On properly stratified Gorenstein algebras
OON PROPERLY STRATIFIED GORENSTEIN ALGEBRAS
TIAGO CRUZ AND REN´E MARCZINZIK
Abstract.
We show that a properly stratified algebra is Gorenstein if and only if the characteristictilting module coincides with the characteristic cotilting module. We further show that properly stratifiedGorenstein algebras A enjoy strong homological properties such as all Gorenstein projective modulesbeing properly stratified and all endomorphism rings End A p ∆ p i qq being Frobenius algebras. We applyour results to the study of properly stratified algebras that are minimal Auslander-Gorenstein algebrasin the sense of Iyama-Solberg and calculate under suitable conditions their Ringel duals. This appliesin particular to all centraliser algebras of nilpotent matrices. Introduction
Quasi-hereditary algebras constitute an important class of finite dimensional algebras including manywell-studied algebras such as algebras of global dimension at most two, Schur algebras, see for example[Don98] or [Gre07], and blocks of category O , see for example [Hum08].Standardly stratified algebras were introduced as a generalisation of quasi-hereditary algebras in[CPS96]. It can be shown that a standardly stratified algebra A is quasi-hereditary if and only if A has finite global dimension and standardly stratified algebras include many important algebras with in-finite global dimension arising for example in Lie theory, see [Maz04b]. Standardly stratified algebrasalways have a characteristic tilting module but in general no characteristic cotilting module. When theopposite algebra of a standardly stratified algebra is also standardly stratified, the algebra is properlystratified. The properly stratified algebras where the characteristic tilting and cotilting modules coincideare of particular importance as they have many strong homological properties that are in general missingin the more general case of standardly stratified algebras, see for example [FM06]. In [MP04], Mazorchukand Parker conjectured that the finitistic dimension of any properly stratified algebra with a simple pre-serving duality is equal to two times the projective dimension of the characteristic tilting module. Thisconjecture was proven by Mazorchuk and Ovsienko in [MO04] under the additional assumption that thecharacteristic tilting module coincides with the characteristic cotilting module and in [Maz04a] it wasshown that the conjecture is wrong without this assumption. Our main result gives a new homologicalcharacterisation when the characteristic tilting module coincides with the characteristic cotilting moduleand implies for example that the main theorem of Mazorchuk and Ovsienko can be simplified as thefinitistic dimension coincides with the Gorenstein dimension of the algebra under their assumptions. Theorem (2.2) . For a properly stratified algebra A , the characteristic tilting module coincides with thecharacteristic cotilting module if and only if A is Gorenstein. We show that properly stratified Gorenstein algebras enjoy strong homological properties. The nexttheorem collects the most important properties.
Theorem (2.4, 2.5) . Let A be a properly stratified Gorenstein algebra with standard modules ∆ .(1) Every Gorenstein projective A -module is properly stratified (i.e. it belongs to F p ∆ q ).(2) A module M is in F p ∆ q if and only if M is in F p ∆ q with finite projective dimension.(3) All endomorphism rings End A p ∆ p i qq are Frobenius algebras. We introduce GIGS algebras as gendo-symmetric properly stratified Gorenstein algebras having aduality. Those algebras generalise the algebras introduced by Fang and Koenig in [FK11b] from thequasi-hereditary scenario to the more general case of properly stratified algebras. We will see that central
Date : January 29, 2021.2010
Mathematics Subject Classification.
Primary 16G10, 16E10.
Key words and phrases. properly stratified algebras, minimal Auslander-Gorenstein algebras, Gorenstein algebras, tiltingmodules, dominant dimension. a r X i v : . [ m a t h . R T ] J a n TIAGO CRUZ AND REN´E MARCZINZIK results obtained by Fang and Koenig still hold even when the algebras are not necessarily quasi-hereditary.The class of GIGS algebras contain all Schur algebras S p n, k q for n ě k , blocks of category O and we willshow that they also contain centraliser algebras of nilpotent matrices, which are properly stratified butin general not quasi-hereditary. By the result of Mazorchuk-Ovsienko, GIGS algebras always have evenGorenstein dimension and since they are gendo-symmetric by definition they are always isomorphic to analgebra of the form End U p X q for a symmetric algebra U and a generator X of mod- U . One of the mostimportant subclasses of Gorenstein algebras are the minimal Auslander-Gorenstein algebras introducedby Iyama and Solberg in [IS18] as a generalisation of higher Auslander algebras that were introduced in[Iya07]. Our main result for GIGS algebras that are minimal Auslander-Gorenstein is as follows: Theorem (3.5) . Let A “ End U p U ‘ M q be a GIGS algebra with a symmetric algebra U and a generator U ‘ M of mod- U , where we can assume that M has no projective direct summands. Assume A isfurthermore a minimal Auslander-Gorenstein algebra with Gorenstein dimension equal to d for some d ě . Then the Ringel dual R A of A is isomorphic to End U p U ‘ Ω d p M qq . In particular, R A is again aGIGS algebra that is minimal Auslander-Gorenstein with Gorenstein dimension d . Recall that for an n ˆ n -matrix M with entries in a field K , the centralizer algebra of M is defined asthe K -algebra of all n ˆ n -matrices X with XM “ M X . In Subsection 3.1, we will apply the previoustheorem to calculate the Ringel duals of centraliser algebras of nilpotent matrices and determine whenthey are Ringel self-dual. 1.
Preliminaries
In this paper, unless stated otherwise, all algebras under consideration are finite dimensional over afield K and all modules are right modules. We assume the reader is familiar with the basic representationtheory and homological algebra of finite dimensional algebras and refer for example to [SY11] and [ARS97]for a basic introduction. For a subcategory C of mod- A for a finite dimensional algebra A , we denoteby add p C q the full subcategory of direct sums of direct summands of a module X P C . For a module X , add p X q simply denotes the full subcategory of modules that are direct summands of X n for some n . By D we denote the functor Hom K p´ , K q and J denotes the Jacobson radical of an algebra A . Fora subcategory U of mod- A we denote by p U (or q U ) the full subcategory of mod- A consisting of modules X such that there is an exact sequence 0 Ñ C n Ñ ¨ ¨ ¨ Ñ C Ñ X Ñ Ñ X Ñ C Ñ ¨ ¨ ¨ Ñ C n Ñ
0) with C i P U . For a module X we define K X : “ t Y P mod- A | Ext iA p Y, X q “ i ą u and X K : “ t Y P mod- A | Ext iA p X, Y q “ i ą u . By Fac p X q we denote the full subcategory of modulesthat are an epimorphic image of X . We denote by P ă8 the full subcategory of modules having finiteprojective dimension and by I ă8 the full subcategory of modules having finite injective dimension. Asubcategory U of mod- A is called resolving if it contains all projective A -modules and is closed underdirect summands, extensions and kernels of epimorphisms. A coresolving subcategory is defined dually.1.1. Gorenstein algebras and cotilting modules.
