Higher dualizability and singly-generated Grothendieck categories
aa r X i v : . [ m a t h . C T ] F e b Higher dualizability and singly-generated Grothendieck categories
Alexandru Chirvasitu
Abstract
Let k be a field. We show that locally presentable, k -linear categories C dualizable in thesense that the identity functor can be recovered as ` i x i ⊗ f i for objects x i ∈ C and left adjoints f i from C to Vect k are products of copies of Vect k . This partially confirms a conjecture byBrandenburg, the author and T. Johnson-Freyd.Motivated by this, we also characterize the Grothendieck categories containing an object x with the property that every object is a copower of x : they are precisely the categories ofnon-singular injective right modules over simple, regular, right self-injective rings of type I orIII. Key words: locally presentable category; dualizable; abelian category; Grothendieck category; regularring; self-injective ring; non-singular module; type
MSC 2020: 18C35; 16D50; 18A30; 18A35
Introduction
The initial motivation for the present note stems from [4], which studies the notion of dualizabil-ity for particularly well-behaved categories: a property capturing a higher-categorical version ofthe “rigidity” specific to finite-dimensional vector spaces or, more generally, projective finitely-generated modules. The concept is of interest at least in part due to the pervasiveness of dualities(of objects, morphisms, functors, etc.) in topological quantum field theory: see e.g. [3, § VII] or[11, § dualizable and fully dualizable objects in a higher category are introduced.To recall the main characters, briefly (with more background and references in the main text),fixing a field k throughout: Definition 0.1 A k -module is a k -linear locally presentable category in the sense of [1, Definition1.17].2- k -modules form a complete and cocomplete self-enriched 2-category Vect M with k -linear leftadjoints as 1-morphisms and k -linear natural transformations as 2-morphisms; the notation is meantto suggest the higher-categorical setting: 2- k -modules are modules over Vect.The dual C ∗ is the 2- k -module Hom( C , Vect).A 2- k -module C is (or just plain dualizable , for brevity) if the canonical functor C ⊠ C ∗ → End( C ) (0-1)is an equivalence. (cid:7) (see Section 1 for more on the categorical tensor product ‘ ⊠ ’).There is a bit of a dearth of dualizable 2- k -modules not of the formlinear functors Γ → Vect1or small k -linear categories Γ, leading us to conjecture in passing in [4, Remark 3.6] that they areall of this form. This paper was initially an attempt to confirm this in particularly simple cases,i.e. the simplest one would try (see Definition 1.2 and theorem 2.1 below): Theorem 0.2
Suppose the 2- k -module C admits sets • x i ∈ C of objects and • f i ∈ C ∗ of functorssuch that the identity functor decomposes as ` i x i ⊗ f i . Then, C is a coproduct of copies of thecategory Vect . Definition 1.2 terms the categories in Theorem 0.2 naively dualizable , because they achievedualizability in the most straightforward way imaginable. In particular, one might consider thecase where the set { i } of indices in Theorem 0.2 is a singleton: the identity functor is of the form x ⊗ f for an object x ∈ C and a left adjoint functor C →
Vect. In that case, the result specializesfurther (Corollary 2.2):
Corollary 0.3
If for the non-trivial 2- k -ring we have id ∼ = x ⊗ f for an object x and a functor f ∈ C ∗ , then C ≃
Vect . An earlier version of the paper dealt with Corollary 0.3 directly (rather than as a consequence ofTheorem 0.2), by first noting that in every category C as in Corollary 0.3 every object is a coproduct(direct sum) of copies of x . This seems to be a peculiar property of some interest of its own, whichmotivates Section 3, which classifies Grothendieck categories with this property (Theorem 3.18):
Theorem 0.4
For a Grothendieck category C the following conditions are equivalent:(a) There is an object x ∈ C such that all objects of C are copowers (i.e. coproducts of copies) of x .(b) C is equivalent to the category M rA of non-singular right injective modules over a simple, regular,right self-injective ring that is either • a division ring, or • of type III in the sense of [7, Definition following Theorem 10.10]. Recall (e.g. [2, § non-singular modules are those containing no elements annihilated byessential ideals. Categories of non-singular injective modules feature prominently in the study of regular rings ( § M rA for regularself-injective A . M rA can be characterized in a number of other ways: it is also, for instance, the category ofinjectives over A that are summands of direct products A I for various index sets I ([15, ChapterXII, Theorem 1.3]).Note the contrast between Corollary 0.3 and Theorem 0.4: the dualizability requirement of theformer appears to further “rigidify” the category, imposing the condition that objects be copowersof x functorially. On the other hand, the second, type-III branch of Theorem 0.4 can be regardedas a loosening of that constraint, allowing for the less intuitive behavior characteristic of type-IIIrings: objects x isomorphic to their own double copowers x ⊕ x [7, Corollary 10.17], etc.2 cknowledgements This work was partially supported through NSF grant DMS-2001128.I am grateful for enlightening comments from Martin Brandenburg (who suggested the originalproblem) and Theo Johnson-Freyd on the contents of Section 2 and Ken Goodearl on those ofSection 3.
