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Journal of Algebra
www.elsevier.com/locate/jalgebra
A generalized Koszul theory and its relation to the classical theory
Liping Li1,2
SchoolofMathematics,UniversityofMinnesota,Minneapolis,MN55455, United States
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received3December2013 Availableonline15September2014 CommunicatedbyChangchangXi
Keywords:
Koszulalgebras Koszulmodules Directedcategories
Standardlystratifiedalgebras
LetA=
i0Ai beagradedlocallyfinite k-algebrawhere A0 isafinitedimensional algebra whosefinitistic dimension is 0. In this paper we develop a generalized Koszul theory preservingmanyclassicalresults,andshowanexplicitcorre- spondencebetweenthisgeneralizedtheoryandtheclassical theory. Applicationsinrepresentations of certain categories and extension algebras of standard modules of standardly stratifiedalgebrasaredescribed.
© 2014ElsevierInc.All rights reserved.
1. Introduction
The classicalKoszul theory plays animportant role in the representation theory of graded algebras.However,there are alotof structures (algebras, categories, etc.) hav- ing naturalgradings withnon-semisimple degree 0 parts,to which theclassical theory
E-mailaddresses:lixxx480@math.umn.edu,lipingli@math.ucr.edu.
1 Currentaddress: Department ofMathematics,University ofCalifornia, Riverside,CA92507, United States.
2 Theauthorwouldliketothanktherefereeforcarefullyreadingandcheckingthemanuscript.He/She pointed outaseriousmistakeinLemma 3.4inapreviousversion,whichiscrucialtoamainresultinthis paper.Thecounterexampleprovidedbytherefereemotivatestheauthortofigureoutacorrectionversion.
http://dx.doi.org/10.1016/j.jalgebra.2014.08.006 0021-8693/© 2014ElsevierInc.All rights reserved.
cannot apply. Particular examples of such structures include tensor algebras gener- ated by non-semisimple algebras A0 and (A0,A0)-bimodules A1, extension algebras of finitelygeneratedmodules(amongwhichwearemostinterestedinextensionalgebrasof standard modulesof standardlystratified algebras [8,15]),graded modularskew group algebras,categoryalgebras offinite EIcategories[14,27,28],andcertaingradedk-linear categories. Therefore, it is reasonable to develop a generalizedKoszul theory to study representations andhomologicalpropertiesoftheabovestructures.
In [11,18,19,29] several generalized Koszul theories have been described, where the degree 0 part A0 of a graded algebraA is notrequired to be semisimple. In [29], A is supposed to be both a left projective A0-module and a right projective A0-module.
However,inmanycasesAisindeedaleftprojectiveA0-module,butnotarightprojective A0-module.InMadsen’spaper[19],A0issupposedtohavefiniteglobaldimension.This requirement istoostrongforussinceinmanyapplicationsA0 isaself-injectivealgebra oradirectsumoflocalalgebras,andhenceA0hasfiniteglobaldimensionifandonlyifit issemisimple,fallingintotheframeworkoftheclassicaltheory.Thetheorydevelopedby Green,ReitenandSolbergin[11]worksinaverygeneralframework,andwewanttofind someconditionswhichare easyto checkinpractice.Theauthor hasalreadydeveloped a generalizedKoszul theoryin [17] underthe assumption thatA0 isself-injective, and used ittostudyrepresentationsand homologicalpropertiesofcertaincategories.
The goal of the work described in this paper is to loosen the assumption that A0 is self-injective (as required in [17]) and replace it by a weaker condition so that the generalized theorycanapply to moresituations.Specifically,since weare interestedin theextension algebrasof modules,categoryalgebras offinite EIcategories, andgraded k-linearcategoriesforwhichtheendomorphismalgebraofeachobjectisafinitedimen- sional local algebra, this weakercondition shouldbe satisfied byself-injective algebras andfinitedimensionallocalalgebras.Ontheotherhand,wealsoexpectthatmanyclas- sical results as the Koszul duality canbe preserved. Moreover, we hope to get a close relation betweenthis generalizedtheoryand theclassicaltheory.
AtrivialobservationtellsusthatintheclassicalsetupA0issemisimpleifandonlyif gl.dimA0,theglobaldimension ofA0,is0.Therefore,itisnaturaltoconsider thecon- dition thatfin.dimA0,thefinitisticdimensionofA0,is0.Obviously,finite dimensional local algebras and self-injective algebras do have this property. It turns out that this weakerconditionissuitableforourapplications,andmanyclassicalresultsstillhold.
Explicitly, letA=
i0Ai beagraded locallyfinite k-algebragenerated indegrees 0 and 1, i.e., dimkAi < ∞ and A1·Ai = Ai+1 for all i 0. We assume that both fin.dimA0 and fin.dimAop0 are 0, where Aop0 is the opposite algebra of A0. We then define generalized Koszul modules and generalized Koszul algebras by linear projective resolutions, aspeopledidforclassicalKoszulmodulesandclassicalKoszulalgebras.
Itiswellknownthatintheclassicaltheorylinearmodules(definedbylinearprojective resolutions) andKoszulmodules (definedbyacertainextensionproperty) coincide.We haveasimilarresult:
Theorem 1.1. Let A =
i0Ai be a locally finite graded algebra with fin.dimA0 = fin.dimAop0 = 0.If A isaprojective A0-module,then agradedmoduleM isgeneralized Koszulifandonly if itisaprojectiveA0-moduleandforalli0,
Ext1A(A0, A0)·ExtiA(M, A0) = Exti+1A (M, A0).