An algebra A is called Gorenstein in case the leftand right injective dimensions of the regular module A are finite, in which case they coincide and thecommon number is the Gorenstein dimension of A . A selfinjective algebra is a Gorenstein algebra withGorenstein dimension zero, or equivalently an algebra where all projective modules are injective. A iscalled Frobenius algebra when D p A q – A as A -right modules and it is called symmetric when D p A q – A as A -bimodules. Every Frobenius algebra is selfinjective and every selfinjective quiver algebra is a Frobeniusalgebra. An algebra A is called gendo-symmetric if D p A q b A D p A q – D p A q or equivalently when A isisomorphic to End B p M q for a symmetric algebra B with a generator M of mod- B . We refer to [FK16] and[FK11a] for more information and properties of gendo-symmetric algebras. The finitistic dimension ofan algebra A is defined as the supremum of all projective dimensions of modules having finite projectivedimension. A module C P mod- A is called cotilting if it has no self-extensions, finite injective dimensionand DA P { add C . Tilting modules are defined, analogously. Theorem 1.1.
The following are equivalent:(1) A is Gorenstein.(2) A module is a tilting module if and only if it is a cotilting module.(3) P ă8 “ I ă8 . N PROPERLY STRATIFIED GORENSTEIN ALGEBRAS 3
Proof.
See [HU96, Lemma 1.3]. (cid:3)
Proposition 1.2.
Let A be an algebra such that A and A op have finite finitistic dimension. In case theleft or right injective dimension of A is finite, A is Gorenstein. In this case the Gorenstein dimensioncoincides with the finitistic dimension of A and A op .Proof. See [AR91, Proposition 6.10]. That the finitistic dimension coincides with the Gorenstein dimen-sion for Gorenstein algebras can be found for example in [Che17, Lemma 2.3.2]. (cid:3)
Recall that a pair p X , Y q of subcategories of mod- A is called a cotorsion pair when Y “ t M P mod- A | Ext A p X , M q “ u and X “ t N P mod- A | Ext A p N, Y q “ u . A cotorsion pair is called complete when X is contravariantly finite. The next result gives a correspondence between basic cotilting modulesand certain complete cotorsion pairs. Theorem 1.3.
Let A be a general finite dimensional algebra. There is a one-one correspondence betweenbasic cotilting modules U and complete cotorsion pairs p X , Y q with X resolving and p X “ mod- A , given by U Ñ p K U, { add p U qq and p X , Y q Ñ the direct sum of all indecomposable modules in X X Y .Proof. See [Rei07, 2.3 (b)]. (cid:3)
Standardly and properly stratified algebras.
For the convenience of the reader, we will recallsome definitions and elementary properties involving stratified algebras.Let e “ t e , . . . , e n u be a linearly ordered complete set of primitive orthogonal idempotents. For each i “ , . . . , n , we write P p i q “ e i A, S p i q “ top p e i A q , I p i q “ D p Ae i q , ∆ p i q “ P p i q{ e i A p e i ` ` ¨ ¨ ¨ ` e n q A. That is, ∆ p i q is the maximal quotient of P p i q without composition factors S p j q , j ą i . Define ∆ p i q to bethe maximal quotient of ∆ p i q having only once S p i q as composition factor. We call ∆ “ t ∆ p q , . . . , ∆ p n qu the (right) standard modules of A and ∆ “ t ∆ p q , . . . , ∆ p n qu the proper standard modules. Let ∆ op and∆ op be the standard and proper standard modules of A op , respectively. By the costandard and propercostandard (right) modules of A we mean ∇ p i q “ D ∆ op p i q and ∇ p i q “ D ∆ op p i q .For a given set of modules Θ, by F p Θ q we mean the full subcategory of mod- A having a filtration bythe modules in the set Θ. Definition 1.4.
An algebra together with a linearly ordered complete set of orthogonal primitive idem-potents p A, e q is called ‚ standardly stratified if A A P F p ∆ q ; ‚ properly stratified if A A P F p ∆ q X F p ∆ q .Note that A A P F p ∆ q if and only if DA A P F p ∇ op q . This last condition is equivalent to A op beingstandardly stratified (see [AHLU00, Theorem 1.1], [ADL98, 2.2] and [Lak00]).The next theorem collects some results on properly stratified algebras: Theorem 1.5.
Let p A, e q be a properly stratified algebra. Then the following assertions hold: (a) F p ∆ q “ t M P mod- A | Ext A p M, F p ∇ qq “ u is a resolving subcategory of mod- A . (b) F p ∇ q “ t N P mod- A | Ext A p F p ∆ q , N q “ u is a coresolving subcategory of mod- A . (c) F p ∆ q Ď P ă8 and F p ∇ q Ď I ă8 . (d) Ext kA p F p ∆ q , F p ∇ qq “ for all k ě . (e) There exists a (unique) tilting module T such that F p ∇ q “ T K and F p ∇ q “ mod- A . (f) There exists a unique basic tilting module T such that add T “ F p ∆ q X F p ∇ q . (g) There exists a (unique) cotilting module C such that F p ∆ q “ K C and { F p ∆ q “ mod- A . (h) There exists a unique basic tilting module C such that add C “ F p ∆ q X F p ∇ q . (i) T K Ď F ac p T q .Proof. Assertions (a) and (b) are the content of Theorem 1.6 of [AHLU00]. For (c), see [PR01, Proposition1.3]. Assertion (d) follows from A op being standardly stratified and from [ADL98, Theorem 3.1]. Forassertions (e) and (f) see Theorem 2.1 and Proposition 2.2 of [AHLU00]. See also Theorem 3.3 of [Xi02].Assertions (g) and (h) follow from assertions (e) and (f) since A op is standardly stratified. For (i), see[AHLU00] on page 151. (cid:3) TIAGO CRUZ AND REN´E MARCZINZIK
We call the tilting module T in the conditions of Theorem 1 . p e q and 1 . p f q the characteristic tiltingmodule of the properly stratified algebra p A, e q . Dually, we call the cotilting module C in the conditions1 . p g q and 1 . p h q the characteristic cotilting module of the properly stratified algebra p A, e q . By theprevious theorem we have that F p ∆ q “ add T , F p ∇ q “ T K and p F p ∆ q , F p ∇ qq is a complete cotorsionpair, see also [Rei07, Theorem 3.6] for this and the dual result.The following lemma although being elementary and in some sense folklore in the literature of stratifiedalgebras will be useful afterwards to characterize Gorenstein properly stratified algebras. Lemma 1.6.