All categories and functors are assumed k -linear for a fixed field k . We writeVect = k Vectfor the category of k -vector spaces.We denote categories of modules by M , decorated with the base ring on the left or right,depending on whether they are left (or respectively right) modules. M A , for instance, is thecategory of right A -modules over a ring A .Inequality symbols between objects typically denote the relation of being a subobject, and ≤ ⊕ means ‘is a direct summand of’. For much of the material below we refer to the more extensive [4, §
2] as well as [1, Chapter 1], whichcovers valuable background on locally presentable categories. Taking that for granted, we pausehere only to recall that locally presentable categories are a particularly pleasant class of categoryto work with; a sample: • (One version of) Freyd’s adjoint functor theorem ([12, § V.8, Theorem 2]) holds for any suchcategory C , in the sense that a cocontinuous functor with domain C is automatically leftadjoint (as follows for instance from the cited theorem and [1, Remark 1.56]); • They form a complete and cocomplete 2-category, with left adjoints as 1-morphisms andnatural transformations as 2-morphisms ([5, Proposition 2.1.11]).All of the above goes through in the k -linear setting, as recalled in Definition 0.1. We note (e.g.[5, Proposition 2.1.11] ) that the (co)product of a family of 2- k -modules C i is simply their productas categories. Remark 1.1
Writing hom( C , D ) for the category of 1-morphisms between 2- k -modules, Vect M issymmetric monoidal with respect to a 2-categorical tensor product ‘ ⊠ ’ with the property that C ⊠ D receives a universal bifunctor C × D → C ⊠ D that is separately linear and cocontinuous.See for instance [4, Lemma 2.7], which in turn cites [1, Exercise 1.1] and [10, § (cid:7) Definition 1.2 C is simply (1-)dualizable if the identity functor id ∈ End( C ) is the image x ⊠ f id (1-1)through (0-1) of a simple tensor for some x ∈ C and f ∈ C ∗ . C is naively (1-)dualizable if the identity functor id ∈ End( C ) is the image a i ∈ I x i ⊠ f i id (1-2)through (0-1) of a coproduct of simple tensors for some family of x i ∈ C and f i ∈ C ∗ . (cid:7) .2 Regular self-injective rings We will have to dabble in the general theory of von Neumann regular rings, for which [7] will serveas our main source. Recall (e.g. [7, p.1, Definition]) that these are the (always unital) rings A with the property that for every x ∈ A there is some y ∈ A with x = xyx . Many equivalentcharacterizations exist: • every one-sided (left or right) principal ideal is generated by an idempotent; • every one-sided (left or right) finitely generated ideal is generated by an idempotent; • every (left or right) module is flat (for which reason the rings are also termed absolutely flat ); • finitely generated submodules of projective (left or right) A -modules are direct summands.We compress the phrase ‘von Neumann regular’ to just plain ‘regular’ for brevity. Of specialinterest will be right self-injective regular rings, i.e. those regular A which are injective as rightmodules over themselves. Unless specified otherwise, ‘self-injective’ means ‘right self-injective’.Regular right self-injective rings have received much attention in the literature; for our presentpurposes [7, Chapters 9-12] provide ample background, with [7, Chapter 10] of particular interest. Theorem 2.1
A naively 1-dualizable 2- k -module C is equivalent as a 2- k -module to a (co)product Vect ⊕ S of copies of Vect . Proof
Suppose we have a decomposition (1-2) of the identity functor on C . This then provides uswith • a functor π : Vect ⊕ I → C defined byVect ⊕ I ∋ ( V i ) i ∈ I M i ∈ I x i ⊗ V i ∈ C , and • a functor ι : C →
