ( p ) is direct, so that ⊕ p ∈ Ass( M ) M ( p ) is a Γ -submodule of M . Is it a U -submodule (so thatthe equality M = ⊕ p ∈ Ass( M ) M ( p ) holds)? Given a simple U -module, one needs recipes to calculate the i ≥ t i ( M ) = M and t i − ( M ) = 0. Recall that a subset { g , . . . , g r } ⊂ Γ is called a regular sequence in case P ≤ i ≤ n Γ g i = Γand ¯ g k := g k + P ≤ i Suppose that Γ is Cohen-Macaulay and equidimen-sional and let d = Kdim (Γ) be its Krull dimension. If M is a U -modulesuch that all ideals in Ass( M ) have the same coheight (e.g. a simple U -module), then the following assertions are equivalent: (1) t i ( M ) = M and t i − ( M ) = 0(2) There is a regular sequence in Γ , maximal with the property ofannihilating some x ∈ M \ { } , which has length d − i .Proof. The equidimensionality guarantees that ht( p ) + cht( p ) = d , forall p ∈ Spec Γ (cf. [Ku][Corollary II.3.6]). Note also that if { g , . . . , g k } is a regular sequence contained in ann Γ ( x ), for some x ∈ M \ { } , then,replacing if necessary x by some gx = 0 with g ∈ G , it is not restrictiveo assume that ann Γ ( x ) = q , for some prime ideal q ∈ Ass( M ). Soassertion (2) is equivalent to the following:(2’) There is a regular sequence in Γ of length d − i contained insome q ∈ Ass( M ) and maximal with that property.By [Ku][Theorem VI.3.14] and the fact that all prime ideals in Ass( M )have the same (co)height, this condition 2’ is in turn equivalent to saythat d − i = ht( q ), for every q ∈ Ass( M ). Therefore assertion 2) holdsif, and only if, cht( q ) = i for all q ∈ Ass( M ). By Corollary 2.5, this isequivalent to assertion (1). (cid:3) An approximation to the assassin of a U -module The preceding section shows that, given a simple U -module, its as-sassin as Γ-module, Ass( M ), is an important invariant. Therefore it isnatural to give recipes to approximate this subset of Spec(Γ). We willsee in this section that, knowing a prime p ∈ Ass( M ) and the finitesubset { u , . . . , u n } ⊂ U of our setup (see 1.1), one can give a precisesubset of Spec Γ in which Ass( M ) is contained.We will follow the terminology used for maximal ideals in [DFO2]and, given u ∈ U , we denote by X u the set of pairs ( q , p ) ∈ Spec Γ × Spec Γ such that Γ u Γ q u Γ+Γ u p = 0 (or equivalently Γ q ⊗ Γ Γ u Γ ⊗ Γ Γ p = 0).For simplicity, we shall write q ≡ u p whenever ( q , p ) ∈ X u .Note that, due to Nakayama lemma, if H if a finitely generated Γ-module and q ∈ Supp( H ) then q H = H . We will use this fact in theproof of the following result, which is a crucial tool for our purposes. Lemma 4.1. Let M be a U -module. The following assertions hold: (1) If u ∈ U , x ∈ M and q ∈ Supp(Γ ux ) , then there exists p ∈ Ass(Γ x ) such that q ≡ u p (2) If all prime ideals in Ass( M ) have the same coheight, then thereis an inclusion Ass(Γ( x + y )) ⊆ Ass(Γ x ) ∪ Ass(Γ y ) ,for all x, y ∈ M .Proof. 