Let p A, e q be a standardly stratified algebra. Then the following assertions hold.(1) Ae n A is projective as right A -module and p A { Ae n A, t e , . . . , e n ´ uq is standardly stratified, where e is the image of e in the quotient A { Ae n A ;(2) Ae n is a projective generator of e n Ae n ;Moreover, if p A, e q is properly stratified then ∆ p i q P F p ∆ p i qq , for each i “ , . . . , n .Proof. Taking into account that each ∆ p j q with j ‰ n belongs to mod- A { Ae n A and filtrations of A canbe chosen so that higher indexes appear at the bottom, A having a filtration by standard modules givesthat A { Ae n A P F p ∆ p i q i ‰ n q (see for example [DK94, Lemma A2.2]).Since F p ∆ q is a resolving subcategory of mod- A , the exact sequence0 Ñ Ae n A Ñ A Ñ A { Ae n A Ñ Ae n A P F p ∆ q . Since Ae n A P F ac p e n A q and all ∆ p j q have no composition factors of the form S p n q we must have Ae n A P F p ∆ p n qq . It follows that Ae n A is projective and add Ae n A “ add e n A . Inparticular, Hom A p e n A, A { Ae n A q “
0. This shows (1).Write e “ e n . Since AeA is projective we can write Ae b eAe eA – AeA . Therefore,add eAe Ae “ add eAe Ae b eAe eAe “ add eAe AeAe “ add eAe Hom A p eA, AeA q “ add eAe End A p eA q . This shows (2).Assume now that p A, e q is properly stratified. Then p A op , e q is standardly stratified and F p ∇ op q is acoresolving subcategory of mod- A op (see [AHLU00, Theorem 1.6]). Thus, F p ∆ q is a resolving subcategoryof mod- A . Therefore, ∆ p n q P F p ∆ q . Further, since Ext p ∆ p i q , ∆ p j qq “
0, for i ą j (see [AHLU00, Lemma1.2] ) we can rearrange, if necessary, the filtration of modules in F p ∆ q so that proper standard moduleswith lower index appear in the top of the filtration. So, there exists a surjective map ∆ p n q Ñ ∆ p j q , where j is the lowest index in the filtration of ∆ p n q . Then S p n q “ top ∆ p n q Ñ top ∆ p j q “ S p j q is surjective.So j must be n . Hence, ∆ p n q P F p ∆ p n qq . Since each ∆ p j q has no composition factors of the form S p i q with i ą j the last claim follows by induction. (cid:3) Proposition 1.7.
Let A be a finite dimensional algebra with an idempotent e such that the ideal AeA isprojective as a right A -module. Then Ext iA { AeA p M, N q “
Ext iA p M, N q for all M, N P mod- A { AeA .Proof.
See for example [APT92, Example 1]. (cid:3)
Theorem 1.8.
Let p A, e q be a standardly stratified algebra. Then A has finite finitistic dimension.Proof. See [AHLU00, Corollary 2.7]. (cid:3)
Let A be a properly stratified algebra with the associated subcategory F p ∆ A q and B a properlystratified algebra with the associated subcategory F p ∆ B q . Then A and B are said to be equivalent asproperly stratified algebras when F p ∆ A q and F p ∆ B q are exact equivalent. When T is the characteristictilting module of A and R A : “ End A p T q the Ringel dual of A (which is also properly stratified by [Rei07,Theorem 3.7]), then A is said to be Ringel self-dual if A and R A are equivalent as properly stratifiedalgebras. This generalises the classical notion of being Ringel self-dual for quasi-hereditary algebras, seefor example [CE18, Seciton 5]. N PROPERLY STRATIFIED GORENSTEIN ALGEBRAS 5
Standardly stratified algebras with a duality.
The algebra p A, e q is said to have a duality ι if ι is an anti-automorphism of A fixing the complete set of primitive orthogonal idempotents e and ι “ id A .Algebras with a duality were studied in [FK11b]. The existence of the duality implies the existence ofexact functors p´q ι : mod- A Ñ A -mod and ι p´q : A -mod Ñ mod- A satisfying, in particular, p e i A q ι – Ae i and ι p M ι q – M , for every M P mod- A . Denote by p´q : mod- A Ñ mod- A the composition of functors D ˝ p´q ι . In addition, assume in the remaining of this subsection that p A, e q is a standardly stratifiedalgebra. Then P p i q – D p Ae i q “ I p i q and S p i q – S p i q . So, p´q is a simple preserving duality of A .Moreover, ∆ p i q – ∇ p i q and ∆ p i q – ∇ p i q , for all i . Applying D to the isomorphism ∆ p i q – ∇ p i q weobtain ∆ p i q ι – D ∇ p i q – ∆ op p i q . It follows that A A – p A A q ι P F p ∆ ι q “ F p ∆ op q . Therefore, a standardlystratified algebra p A, e q with a duality is a properly stratified algebra (see also [CD05, Proposition 2.3]). Theorem 1.9.
Let p A, e q be a properly stratified algebra having a duality with characteristic tilting module T coinciding with the characteristic cotilting module. Then findim p A q “ p T q .Proof. See [MO04]. (cid:3) A characterisation of Gorenstein properly stratified algebras
We will need the following lemma on local algebras in this section:
Lemma 2.1.