Vect ⊕ I going in the opposite direction, defined as the product ι := Y i f i : C → Y i ∈ I Vect ≃ Vect ⊕ I . The dualizability assumption means precisely that ι splits π in the sense that π ◦ ι ∼ = id.Since furthermore π is a left adjoint (as are all morphisms of 2- k -modules), it must be rightexact. On the other hand its domain Vect ⊕ I is a spectral Grothendieck category [14, Chapter 4,Proposition 2.3]: all of its objects are projective (or equivalently, injective). This means that allshort exact sequences split and hence π is in fact exact (i.e. preserves both kernels and cokernels).Now denote by ker π the full subcategory of Vect ⊕ I consisting of objects annihilated by π (cf.[6, Chapitre III, Proposition 5] or [14, § thick or dense subcategory of Vect ⊕ I in the sense of [6, § III.1] (there called ‘´epaisse’) or [14, § → X ′ → X → X ′′ → ,X belongs to ker π if and only if both X ′ and X ′′ do.4e can then form the quotient Vect ⊕ I / ker π of Vect ⊕ I by ker π in the sense of [14, discussionpreceding Lemma 3.4]. Moreover, the canonical functor T : Vect ⊕ I → Vect ⊕ I / ker π (of [14, § π is a localizing subcategory of Vect ⊕ I ([6, p.372]or [14, § § III.4, Proposition 8]: ker π is clearly closed undertaking colimits.According to [14, § ⊕ I Vect ⊕ I / ker π C πT π (2-1)for some (unique, up to natural isomorphism) exact faithful functor π .We do not quite know that π is an equivalence yet, because we do not, at the moment, havefullness; in fact, π will not , in general, be an equivalence: see Example 2.3 below. It is, however, • essentially surjective on objects, because it is right-invertible: π ◦ T ◦ ι ∼ = id; • faithful by construction.According to [6, § III.4, Proposition 8] a localizing subcategory
L ⊆
Vect ⊕ I is full on the objects { x ∈ Vect ⊕ I | hom( x, Q ) = 0 , ∀ Q } (2-2)where Q ∈ Vect ⊕ I ranges over some family F of injective objects. The defining condition for x in (2-2) can be recast as requiring that the i -component of x vanish for all i ∈ I for which some Q ∈ F has non-vanishing i -component. In short, L must be of the formVect ⊕ I ′ ⊆ Vect ⊕ I for some subset I ′ ⊆ I and hence we can identifyVect ⊕ I /kerπ ≃ Vect ⊕ ( I \ I ′ ) . To avoid overburdening the notation, we may as well assume (as we will for the duration of theproof) that π : Vect ⊕ I → C is faithful. Together with πι ∼ = id C , that faithfulness then implies that ι : C →
Vect ⊕ I (2-3)is full (in addition to being a faithful left adjoint). In other words, (2-3) identifies C with a fullsubcategory of Vect ⊕ I , closed under forming colimits.With all of this in hand, we define the support supp I ( C ) of C in I to be the subset of those i ∈ I for which there is some object x ∈ C whose i -component (a vector space) is non-zero. It is easyto construct, for each i ∈ supp I ( C ), an endomorphism x → x in C whose cokernel is precisely theone-dimensional i -indexed vector space k i (regarded as an object of Vect ⊕ I ). But then, since (2-3)is • full and 5 colimit-closed, k i ∈ C . Taking colimits now shows that in fact C (regarded as a full subcategory of Vect ⊕ I via(2-3)) is precisely Vect ⊕ supp I ( C ) ⊆ Vect ⊕ I . This finishes the proof, taking S = supp I ( C ). (cid:4) In particular, taking I to be a singleton in the proof of Theorem 2.1, we obtain the followingcharacterization of simply-dualizable 2- k -modules. Corollary 2.2
A non-trivial simply 1-dualizable 2- k -module C is equivalent to Vect as a 2- k -module. (cid:4) Example 2.3
In (2-1) let • I = { , } be a two-elements set, so that Vect ⊕ I is equivalent to M k × k ; • C ≃ vect; • π : M k × k → Vect be scalar restriction via the diagonal map k → k × k ; • ι : Vect → M k × k be scalar restriction via the projection k × k → k on, say, the first component.Then π is clearly faithful (so that T in (2-1) is an equivalence), π and ι are both left adjoints, wehave πι ∼ = id C , but π is not an equivalence. (cid:7) In the context of Corollary 2.2 one considers 1-dualizable 2- k -modules where every object is acoproduct of copies of a single object x . The result concludes that C is equivalent to Vect buit seems pertinent, in view of that setup, to examine such “singly-generated” categories in moredetail. Definition 3.1 A singleton category is a (usually linear) category C with a distinguished object x such that all objects of C are (possibly empty) coproducts of copies of x . The object x itself is thena singleton for C .The same term applies to narrower classes of categories: singleton 2- k -modules, singletonGrothendieck categories, etc. (cid:7) As seen above, being singleton is a weakening of the notion of simple 1-dualizability. First,recall the following notion (e.g. [14, Chapter 4, Proposition 2.3]).