1) We have q ∈ Supp(Γ ux ) ⊆ Supp(Γ u Γ x ). It follows that Γ u Γ x q u Γ x = 0 since our setup 1.1 guarantees that Γ u Γ x is a finitely gener-ated Γ-module. The assignment ¯ v ⊗ y vy gives a surjective K -linearmap Γ u Γ q u Γ ⊗ Γ Γ x ։ Γ u Γ x q u Γ x = 0.It follows that Γ u Γ q u Γ ⊗ Γ Γ x = 0. But Γ x admits a finite filtrationwith successive factors isomorphic to Γ / p , with p ∈ Supp(Γ x ) (see[Ku][Proposition VI.2.6]). We conclude that there is a p ′ ∈ Supp(Γ x )such that Γ u Γ q u Γ ⊗ Γ Γ p ′ = 0. Choosing now p ∈ Ass(Γ x ) such that p ⊆ p ′ ,we get that Γ u Γ q u Γ ⊗ Γ Γ p = 0 and hence q ≡ u p .) Since we have an inclusion Γ( x + y ) ⊆ Γ x + Γ y it will be enough tocheck that Ass(Γ x + Γ y ) ⊆ Ass(Γ x ) ∪ Ass(Γ y ). To do that, we considerthe canonical exact sequence in Γ − Mod:0 → Γ x ∩ Γ y −→ Γ x ⊕ Γ y −→ Γ x + Γ y → x + Γ y ) ⊆ Supp(Γ x ⊕ Γ y ) = Supp(Γ x ) ∪ Supp(Γ y ).By hypothesis, all prime ideals in Ass( M ) have the same coheight,which implies that all of them are minimal in Supp( M ). As a conse-quence, if q ∈ Ass(Γ x + Γ y ) and we assume that q ∈ Supp(Γ x ), then q is minimal in Supp(Γ x ). This implies that q ∈ Min Supp(Γ x ) =Min Ass(Γ x ) ⊆ Ass(Γ x ). We replace x by y in case q ∈ Supp(Γ y ), andthe proof is finished. (cid:3) Proof of Theorem B. We are now in the position to prove The-orem B.If q ∈ Ass( M ) then we have q = ann Γ ( ux ), for some u ∈ U . If u ∈ Γthen q = p and there is nothing to prove. So we assume u Γ, inwhich case u is a sum of products of the form g u k g . . . g r u k r g r +1 ,where the g k belong to Γ and the k , . . . , k r belong to { , . . . , n } .Lemma 4.1 allows us to assume, without loss of generality, that u = g u k g . . . g r u k r g r +1 ,something that we do from now on in this proof.We then have q ∈ Ass(Γ ux ) ⊆ Ass(Γ u k g ...g r u k r g r +1 x ).By Lemma 4.1(1), there is a q ∈ Ass(Γ g u k ...g r u k r g r +1 x ) such that q ≡ u k q . By induction we get a sequence q = q , q , ..., q r of primeideals in Ass( M ), whence of coheight exactly cht( p ) (see Theorem 3.2,such that q r ∈ Ass(Γ g r +1 x ) and q i − ≡ u ki q i for i = 1 , ..., r . ButAss(Γ g r +1 x ) = { p } since ann Γ ( x ) = p is a prime ideal and g r +1 x = 0.Then q r = p and the proof is finished.Theorem B suggests to define, for each 0 ≤ i ≤ d , a (not necessarilysymmetric) relation ≡ in the set Min Z i of prime ideals of coheight i by saying that q ≡ p if, and only if, there are a sequence q = q , q , . . . , q s = p in Min Z i and a sequence of indices k , . . . , k s ∈{ , . . . , n } such that q i − ≡ u ki q i , for all i = 1 , . . . , s . Corollary 4.2. If M is a simple U -module and p , q ∈ Ass( M ) then q ≡ p .Proof. As U -module, M is generated by any of its nonzero elements.Choose 0 = x ∈ M such that ann Γ ( x ) = p and apply Theorem 3.2. (cid:3) We obtain immediately the following refinement of Proposition 2.7. orollary 4.3. Let M be a simple U -module and take p ∈ Ass( M ) ,with cht( p ) = i . Suppose that q and q ′ are coprime whenever q = q ′ are distinct prime ideals of Γ of coheight i such that q ≡ p and q ′ ≡ p .Then we have a decomposition M = ⊕ q ∈ Ass( M ) M ( q ) as Γ -module.Proof. By Theorem B, we have an inclusion Ass( M ) ⊆ { q ∈ Spec Γ :cht( q ) = i and q ≡ p } . Therefore the elements of Ass( M ) are pairwisecoprime and Proposition 2.7 applies. (cid:3) The following example shows that in some circumstances (usuallywhen the coheight is large), Theorem B is not sufficient to approximateAss(M). Example 4.4. Let U = A n ( K ) be the Weyl algebra given by generators X , . . . , X n , Y , . . . , Y n subject to the relations X i X j − X j X i = 0 = Y i Y j − Y j Y i X i Y j − Y j X i = δ ij ,for all i, j ∈ { , . . . , n } , where δ ij is the Kronecker symbol. Assume n > , put t i = X i Y i and put Γ = K [ t , . . . , t n ] . Then Γ and U satisfythe conditions of our setup 1.1 by taking u j ∈ { X σ ( j ) , Y σ ( j ) } for all j = 1 , . . . , n , where σ ∈ S n is any permutation. If p = Γ( t − then q ≡ p , for every prime ideal q ∈ Spec(Γ) of height .Proof. For simplicity put u i = Y i ( i = 1 , ..., n ), the other choices beingtreated similarly. Then one readily shows the equalities Y i t j = t j Y i ( i = j ) Y i t i = ( t i − Y i (equivalently t i Y i = Y i ( t i + 1)),for all i = 1 , ..., n . If f, g ∈ Γ are irreducible polynomials we derive fromthese equalities that f ≡ Y i g if and only if the polynomials s i ( f ) := f ( t , ..., t i − , t i + 1 , t i +1 , ..., t n ) and g are not coprime (i.e. the primeideals of Γ generated by them are not coprime). Indeed we have that f Y i Γ = Y i s i ( f )Γ and Γ Y i Γ = Y i Γ using the above equalities. Butthen the obvious isomorphism of ’right’ Γ-modules Γ ∼ = Y i Γ induces anisomorphism Γ Y i Γ fY i Γ = Y i Γ Y i s i ( f )Γ ∼ = ←→ Γ( s i ( f )) .It follows that Γ Y i Γ fY i Γ+Γ Y i g ∼ = Γ Y i Γ fY i Γ ⊗ Γ Γ( g ) is nonzero if an only if Γ( s i ( f )) ⊗ ΓΓ( g ) = 0. This happens exactly when s i ( f ) and g are not coprime.We pass now to prove the statement. If s i ( f ) is not coprime with t − 1, for some i = 1 , ..., n , then last paragraph applies with g = t − s i ( f ) is coprime with t − i = 1 , ..., n .( Note that this situation can actually happen. For instance if f = a + b ( t − m , with m > a, b ∈ K and a = 0 = a + ( − m b ). We thenput f ′ := s ( f ) and express it as a sum P ≤ k ≤ r g k ( t , ..., t n )( t − k .Then we get = f ′ Γ + ( t − g Γ + ( t − g is a constant polynomial, sothat we can rewrite f ′ ( t , ..., t n ) = a + ( t − m g ( t , ..., t ),where g ∈ Γ \ { } and a ∈ K \ { } . Note that, given any index i = 2 , . . . , n , we cannot have g ( t , . . . , t i − , α, t i +1 , . . . , t n ) = 0, for all α ∈ K . Indeed in that case the polynomial g would be zero. We thenchoose α ∈ K such that g ( t , α, t , . . . , t n ) = 0 and claim that f ′ and t − α are not coprimes. To see that, note that f ′ and t − α are coprimeif, and only if, ¯ f ′ := f ′ + ( t − α ) is invertible in Γ / ( t − α ). Using thecanonical isomorphism K [ t , . . . , t n ] / ( t − α ) ∼ = −→ K [ t , t , . . . , t n ](¯ h h ( t , α, t , . . . , t n )),we immediately find a polynomial u ∈ K [ t , t , . . . , t n ] satisfying theequality f ′ ( t , α, t , . . . , t n ) u ( t , t , . . . , t n ) = 1in K [ t , t , . . . , t n ]. It follows that f ′ ( t , α, t , . . . , t n ) = 1 + ( t − m g ( t , α, t , . . . , t n )is a constant polynomial, something which can only happen when g ( t , α, t , . . . , t n ) = 0. But this contradicts our choice of α .Put now h := t − α . We then get that f ≡ Y h since f ′ = s ( f ) is notcoprime with h = t − α . On the other hand, we also have h ≡ Y t − s ( h ) = h ( t + 1) = t + 1 − α is not coprime with t − 1. Wethen conclude that f ≡ t − (cid:3) We end the section with a result on extensions of U -modules. Proposition 4.5. Let M and N be nonzero U -modules and supposethat Γ u Γ q u Γ+Γ u p = 0 , for all u ∈ U , q ∈ Ass( M ) and p ∈ Ass( N ) . Thefollowing assertions hold: (1) Ext i Γ ( M, N ) = 0 = Ext i Γ ( N, M ) , for all i ≥ U ( N, M ) = 0 Proof. 