Let A be a local algebra.(1) A has finitistic dimension equal to zero.(2) A is Gorenstein if and only if A is selfinjective.Proof. Note that in a local algebra there is a unique indecomposable projective module and thus allprojective modules have the same Loewy length. Let M be a non-projective module and let f be theprojective cover f : P Ñ M . Then Ω p M q “ Ker p f q has Loewy length strictly smaller than theLoewy length of P and thus Ω p M q cannot be projective. Therefore all Ω i p M q are non-projective andthus M has infinite projective dimension and the finitistic dimension is zero. This shows (1). Clearlyany selfinjective algebra is Gorenstein. Now assume that A is a local Gorenstein algebra. Since A isGorenstein, the finitistic dimension of A coincides with the Gorenstein dimension of A by 1.2. Thus theGorenstein dimension is zero and A is selfinjective, which shows (2). (cid:3) Theorem 2.2.
Let p A, e q be a properly stratified algebra with characteristic tilting module T and char-acteristic cotilting module C . Then the following are equivalent:(1) A is Gorenstein.(2) T “ C .(3) { add p T q “ F p ∇ q .Proof. We first show that (2) is equivalent to (3) and that (3) implies (1). Assume (2). Then T “ C andthus { add p T q “ { add p C q “ F p ∇ q . Now assume (3), so that we have { add p T q “ F p ∇ q . Since A is properlystratified we have D p A q P F p ∇ q “ { add p T q . Thus there is by definition an exact sequence0 Ñ T r Ñ ¨ ¨ ¨ Ñ T Ñ D p A q Ñ T i P add p T q . Now since T is a tilting module, it has finite projective dimension. Let K i ` bedefined as the kernel of the maps T i Ñ T i ´ for i ě T ´ : “ D p A q . Then the exact sequence0 Ñ T r Ñ T r ´ Ñ K r ´ Ñ K r ´ has finite projective dimension since T r , T r ´ P add p T q have finite projective dimension. Inductively we see that also each K i has finite projective dimensionbecause of the short exact sequences 0 Ñ K i ` Ñ T i Ñ K i Ñ i ě
1. Thus also D p A q has finiteprojective dimension because of the short exact sequence 0 Ñ K Ñ T Ñ D p A q Ñ
0. Since the rightmodule D p A q has finite projective dimension, by duality the left module A has finite injective dimension.Since A is assumed to be properly stratified, A and A op have finite finitistic dimension by 1.8 and thusby 1.2 A is Gorenstein. Thus we see already that (3) implies (1) and we go further that (3) implies also(2). To see this, note that since A is Gorenstein we know that the tilting module T is also a cotiltingmodule by 1.1. By the cotilting correspondence 1.3, T as a cotilting module corresponds to the cotorsionpair p K T, { add p T qq . But the characteristic cotilting module C also has { add p T q “ F p ∇ q as the right side inthe cotorsion pair (and the left and right sides in a cotorsion pair determine each other) and this implies T “ C . TIAGO CRUZ AND REN´E MARCZINZIK
Assume now that (1) holds. The characteristic cotilting module is the unique module (up to multi-plicities) satisfying add C “ F p ∆ q X F p ∇ q . Thus, it is enough to prove that add T “ F p ∆ q X F p ∇ q . Thecategory F p ∆ qX F p ∇ q only contains n non-isomorphic indecomposables modules, so we just need to showthat T P F p ∆ q X F p ∇ q . Since T is the characteristic tilting module add T “ F p ∆ q X F p ∇ q . By Lemma1.6, T P F p ∆ q Ă F p ∆ q . Due to T P F p ∇ q and Lemma 1.2 (vi) of [AHLU00], Ext j ą A p ∆ p i q , T q “
0, for all i . In view of Theorem 1.6 of [AHLU00] and A op being standardly stratified, we want0 “ Ext A op p DT, ∇ op p i qq – Ext A p D ∇ op p i q , T q – Ext A p ∆ p i q , T q , for all i . By Theorem 1.1, T has finite injective dimension. Let s i be the smallest non-negative integer suchthat Ext s i A p ∆ p i q , T q ‰ s i ` A p ∆ p i q , T q “
0, for each i . Let i “ t , . . . , n u be arbitrary and assumethat s i ą
0. Since A is properly stratified then ∆ p i q admits a filtration 0 “ X Ă X Ă ¨ ¨ ¨ Ă X t “ ∆ p i q with factors of the form ∆ p i q by Lemma 1.6. Applying Hom A p´ , T q to this filtration yields the exactsequence Ext s i A p X r , T q Ñ Ext s i A p X r ´ , T q Ñ Ext s i ` A p ∆ p i q , T q “ , r “ , . . . , t. (2.2.1)Since X “ ∆ p i q it follows by (2.2.1) that Ext s i A p X r , T q ‰
0, for all r ě
1. In particular, 0 ‰ Ext s i A p X t , T q “ Ext s i A p ∆ p i q , T q . This contradicts T belonging to F p ∇ q . Thus, s i “
0. As i was ar-bitrary, this shows that T P F p ∇ q . (cid:3) Remark 2.3.
Given a characteristic tilting module T of a properly stratified algebra A , to check condition(2) of Theorem 2.2 it is enough to observe whether T P F p ∇ q . For a characteristic cotilting module C ofa properly stratified algebra A , it is enough to observe that C P F p ∆ q .2.1. Properties of Gorenstein properly stratified algebras.
We record some further properties ofGorenstein properly stratified algebras. Recall that a module M over a Gorenstein algebra A is called Gorenstein projective (or also maximal Cohen-Macaulay in the literature) if Ext iA p M, A q “ i ą N is called Gorenstein injective if Ext iA p D p A q , N q “ i ą Proposition 2.4.