Definition 3.2
A Grothendieck category is spectral if all of its objects are injective or, equivalently,projective. (cid:7)
Our first observation is
Lemma 3.3
A singleton Grothendieck category is spectral in the sense of Definition 3.2. roof Let C be a singleton Grothendieck category and x ∈ C an object with the property thatevery other object of C is a copower x ⊕ S (for some set S ).Note that there are arbitrarily large injective copowers x ⊕ S (i.e. for arbitrarily large sets S ): simply take an injective envelope E of some copower of x whose set of subobjects has somearbitrarily large cardinality, and use the fact that by assumption, E must itself be a copower of x .Finally, embed an arbitrary copower x ⊕ S as a summand into an injective x ⊕ T , | S | ≤ | T | , henceconcluding that the arbitrary copower x ⊕ S must again be injective. (cid:4) Lemma 3.3 is what first goes wrong as soon as one tries to generalize these considerations tomore than a single object x : Example 3.4
Let k be a field. The category M A of modules over the ring A = k [ ε ] / ( ε ) of dualnumbers has the property that every object is a direct sum of copies of k and A (the indecomposableobjects of M A ). M A is not spectral, showing that even two “base” objects will undo Lemma 3.3. (cid:7) The following remark justifies the phrase “ the singleton”.
Lemma 3.5
In a singleton Grothendieck category all singletons are isomorphic.
Proof If x and y are singletons, each is a summand of the other, and they are moreover injectiveby Lemma 3.3. It follows that they are isomorphic, e.g. by [7, Theorem 10.14] (which is phrasedfor modules, but goes through in arbitrary Grothendieck categories). (cid:4) Lemma 3.6
Let C be a singleton Grothendieck category and = x ∈ C a singleton. Then, allnon-zero subobjects of x are isomorphic. Proof
The argument is similar to that used in the proof of Lemma 3.5: if y ≤ x is a subobjectthen x and y are injectives realizable as summands in each other, and hence are isomorphic by [7,Theorem 10.14]. (cid:4) Now let C be a singleton Grothendieck category. Lemma 3.3 imposes strong restrictions on thestructure of C . Specifically, [14, Chapter 4, Theorem 2.8] says that there is a regular self-injectivering A such that C ≃ M rA , (3-1)where the latter (with ‘ r ’ standing for ‘reduced’) denoting the full subcategory M rA ⊂ M A consisting of right modules appearing as direct summands of powers A I for sets I (see also [15,Chapter XII, Theorem 1.3]). Furthermore, the equivalence (3-1) can be described concretely asHom C ( x, − ) : C → M rA (3-2)for some generator x ∈ C .Regarded as a functor to the category M A of all modules Hom C ( x, − ) is a right adjoint andhence preserves products. It does not , in general, preserve coproducts: although the category M rA is cocomplete (indeed, it is Grothendieck), direct sums are somewhat more involved that directproducts, which are simply the usual Cartesian products of modules. The following result follows,for instance, from [7, Proposition 9.1], and we omit the proof.7 emma 3.7 For a regular self-injective ring A , the coproduct in M rA of a family of objects X j ∈M rA , j ∈ J is the unique summand E of the product Q j X j fitting into the chain M j X j ≤ E ≤ ⊕ Y j X j so that the left hand inclusion is essential. (cid:4) Remark 3.8
In particular, the uniqueness of such a summand is part of the statement; it is thatuniqueness that follows from the cited result, specifically [7, Proposition 9.1 (e)]. (cid:7)
Throughout the discussion we fix a singleton Grothendieck category C with a singleton 0 = x ∈C . A typically denotes a regular right self-injective ring with C ≃ M rA but we do not assume, ingeneral, that the equivalence identifies A to x (or to a singleton, in general).The goal is to understand the structural restrictions imposed on A by the demand that C besingleton. A first observation, Lemma 3.10 below, borrows the following operator-algebraic term(e.g. [9, Chapter 1, Exercise 7], [13, Definition 3.1.2], [16, Definition II.3.2], etc.). Definition 3.9