1) By taking u = 1 above, we see that p and q are coprimewhenever p ∈ Ass( M ) and q ∈ Ass( N ). The assertion follows fromProposition 2.8.2) Let 0 → M −→ X −→ N → U − Mod . By assertion 1 we know that it split in Γ − Mod. Then we shall identify X = M ⊕ N , in which case the external multiplication map U × X −→ X (( u, x ) u · x ) is entirely determined by the U -module structureson M and N and by a K -bilinear map µ : U × N −→ M satisfying thefollowing three properties for all u, u ′ ∈ U , g ∈ Γ and y ∈ N :1) µ ( uu ′ , y ) = uµ ( u ′ , y ) + µ ( u, u ′ y ) (this guarantees that ( uu ′ ) · y = u · ( u ′ · y ))(2) µ ( g, y ) = 0, for all g ∈ Γ (this guarantees that the structureof Γ-module on M ⊕ N given by restriction of scalars via theinclusion j : Γ ֒ → U is that of the direct sum)(3) u · y = µ ( u, y ) + uy (this guarantees that the projection (cid:0) (cid:1) : X = M ⊕ N −→ N is a U -homomorphism)It follows that the assignment u ⊗ y µ ( u, y ) defines a homomorphismof Γ-modules µ ′ : U ⊗ Γ N −→ M .We claim that µ ′ = 0. Suppose not and take q ∈ Ass(Im( µ ′ )) ⊆ Ass( M ). The surjective Γ-homomorphism U ⊗ Γ N ։ Im( µ ′ ) inducesanother surjective Γ-homomorphism ⊕ u ∈ U,y ∈ N Γ u Γ ⊗ Γ Γ y ։ Im( µ ′ ).In particular, we get that q ∈ Supp(Γ u Γ ⊗ Γ Γ y ), for some u ∈ U and y ∈ N . Since Γ u Γ ⊗ Γ Γ y is an epimorphic image in Γ − Mod of Γ u Γ,which is finitely generated as ’left’ Γ-modules, it follows that Γ u Γ ⊗ Γ Γ y is a finitely generated Γ-module and thereby that q (Γ u Γ ⊗ Γ Γ y ) =Γ u Γ ⊗ Γ Γ y . That means that the left arrow in the exact sequence q u Γ ⊗ Γ Γ y −→ Γ u Γ ⊗ Γ Γ y −→ Γ u Γ q u Γ ⊗ Γ Γ y → Γ u Γ q u Γ ⊗ Γ Γ y = 0. The argument ofLemma 4.1(1) shows that there exists a p ∈ Ass(Γ y ) ⊆ Ass( N ) suchthat Γ u Γ q u Γ ⊗ Γ Γ p = 0. We then get Γ u Γ q u Γ+Γ u p = 0, which contradicts thehypothesis. (cid:3) Applications and some open questions We start with a proposition which will be useful in the sequel for itshypotheses are satisfied by all examples of this final section. Proposition 5.1. In the setup 1.1 suppose in addition that the follow-ing conditions hold: (1) If Z = Z ( U ) is the center of U then Z ∩ Γ is equidimensional(see [Mat] , p. 250) (2) Γ is flat as a Z ∩ Γ -module (3) For each simple U -module, the endomorphism algebra End U ( M ) has dimension equal to as a K -vector space.If U − f l denotes the subcategory of U -modules of finite length, then T i ∩ U − f l = T j ∩ U − f l , for all i, j ≥ Kdim (Γ) − Kdim ( Z ∩ Γ) .Proof. Let M be a simple U -module. Then the structural map K −→ End U ( M ) is an algebra isomorphism, which we view as an identifica-tion. On the other hand, every element z ∈ Z induces by multiplicationan endomorphism λ z ∈ End U ( M ). Put Z ′ = Z ∩ Γ. The assignment z λ z gives then an isomorphism ′ /ann Z ′ ( M ) ∼ = −→ End U ( M ) = K ,thus showing that m := ann Z ′ ( M ) is a maximal ideal of Z ′ . Let now p ∈ Spec (Γ) be minimal over Γ m . We clearly have m = Z ′ ∩ p and wehave an equality ht ( p ) = ht ( m ) + Kdim ( Γ p Γ p m )(cf. [Mat][Theorem 15.1]). But the prime spectrum of Γ p Γ p m is in bi-jection with the set of q ∈ Spec (Γ) such that Γ m ⊆ q ⊆ p . By ourchoice of p , this implies that Spec ( Γ p Γ p m ) has one element. It followsthat Kdim ( Γ p Γ p m ) = 0, so that ht ( p ) = ht ( m ), for all m ∈ Specm ( Z ′ )and all p ∈ Spec (Γ) minimal over Γ m .Put d := Kdim (Γ) and e := Kdim ( Z ′ ). Equidimensionality of Z ′ gives that ht ( m ) = e (cf. [Ku][Corollary II.3.6]). Then from the lastparagraph and the inequality ht ( p ) + Kdim (Γ / p ) ≤ Kdim (Γ)we readily derive that Kdim ( ΓΓ m ) = Sup { Kdim (Γ / p ) : p ∈ Spec (Γ) minimal over Γ m } ≤ d − e .This says that the coheight of any p ∈ Spec (Γ) containing a maximalideal of Z ′ is always ≤ d − e . In particular that happens for all p ∈ Ass ( M ), for every simple U -module M . It follows that the simple U -modules in T i are the same for all d − e ≤ i ≤ d , which implies thestatement. (cid:3) Remark 5.2. Bearing in mind that our field is algebraically closed,condition (3) in Proposition 5.1 is satisfied whenever U admits an ex-haustive filtration U ⊂ U ⊂ ... such that the associated graded algebra gr ( U ) is a commutative finitely generated algebra (cf. [Dix] [Lemma2.6.4]). It is the case for all finite W -algebras (cf. [BK1] ,Theorem10.1 or [GG] ,4.4). The following problems are of special interest in the case of envelop-ing algebras of Lie algebras and finite W-algebras. Problems 5.3. Suppose that Γ and U satisfy the conditions of Setup1.1 and also the hypotheses of Proposition 5.1. We propose the follow-ing problems: (1) To identify the set N U of natural numbers ≤ j ≤ d − e forwhich there exists a simple U -module M such that t j ( M ) = M and t j − ( M ) = 0 (convening that t − ( M ) = 0 ). (2) Given j ∈ N U , to identify the set of p ∈ Spec (Γ) such that cht ( p ) = j and p ∈ Ass ( M ) , for some simple U -module M (Local version) Given a character χ : Z ′ = Z ∩ Γ −→ K , toidentify the set N ( χ ) of natural numbers ≤ j ≤ d − e forwhich there exists a simple U -module M annihilated by Ker ( χ ) with t j ( M ) = M and t j − ( M ) = 0 . For any j ∈ N ( χ ) , toidentify all p ∈ Spec (Γ) such that cht ( p ) = j , Ker ( ξ ) ⊂ p and p ∈ Ass ( M ) for some simple U -module M . We move now to the announced classical examples.5.1. Finite W-algebras. Associated with a nilpotent element and agood grading in the Lie algebra gl n , there is associated a finite W -algebra (see [EK] for the definition and details). Each finite W-algebraof type A is determined by a sequence of integers τ = ( p , . . . , p m ) suchthat 1 ≤ p ≤ . . . ≤ p m and p + . . . + p m = n . We denote such analgebra by W ( τ ). If for each k = 1 , . . . , m we put τ k = ( p , . . . , p k ),then we obtain a chain of subalgebras W ( τ ) ⊂ . . . ⊂ W ( τ m ) = W ( τ ).The subalgebra Γ of W ( τ ) generated by the centers of the W ( τ k ) isa commutative algebra usually called the Gelfand-Tselin subalgebra of W ( τ ).As shown in [FO1] and [FO2], the algebra U = W ( τ ) and the com-mutative subalgebra Γ satisfy all the conditions of Setup 1.1 and allthe hypothesis of Proposition 5.1, actually with Z ⊂ Γ and hence Z ∩ Γ = Z . Moreover, we have d = mp + ( m − p + ... + 2 p m − + p m and e = p + . . . + p m (see [FMO] and [BK1]), where d and e are as inProposition 5.1. In particular we get: Corollary 5.4. Let us consider the natural number r = ( m − p +( m − p + . . . + p m − . The following assertions hold: (1) The torsion theories ( T i , F i ) ( i = 0 , , . . . , d ) are liftable from Γ − Mod to W ( τ ) − Mod . (2) If M is a simple W ( τ ) -module then there is a unique naturalnumber ≤ j ≤ r such that t j ( M ) = M and t j − ( M ) = 0 . Inthis case all prime ideals in Ass( M ) have coheight exactly j . Note that in the case m = n and p = . . . = p m = 1 the correspond-ing W -algebra is isomorphic to U (gl n ).5.2. The Lie algebra gl n . Given any positive integer n and any basis π = { α , . . . , α n } of the root system of the Lie algebra gl n , we denoteby gl i the Lie subalgebra corresponding to the simple roots α , . . . , α i .We then have inclusions of Lie algebrasgl ⊂ gl ⊂ . . . ⊂ gl n inducing corresponding inclusions of associative algebras U ⊂ U ⊂ . . . ⊂ U n ,here U k = U (gl k ) is the universal enveloping algebra of gl k for each k > 0. If we put U = U n then the subalgebra Γ( π ) of U generated bythe centers of U , . . . , U n is a maximal commutative subalgebra, calledthe Gelfand-Tsetlin subalgebra of U associated to the root system π .The inclusion Γ( π ) ⊂ U satisfies all the requirements of Setup 1.1and the hypotheses of Proposition 5.1, again with Z ⊆ Γ. ConcretelyΓ( π ) is isomorphic to a polynomial algebra on n ( n +1)2 variables (cf.[FO1], [FO2]) while the center Z = Z ( U ) is a polynomial algebra on n variables. We therefore have: Corollary 5.5. The following assertions hold: (1) The torsion theories ( T i , F i ) ( i = 0 , , . . . , n ( n +1)2 ) are liftablefrom Γ( π ) − Mod to U (gl n ) − Mod . (2) If M is a simple gl n -module then there is a unique natural num-ber ≤ j ≤ n ( n − such that t j ( M ) = M and t j − ( M ) = 0 . Inthis case all prime ideals in Ass( M ) have coheight exactly j . An interesting phenomenon for U n = U ( gl n ) is that there are severalGelfand-Tsetlin subalgebras to which we can apply our general theory,namely, one per each choice of a basis of the root system. We denoteby T i ( π ) the class of U n -modules M such that, viewed as Γ( π )-module, M belongs to T i . Since different root systems are conjugated by theWeyl group, one immediately gets: Proposition 5.6. Let π and π ′ be two bases of the root systems of gl n .The categories T i ( π ) and T i ( π ′ ) are equivalent for any i . Concerning Problem 5.3(1), it is well-known that 0 ∈ N U when U isa finite W -algebra of type A . For the particular case U = U ( gl n ) wehave that 1 ∈ N U , as the following example show. Example 5.7. There are simple gl n -modules which are not in T forall n > .Proof. Consider any generic simple non-weight (with respect to anyCartan subalgebra) gl -module V , such modules exist by [Bl]. Then V ∈ T and is not Gelfand-Tsetlin. Let H be a Cartan subalgebra ofgl . Fix a ∈ C . Let ( c , c ) be the central character of V ( c is aneigenvalue of e + e and c is an eigenvalue of the quadratic Casimirelement). Let P be a parabolic subalgebra of gl whose Levi factoris gl + H . Now consider the induced module M ( V, a ) = U (gl ) ⊗ U ( P ) V where V is naturally viewed as a P -module with a trivial actionof the radical and e + e + e acts by multiplication by a . Then M ( V, a ) has a unique simple quotient L ( V, a ) which belongs to thesubcategory T ⊂ gl − Mod and is not Gelfand-Tsetlin. Similarly,one can induce now from L ( V, a ) to get a gl -module with a uniquesimple quotient in T ⊂ gl − Mod which is not Gelfand-Tsetlin. Onecontinues inductively. Hence, for each n ≥ n -module in T which is not Gelfand-Tsetlin. (cid:3) . Acknowledgment The first author is supported in part by the CNPq grant (301743/2007-0) and by the Fapesp grant (2005/ 60337-2). 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