Let p A, e q be a properly stratified Gorenstein algebra.(1) Every Gorenstein injective module is in F p ∇ q .(2) Every Gorenstein projective module is in F p ∆ q .(3) P ă8 X F p ∆ q “ F p ∆ q .(4) I ă8 X F p ∇ q “ F p ∇ q .Proof. We prove (1), the proof of (2) is dual. Let T be a tilting module. T has by definition finiteprojective dimension and since A is Gorenstein, T also has finite injective dimension. Let0 Ñ T Ñ I Ñ ¨ ¨ ¨ Ñ I r ` Ñ T . Then we have the short exact sequence 0 Ñ Ω ´ r p T q Ñ I r Ñ I r ` Ñ . Now let X be an A -module with X P D p A q K , then we apply the functor Hom A p´ , X q to theabove short exact sequence and obtain the following exact sequence as a part of the resulting long exactExt-sequence for every t ě ¨ ¨ ¨ Ñ Ext tA p I r ` , X q Ñ Ext tA p I r , X q Ñ Ext tA p Ω ´ r p T q , X q Ñ Ext t ` A p I r ` , X q Ñ ¨ ¨ ¨ . Since we assume X P D p A q K , we obtain Ext tA p Ω ´ r p T q , X q “ t ě
1. Now we apply the functorHom A p´ , X q to the short exact sequence 0 Ñ Ω ´p r ´ q p T q Ñ I r ´ Ñ Ω ´ r p T q Ñ tA p Ω ´p r ´ q p T q , X q “ t ě tA p T, X q “ t ě
1. Thus we have D p A q K Ď T K for every tilting module T assuming that A is Gorenstein.Choosing for T the characteristic tilting module, we obtain D p A q K Ď F p ∇ q .We prove now (3), the proof of (4) is dual. F p ∆ q Ď P ă8 X F p ∆ q is clear by 1.5(c). Now assume X P P ă8 X F p ∆ q . By the dual of Lemma 2.5. (iii) of [AHLU00], every module M P F p ∆ q has a (possiblyinfinite) add p T q -coresolution. Thus X has such a coresolution as follows:0 Ñ X Ñ T Ñ T Ñ ¨ ¨ ¨ Ñ T i Ñ ¨ ¨ ¨ , with T i P add p T q . Now since A is Gorenstein, X P P ă8 has finite injective dimension. If the add p T q -coresolution would be infinite, we would have Ext iA p K i , X q ‰ i ą K i is the kernel N PROPERLY STRATIFIED GORENSTEIN ALGEBRAS 7 of T i Ñ T i ` and X would have infinite injective dimension. Thus the coresolution is finite and X P add p T q “ F p ∆ q . (cid:3) Theorem 2.5.
Let p A, e q be a properly stratified Gorenstein algebra. Then the following hold for any i :(1) End A p ∆ p i qq is a Frobenius algebra.(2) End A p ∇ p i qq is a Frobenius algebra.(3) The quotient p A { A p e i ` ` ¨ ¨ ¨ ` e n q A, e i q is a properly stratified Gorenstein algebra, with e i “t e i , . . . , e ii u where e ij denotes the image of e j in A { A p e i ` ` ¨ ¨ ¨ ` e n q A .Proof. We prove (1), the proof of (2) is dual. Assume A is properly stratified Gorenstein with linearorder 1 , ..., n of the n simple A -modules. Let I be an injective A -module. Since F p ∆ q is contravariantlyfinite, there exists a minimal right F p ∆ q -approximation of I :0 Ñ K I Ñ F Ñ I Ñ , where F P F p ∆ q . Since F p ∆ q is extension-closed we obtain Ext A p F p ∆ q , K I q “ K I P F p ∇ q . Now we apply the functor Hom A p X, ´q for X P F p ∆ q to the above shortexact sequence to obtain from the long exact Ext-sequence: ¨ ¨ ¨ Ñ “ Ext A p X, K I q Ñ Ext A p X, F q Ñ Ext A p X, I q “ Ñ ¨ ¨ ¨ . Thus Ext A p X, F q “ X P F p ∆ q and thus F P F p ∆ q X F p ∇ q “ add p T q . Since K I P F p ∇ q wecan continue this process to obtain an add p T q -resolution of I as follows: ¨ ¨ ¨ Ñ F t Ñ F t ´ Ñ ¨ ¨ ¨ Ñ F Ñ I Ñ . Let K t denote the kernel of the map F t Ñ F t ´ then if this resolution would be infinite, we would haveExt t p I, K t q ‰ t ě I would have infinite projective dimension, contradicting ourassumption that A is Gorenstein. Thus there exists a finite add p T q -resolution of I :0 Ñ T r Ñ T r ´ Ñ ¨ ¨ ¨ Ñ T Ñ I Ñ , with T i P add p T q for all injective A -modules I . Since ∆ p n q is projective, the module I : “ D Hom A p ∆ p n q , A q is injective. We apply the functor Hom A p ∆ p n q , ´q to the above finite add p T q -resolutionof this I to obtain the exact sequence:(2.5.1) 0 Ñ Hom A p ∆ p n q , T r q Ñ ¨ ¨ ¨ Ñ Hom A p ∆ p n q , T q Ñ Hom A p ∆ p n q , I q Ñ . By [AHLU00, Lemma 2.5], there is a left F p ∇ q -approximation of ∆ p i q as follows:0 Ñ ∆ p i q Ñ T p i q Ñ Y i Ñ , where Y i P F p ∆ j ă i q and T p i q is the indecomposable i -th summand of the characteristic tilting module T . We apply Hom A p ∆ p n q , ´q to the left approximation as above for i “ n to obtain:0 Ñ Hom A p ∆ p n q , ∆ p n qq Ñ Hom A p ∆ p n q , T p n qq Ñ Hom A p ∆ p n q , Y n q “ Ñ . Thus Hom A p ∆ p n q , T p n qq – Hom A p ∆ p n q , ∆ p n qq is a projective R -module, where R : “ End A p ∆ p n qq .Since Hom A p ∆ p n q , T p j qq “ j ă n , we have that Hom A p ∆ p n q , T q is a projective R -module for any T P add p T q . Thus the exact sequence 2.5.1 gives us that the module Hom A p ∆ p n q , I q has finite projectivedimension as an R -module. But since R is local and therefore has finitistic dimension equal to zero by2.1 (1), Hom A p ∆ p n q , I q must be even projective. Now, since ∆ p n q is A -projectiveHom A p ∆ p n q , I q “ Hom A p ∆ p n q , D Hom A p ∆ p n q , A qq – D p Hom A p ∆ p n q , A q b A ∆ p n qq “ DR is injective as an R -module. Thus Hom A p ∆ p n q , I q is a projective-injective non-zero module for the localalgebra R and thus R is selfinjective and therefore also a Frobenius algebra (using that every selfinjectivelocal algebra is automatically Frobenius). We proved that R “ End A p ∆ p n qq is Frobenius and now we canuse Ext iA { Ae n A p M, N q –
Ext iA p M, N q for every A { Ae n A -modules M and N , which is true by 1.7. Thisgives us that Ext iA { Ae n A p M, A { Ae n A q – Ext iA p M, A { Ae n A q “ i ą p for some finite number p since as an A -module A { Ae n A has finite projective dimension (since Ae n A has finite projective dimension)and therefore also finite injective dimension, using that A is Gorenstein. Thus with A , also the properlystratified algebra A { Ae n A is Gorenstein. Now we can use induction to conclude that End A p ∆ p i qq isFrobenius for all i “ , ..., n and the remaining cases of (3). (cid:3) TIAGO CRUZ AND REN´E MARCZINZIK
The next example shows that there are non-Gorenstein properly stratified algebras such that P ă8 X F p ∆ q “ F p ∆ q and such that all endomorphism rings of the ∆ p i q are Frobenius algebras. This shows thatthose properties can not be used to characterise the Gorenstein property for properly stratified algebras. Example 2.6.