A regular ring is a factor if its only central idempotents are 0 and 1. (cid:7)
Lemma 3.10
The regular self-injective ring A in (3-1) is a factor. Proof
Let e ∈ A be a central idempotent. If e were non-trivial (i.e. neither 0 nor 1) then eA and(1 − e ) A would both be objects of M rA ≃ C whose annihilators ((1 − e ) A and eA respectively) arenot both contained in the annihilator of a non-trivial object in M rA .On the other hand, x ∈ C is a summand of every non-zero object in C and hence, as an A -module, its annihilator contains those of every other C -object. This gives the desired contradiction,proving that e ∈ { , } . (cid:4) In particular,
Corollary 3.11
In the context of Lemma 3.10 the center of A is a field. Proof
This follows from Lemma 3.10 and [7, Corollary 1.15]. (cid:4)
In a sense, Lemmas 3.6 and 3.10 completely classify singleton Grothendieck categories. First,we need
Definition 3.12
A right A -module is isominimal if it is isomorphic to all of its non-zero summands.The ring A itself is isominimal if it is isominimal as a right A -module.Finally, x ∈ A is isominimal if xA is. (cid:7) Remark 3.13
Note that isominimal regular rings are automatically factors. (cid:7)
Theorem 3.14
For a non-trivial Grothendieck category C , the following conditions are equivalent:(1) C is singleton.(2) C is equivalent to M rA for a (non-zero) regular isominimal self-injective ring.(3) C is equivalent to M rA for a regular self-injective factor containing some non-zero isominimalidempotent e ∈ A . roof (1) ⇒ (2). This is a consequence of Lemma 3.6 and the surrounding discussion: we havean equivalence (3-2) for a regular self-injective A , identifying a singleton x ∈ C to A ∈ M rA . Itfollows from Lemma 3.6 that all non-zero principal right ideals of A are isomorphic to A , and thusthe latter must be isominimal. (2) ⇒ (3). Trivial. (3) ⇒ (1). We will argue that every summand X ≤ ⊕ A I of a power of A is a direct sum (in M rA ) of copies of eA . We first do this for the one-element set I : Claim: Non-zero summands of A are direct sums in M rA of copies of eA . Let xA ≤ ⊕ A be a non-zero summand X j ≤ xA, j ∈ J (3-3)be a maximal independent family of objects X j ∼ = eA (‘independent’ in the sense that, as before,their sum in A is direct). Given the description of coproducts in M rA provided by Lemma 3.7, thismeans proving that the actual, module-theoretic direct sum L X j is essential in xA .If not, the unique summand E ≤ ⊕ xA that contains L j X j essentially has a non-zero comple-ment F , i.e. xA = E ⊕ F . Since A is a regular self-injective factor, it satisfies the comparabilityaxiom ([7, p.80, Definition]) by [7, Corollary 9.16] and hence one of eA and F must be a summandof the other. Either way, the isominimality of e implies that eA is realizable as a summand of F ,in which case the independence of the family { X j , j ∈ J } ⊔ { F } contradicts the maximality of (3-3). This concludes the proof of the claim.We now resume the main line of reasoning. Once more, consider a maximal independent family X j ∼ = eA of copies of eA . We claim that X = M X j in M rA . As above, we have to show that the module-theoretic direct sum L X j is essential in X . Theargument is entirely parallel: assuming the opposite, the unique summand E ≤ ⊕ X that contains L j X j essentially will be proper in X and hence have a complement: X = E ⊕ F, F = 0 .F embeds as a summand in the original power A I , and hence maps non-trivially to some factor A I → A via a morphism ϕ : F → A . We know from [7, Proposition 9.1, parts (a) and (b)] that0 → ker ϕ → F → im ϕ → A of ϕ . It thus follows that a non-zerosummand Z of A can be realized as a summand of the complement F = X ⊖ E . But this meansthat the maximal family { X j } can be extended by some summand eA ≤ ⊕ Z (at least one exists bythe claim proven above). This contradicts the maximality of the family { X j } and thus finishes theproof. (cid:4) As for a more concrete description of singleton categories, we can resort (via (3-1)) to theclassification theory of regular rings as covered in [7, Chapter 10].9 roposition 3.15