Let Q be the quiver 2 1 β α and A “ KQ { I with I “ă α , βα ą . A isa radical square zero algebra and it is easily checked that A has exactly five indecomposable moduleswith add p P ă8 q “ add p e A ‘ e A q . Thus A has finitistic dimension equal to zero. Since for Gorensteinalgebras, the finitistic dimension coincides with the Gorenstein dimension, A can only be Gorenstein whenit is selfinjective. Since the indecomposable projective modules are not injective, A is not selfinjective andtherefore also not Gorenstein. We have ∆ p q “ e A, ∆ p q “ e A, ∆ p q “ S and ∆ p q “ e A . Thus A isproperly stratified and not Gorenstein. We have End A p ∆ p qq – K r x s{p x q and End A p ∆ p qq – K , whichare Frobenius algebras. We have P ă8 X F p ∆ q “ F p ∆ q since add p P ă8 q “ add p e A ‘ e A q “ F p ∆ q . The following example shows that unlike quasi-hereditary algebras, in general, Gorenstein properlystratified algebras have no recollements of bounded derived categories for A { AeA , A and eAe . Example 2.7.
Let K be an algebraically closed field and let A be the bound quiver K -algebra1 2 3 α β α β with relations 0 “ β α α “ β β α “ β α , α β “ β α β α . A is a properly stratified algebra with ∆ p q “ ∆ p q “ e A , ∆ p q “ ∆ p q “ S p q , ∆ p q “ e A { ∆ p q ,and ∆ p q “
21 . A has injective dimension two and A { Ae A is the bound quiver algebra 1 2 α β with relation β α β α “
0. Note that Hom A p A { Ae A, P p qq – ∆ p q as A { Ae A -modules. Since ∆ p q has infinite injective dimension it follows that the condition (1) of Proposition 3.4 [CK17] holds butcondition (3) fails. 3. GIGS algebras
We first define a generalised version of the algebras studied by Fang and Koenig in [FK11b].
Definition 3.1. A GIGS algebra is a gendo-symmetric properly stratified Gorenstein algebra p A, e q having a duality.In [FK11b], the class of gendo-symmetric and quasi-hereditary algebras p A, e q having a duality is called A . Note that a GIGS algebra has finite global dimension if and only if it is in the class A of [FK11b]. By1.9, GIGS algebras have even Gorenstein dimension equal to two times the projective dimension of thecharacteristic tilting module since the Gorenstein dimension coincides with the finitistic dimension. Ourmain result 2.2 says that being Gorenstein is equivalent to the characteristic tilting module coinciding withthe characteristic cotilting module, which is the central property used by Fang and Koenig in [FK11b] toprove their main results. For a properly stratified algebra A , we will denote by R A the Ringel dual of A .Note that since being Gorenstein is a derived invariant, A is Gorenstein if and only if R A is Gorenstein.We give an example of a GIGS algebra with infinite global dimension. Example 3.2.
Let A “ K r x, y s{p xy, x ´ y q and M “ A ‘ xA ‘ yA ‘ x A and B “ End A p M q . Notethat A is a (commutative) symmetric Frobenius algebra and thus B is gendo-symmetric. By results ofChen and Dlab in [CD05] B is also properly stratified with a duality. B has Gorenstein dimension 4,dominant dimension 2 and infinite global dimension.The next result was proven in [FK11b, Theorem 4.3] for GIGS algebras with finite global dimension,but the arguments remain valid for GIGS algebras once we establish that the Ringel dual is also gendo-symmetric. Theorem 3.3.
Let p A, e q be a GIGS algebra with characteristic tilting module T .(1) The Ringel dual of A , R A , is again a GIGS algebra.(2) p T q “ domdim p A q . N PROPERLY STRATIFIED GORENSTEIN ALGEBRAS 9
Proof.
By Theorem 2.2, T is also the characteristic cotilting module and therefore the Ringel dual of A , R A , is a properly stratified algebra by Theorem 5 of [FM06]. Using the exact sequences for tilting modulesestablished in [AHLU00, Lemma 2.5] we can see that T p i q – T p i q , for all i . In particular, T – T . ByProposition 2.4 of [FK11b], R A has a duality τ which fixes all the idempotents T (cid:16) T p i q ã Ñ T . For thestatement (1), it remains to show that R A is gendo-symmetric. Let eA be the minimal faithful projective-injective module of A . In particular p eA q – eA . Analogously to [FK11b, Lemma 4.2], Hom A p T, eA q is aright minimal faithful projective-injective module over R A and Hom A p DT, Ae q is a left minimal faithfulprojective-injective module over R A . Further, as p eAe, R A q -bimodulesHom R A p Hom A p T, eA q , R A q – Hom R A p Hom A p T, eA q , Hom A p T, T qq (3.3.1) – Hom A p eA, T q – Hom A p DT, D p eA qq – Hom A p DT, Ae q . (3.3.2)The last isomorphism follows from A being gendo-symmetric. Note also that since Hom A p T, ´q is fullyfaithful on F p ∇ q , we also have the isomorphism eAe – End A p eA q – End R A p Hom A p T, eA qq . Since Lemma4.2 of [FK11b] remains true for properly stratified algebras, part (2) of the proof of Theorem 4.3 of [FK11b]also holds for R A . That is, domdim R A ě
2. By Theorem 3.2 of [FK11a], R A is gendo-symmetric and(1) holds. It is now clear that the analogous statement of [FK11b, Lemma 3.2(5)] holds for R A . Finally,since the remaining arguments that are involved in the proof of [FK11b, Theorem 4.3] remain true underthe assumption of p A, e q being properly stratified in place of just being quasi-hereditary, (2) follows. (cid:3) We can now apply our results to calculate the Ringel dual of GIGS algebras that are minimal Auslander-Gorenstein algebras. Recall that an algebra A is called minimal Auslander-Gorenstein if it has finiteGorenstein dimension equal to the dominant dimension and both dimensions are at least two (we excludeselfinjective algebras in the definition for minimal Auslander-Gorenstein algebras here for simplicity asselfinjective algbras are not interesting for standardly stratified algebras). Minimal Auslander-Gorensteinalgebras generalise higher Auslander algebras, which are exactly those minimal Auslander-Gorensteinalgebras with finite global dimension.We will need the following result on minimial Auslander-Gorenstein algebras, see [IS18, Theorem 4.5]. Theorem 3.4.
There is a bijection between Morita equivalence classes of minimal Auslander-Gorensteinalgebras A of Gorenstein dimension n ě and equivalence classes of tuples p B, N q of algebras B witha generator-cogenerator N of mod- B such that Ext iB p N, N q “ for i “ , ..., n ´ and τ p Ω n ´ p N qq P add p N q . The bijection associates to p B, N q the algebra A “ End B p N q . Note that when B is symmetric and N “ B ‘ M with M having no projective direct summands asin the previous theorem, then τ – Ω (see for example [ARS97, Proposition 3.8 in Chapter IV]) and thecondition τ p Ω n ´ p N qq P add p N q simplifies to Ω n p M q – M . Theorem 3.5.
Let A “ End U p U ‘ M q be a GIGS algebra with a symmetric algebra U and a generator U ‘ M of mod- U , where we can assume that M has no projective direct summands. Assume A isfurthermore a minimal Auslander-Gorenstein algebra with Gorenstein dimension equal to d for some d ě . Then the Ringel dual R A of A is isomorphic to End U p U ‘ Ω d p M qq . In particular, R A is again aGIGS algebra that is minimal Auslander-Gorenstein with Gorenstein dimension d .Proof. A is properly stratified and Gorenstein and thus we can apply our main result 2.2 to conclude thatthe characteristic tilting module of A coincides with the characteristic cotilting module, which we denoteby T . By [Mar18, Theorem 2.5 and Theorem 2.6.], T – eA ‘ Ω ´ d p A q when eA denotes the minimalfaithful projective-injective A -module. Now R A – End A p eA ‘ Ω ´ d p A qq . Since d ě
1, the module T has dominant dimension at least one and codominant dimension at least one. By [APT92, Lemma 3.1(2) (ii)] the functor Hom A p eA, ´q induces an isomorphism of algebras between End A p eA ‘ Ω ´ d p A qq andEnd eAe p eAe ‘ Ω ´ d p A q e q . Now since the functor Hom A p eA, ´q is exact and sends projective-injective A -modules to injective eAe -modules, we have Ω ´ d p A q e – Ω ´ d p Ae q . Thanks to the Morita-Tachikawacorrespondence and since eAe is symmetric, we have Ae – U ‘ M as p A, eAe q -bimodules and thus R A – End A p eA ‘ Ω ´ d p A qq – End eAe p eAe ‘ Ω ´ d p Ae qq – End U p U ‘ Ω ´ d p M qq . Since eAe is symmetric and A is minimal Auslander-Gorenstein, M is 2 d -periodic: Ω d p M q – τ p Ω d ´ p M qq – M and thus End U p U ‘ Ω ´ d p M qq – End U p U ‘ Ω d p M qq . When B is a general sym-metric algebra and N a B -module without projective direct summands, the algebras End B p B ‘ N q and End B p B ‘ Ω i p N qq are almost ν -stable derived equivalent in the sense of [HX10] and this derived equiv-alence preserves the Gorenstein and dominant dimensions, see [HX10, Corollary 1.3 (2)] and [HX10,Corollary 1.2]. Now, the Ringel dual has also a duality and is properly stratified by Theorem 3.3 andthus the statement follows. (cid:3) We have the following corollary:
Corollary 3.6.
Let A be a GIGS basic algebra that is minimal Auslander-Gorenstein. Then A is Ringelself-dual if and only if A is isomorphic to its Ringel dual R A .Proof. For any properly stratified algebra A , being Ringel self-dual implies that A and R A are, in par-ticular, Morita equivalent. Since both algebras are basic, A and R A are isomorphic. Now assume that A and R A are isomorphic, then the isomorphism induces a Morita equivalence between mod- A and mod- R A .Now by [Mar18, Main theorem] a GIGS algebra C that is minimal Auslander-Gorenstein has the propertythat F p ∆ q is equal to the subcategory of mod- C consisting of the modules with projective dimension atmost r when r is the Gorenstein dimension of C . Since by 3.5 A and R A have the same Gorensteindimension and a Morita equivalence preserves the projective dimension of modules, the subcategories F p ∆ A q and F p ∆ R A q are isomorphic and A is Ringel self-dual. (cid:3) Centraliser algebras of nilpotent matrices.
Let H be an arbitrary nilpotent matrix in K n ˆ n for a field K and n ě
2. Then it is well known, see for example [AW92, Proposition 4.1], that thecentraliser t X P K n ˆ n | XH “ HX u is isomorphic to Hom K r x s p V H , V H q , where we view V H – K n asan K r x s -module by letting x act as H on K n . Now since H is nilpotent x n annihilates V H and thusHom K r x s p V H , V H q – Hom K r x s{p x n q p V H , V H q is a finite dimensional K -algebra. Since every indecomposable K r x s{p x n q -module is isomorphic to a module of the form p x k q{p x n q for some k “ , , ..., n ´ V H is a direct sum of such indecomposable modules and we can up to Morita equivalenceassume that V H is basic. Furthermore, we can assume that H is nilpotent with H n “ H n ´ ‰ K r x s{p x n q (which is the unique indecomposable projective K r x s{p x n q -module) appears as a direct summand. We call Hom K r x s p V H , V H q – Hom K r x s{p x n q p V H , V H q the centraliser algebra of the nilpotent matrix H . A classical theorem of Frobenius gives the vector spacedimension of a general centraliser algebra of a matrix. For this and more on centraliser algebras we referfor example to [AW92, Section 5.5]. We will see that this algebra satisfies the assumption of 3.5 and wewill use this to determine when the algebra is Ringel self-dual.We now fix notation for our problem. Let U “ K r x s{p x n q and N “ U ‘ M an arbitrary non-projectivegenerator of mod- U , where we can assume that M has no projective direct summands and up to Moritaequivalence we can also assume that N is basic. The algebra A “ End U p N q is properly stratified (see[CD05, Theorem 2.4]) and has a duality (this is explained in [CD05] at the end of page 64). Example 3.7.