Let
C ∼ = M rA be a singleton category for a regular self-injective ring A as thesingleton. Then, A is of type I or III. Proof
Given that • A is a factor by Lemma 3.10 on the one hand and • being regular self-injective, A decomposes uniquely as a product of type-I, II and III rings onthe other ([7, Theorem 10.13]),all we have to do is rule out type II.If A were type-II it would have a non-zero directly finite idempotent e ∈ A [7, Proposition10.8], in which case eA could not be isomorphic to any of its proper direct summands by the verydefinition of direct finiteness for modules ([7, p.49, Definition]). This means that eA must be a finite direct sum of copies of the singleton x , so we may as well assume eA ∼ = x .By direct finiteness again x ∼ = eA cannot be isomorphic to a proper direct summand, so in fact eA must be indecomposable. But then e = 0 is an abelian idempotent in the sense of [7, p.110,Definition]: the regular ring eAe has no non-trivial idempotents at all, and hence is a division ring(and in particular an abelian regular ring). This contradicts the assumption that A is type-II ([7,p.113, Definition] requires that no non-zero abelian idempotents exist), finishing the proof. (cid:4) As expected, the “easy” case is that of type I.
Proposition 3.16
For a Grothendieck category C the following conditions are equivalent:(1) C is equivalent to the category of right vector spaces over a division ring D .(2) C is a singleton Grothendieck category with an indecomposable singleton object.(3) C is singleton and any regular self-injective ring A acting as a singleton in M rA ≃ C is a divisionring.(4) C is singleton and any regular self-injective ring A with M rA ≃ C is a factor of type I.(5) C is equivalent to M rB for some regular self-injective type-I factor B . Proof (1) ⇒ (2). Indeed, in that case the (unique, up to isomorphism) singleton is D itself, whichas indecomposable as a right D -module. (2) ⇒ (3). If there is an indecomposable singleton (which will automatically be unique upto isomorphism) then A ∈ M A must be indecomposable and hence simple as a right A -module(because principal right ideals are summands). It follows that A is a division ring. (3) ⇒ (4). Division rings are type-I. (4) ⇒ (5). Trivial. (5) ⇒ (1). Being of type I, [7, Theorem 10.6] implies that B is of the form End D ( M ) where • D is an abelian regular self-injective ring (‘abelian’ in the sense of [7, Definitions on p.25 andp.110]: idempotents are central); • M is a non-singular injective right D -module.Since B is a factor so is D . Abelian factors are division rings, so B is a full endomorphism rings ofa vector space M over a division ring D . But in that case we have C ≃ M r End D ( M ) ≃ Vect D , finishing the proof. (cid:4)
10s for the type-III arm of Proposition 3.15, not only does it occur, but in fact accounts foreverything Proposition 3.16 doesn’t.
Proposition 3.17
For any type-III regular self-injective factor A , the category M rA is singleton. Proof [8, Theorem 5-3.6] implies that for a type-III regular self-injective factor A the poset formedby the isomorphism classes of finitely-generated A -projectives (with order given by being a directsummand) is isomorphic to an initial segment of ordinal numbers.In particular, such a factor has a non-zero isominimal idempotent in the sense of Definition 3.12and the conclusion follows from Theorem 3.14. (cid:4) Collecting together Propositions 3.15 to 3.17, we obtain the following complete classification ofsingleton Grothendieck categories.
Theorem 3.18
The singleton Grothendieck categories C are precisely those of the form(a) M rA ≃ Vect D for a division ring A = D ;(b) M rA for a type-III regular self-injective factor or, equivalently, simple ring A . Proof
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Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
E-mail address : [email protected]@buffalo.edu