Let U “ K r x s{p x q and N “ U ‘ S , where S is the unique simple U -module. Then A “ End U p N q is isomorphic to the following quiver algebra KQ { I with quiver Q : 1 2 β β β and relations I “ x β β , β β , β β , β β ´ β y . A has infinite global dimension and thus is not quasi-hereditary. Proposition 3.8. A “ End U p N q is a GIGS algebra that is minimal Auslander-Gorenstein of Gorensteindimension 2. It is Ringel self-dual if and only if M – Ω p M q .Proof. That A is properly stratified with a duality follows from [CD05]. Now A is minimal Auslander-Gorenstein of Gorenstein dimension 2 since M is 2-periodic. Thus A is a GIGS algebra that is Auslander-Gorenstein and we can use 3.6 that shows that A is Ringel self-dual if and only if A is isomorphic to itsRingel dual. By 3.5, the Ringel dual of A is isomorphic to End U p U ‘ Ω p M qq . Thus when M – Ω p M q , A is Ringel self-dual.Now assume that A “ End U p N q is Ringel self-dual. We show that then M – Ω p M q . Let M have indecomposable non-projective direct summands U { J p i for i “ , ..., r with p ă p ă ... ă p r , N PROPERLY STRATIFIED GORENSTEIN ALGEBRAS 11 where we recall that J denotes the Jacobson of the algebra U . Then Ω p M q has indecomposable non-projective direct summands U { J n ´ p i . If B : “ End U p U ‘ M q and B : “ End U p U ‘ Ω p M qq are iso-morphic, which is equivalent to being Ringel self-dual, they have the same dimensions of their inde-composable projective modules. Note that we have in general Hom U p U { J k , U { J t q – J max p ,t ´ k q { J t and thus dim p Hom U p U { J k , U { J t qq “ dim J max p ,t ´ k q { J t “ min p k, t q . The indecomposable projec-tive B -modules are given by P iB “ Hom U p U ‘ M, U { J p i q . Similarly, the indecomposable projec-tive B -modules are given by P iB “ Hom U p U ‘ Ω p M q , U { J n ´ p r ´ i q . Here we choose the orderingfor the indecomposable projective B and B modules in increasing order of their vector space di-mensions. When B and B are isomorphic, we have dim p P iB q “ dim p P iB q for i “ , ..., r . Nowdim Hom U p U ‘ M, U { J p i q “ dim Hom U p U, U { J p i q ` dim Hom U p M, U { J p i q “ p i ` r ř k “ min p p k , p i q . In thesame way we obtain dim Hom U p U ‘ Ω p M q , U { J n ´ p r ´ i q “ n ´ p r ´ i ` r ř s “ min p n ´ p s , n ´ p r ´ i q . The con-dition dim p P B q “ dim p P B q gives us that p ` p r ` q p “ n ´ p r ` p r ` qp n ´ p r q and thus p “ n ´ p r and using induction and the equations dim p P iB q “ dim p P iB q we conclude that p i “ n ´ p r ´ i for all i “ , ..., r . This is equivalent to M “ Ω p M q . (cid:3) Thus in the situation of the previous algebra, we have Ringel self-duality if and only if the generatorsare stable 1-periodic.3.2.
Representation-finite blocks of Schur algebras.
The next application shows that therepresentation-finite blocks of Schur algebras are Ringel self-dual even when the generator is not sta-ble d -periodic. For statements without proofs about the representation-finite blocks of Schur algebras,we refer for example to [Mar18, section 4]. Recall that the representation-finite blocks of Schur algebrasare given by quiver and relations as1 2 3 4 ¨ ¨ ¨ m ´ m a b a b a b a m ´ b m ´ , m ě ,b m ´ a m ´ , b i ´ a i ´ ´ a i b i , a i ´ a i , b i b i ´ , i “ , . . . , m ´
1. We denote the quiver algebra of arepresentation-finite block of a Schur algebra with n simples by A n . By P p i q we mean the projec-tive indecomposable of A n associated to the vertex i . We denote by B n the endomorphism algebraEnd A n ` p P p q ‘ ¨ ¨ ¨ ‘ P p n qq if n ě B “ A . These algebras have the same quiver as A n ( A if n “
0) and these correspond to the finite-type blocks of basic algebras of the group algebras of a symmet-ric group. Of course, B corresponds to the simple block. Hence, the interesting case lies in n ě p A n ` , P p q ‘ ¨ ¨ ¨ ‘ P p n qq is a quasi-hereditary cover of B n and A n ` has global dimension and dominant dimension equal to 2 n . In particular, A n ` – End B n p B n ‘ S n q ,where S i denotes the simple module in the quiver of B n corresponding to vertex i . Thus, the Ringel dualof A n ` is isomorphic to End B n p B n ‘ Ω n p S n qq – End B n p B n ‘ S q by Theorem 3.5, since Ω n p S n q – S .Note that due to the symmetry in the relations, we also have A n ` – End B n p B n ‘ S q and A n ` is Ringelself-dual. Thus, in contrast to the previous class of examples, one can have Ringel self-duality even whenthe corresponding modules U ‘ M and U ‘ Ω d p M q are not isomorphic. Acknowledgments
Rene Marczinzik is funded by the DFG with the project number 428999796. Tiago Cruz is fundedby the
Studienstiftung des Deutschen Volkes . The proof of part (1) of Theorem 2.5 was first proven byVolodymyr Mazorchuk, who shared it with the second author via email. We followed his sketch of theproof and thank him for allowing us to use this result in the present article. We thank Steffen Koenigfor helpful comments and suggestions.We profited from the use of the GAP-package [QPA16].
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