Inparticular,thegradedΓ = Ext∗A(A0,A0)-moduleExt∗A(M,A0)isgeneratedindegree 0.
WealsohavethegeneralizedKoszuldualityas follows:
Theorem 1.2. Let A =
i0Ai be a locally finite graded algebra with fin.dimA0 = fin.dimAop0 = 0.If A isageneralizedKoszulalgebra, thenE= Ext∗A(−,A0)gives adu- alitybetweenthecategoryofgeneralizedKoszulA-modulesandthecategoryofgeneralized KoszulΓ = Ext∗A(A0,A0)-modules.Thatis,ifM isageneralizedKoszulA-module,then E(M)is a generalizedKoszul Γ-module,and EΓEM = Ext∗Γ(EM,Γ0)∼=M as graded A-modules.
Letr be the radical of A0 and define R =ArA to be thetwo-sided idealgenerated byr.ForagradedA-moduleM=
i∈ZMisuchthatMi= 0 fori0, wethendefinea quotientalgebraA¯=A/ArA=
i∈ZAi/(ArA)iandM¯ =M/RM=
i∈ZMi/(RM)i. Clearly,M¯ isagradedA-module,¯ andthegradedA-moduleM isgeneratedindegree 0 if and only if the corresponding graded A-module¯ M¯ is generated in degree 0. More- over,inthesituationthatrA1=A1r,we getthefollowingcorrespondencebetweenour generalizedKoszultheoryand theclassicaltheory:
Theorem 1.3. Let A =
i0Ai be a locally finite graded algebra with fin.dimA0 = fin.dimAop0 = 0 andsuppose rA1=A1r.Then:
(1) A isa generalized Koszulalgebra if and only if it isa projective A0-module andA¯ is aclassicalKoszulalgebra.
(2) Suppose that A is a projective A0-module. A graded A-module M is generalized Koszul if and only if it is a projective A0-module and the corresponding graded A-module¯ M¯ isclassicalKoszul.
WethenapplythegeneralizedKoszultheorytosomefinitecategories. Theyplayan importantroleinthetheoryoffinitegroupsandtheirrepresentations.Inadditiontothe correspondence established inthe previous theorem, we give other correspondences in Section4(seeTheorems 4.1 and 4.2),showing thatthegeneralizedKoszul property of thecategoryalgebrasofsuchcategoriesisequivalentto theclassicalKoszulpropertyof quotientalgebrasofcertainfinitedimensionalhereditaryalgebras.Inpractice,thelatter oneismucheasiertochecksinceitisnothardtoconstructgradedprojectiveresolutions forsimplemodules.
Extension algebras of standard modules of standardly stratified algebras havebeen widelystudiedin[1–3,8,13,15,20,22,24–26].Differentfromtheusualapproach,wedonot assumethatthestandardlystratified algebraisgraded,thedegree 0partissemisimple, and itsgrading iscompatiblewith thefiltrationbystandardmodules. Instead,using a combinatorialpropertyofthefiltrationbystandardmodules,weshowthattheextension algebraofstandardmoduleshasgeneralizedKoszulproperty.
The paper is organized as follows: In the next section we develop the generalized Koszul theory and provethe first two theorems. In Section 3we describe the relation between the generalized theory and the classical theory, and prove the thirdtheorem.
Applications ofthis theory and thecorrespondence to certain categories are described inSection4.Inthelastsectionwediscuss thegeneralizedKoszulpropertyofextension algebras ofstandardmodules.
Weintroducesomenotationhere.Throughoutthispaperkisafield.LetA=
i0Ai
be a locally finite graded algebra generated in degrees 0 and 1. An A-module M =
i∈ZMi such that Mi = 0 for i 0 is graded if Ai ·Mj ⊆ Mi+j. It is said to be generated in degree sifM =A·Ms.It islocallyfinite ifdimkMi<∞foralli∈Z. In this paperallgradedmodulesaresupposedtobelocally finite.
GiventwogradedA-modulesMandN,HomA(M,N) andhomA(M,N) arethespaces of all modulehomomorphisms and of all graded module homomorphisms respectively.
The compositeof mapsf :L→M andg :M →N isdenotedbygf.The degreeshift functor [−] isdefinedbylettingM[i]s=Ms−i for i,s∈Z. DenoteJ=
i1Ai,which is atwo-sidedidealof A.We identifyA0 with thequotientmodule A/J and viewitas agradedA-moduleconcentratedindegree 0.Weviewthezeromodule0asaprojective modulesincethiscansimplifytheexpression ofsomestatements.
2. AgeneralizedKoszultheory
We start with some preliminary results, most of which are generalized from those described in [5,9,10,21,23]. The reader is also suggested to look at other generalized Koszultheoriesdescribed in[11,18,19,29].
Thefollowinglemmasareprovedin[17],wherewedidnotusetheconditionthatA0 is self-injective(Remark 2.8in[17]).
Lemma2.1.(SeeLemma2.1in[17].)LetAbeasaboveandletM beagradedA-module.
Then:
(1) J iscontainedinthegradedradical ofA;
(2) M hasa gradedprojectivecover;
(3) the gradedsyzygyΩM isalso locallyfinite.
Lemma 2.2. (SeeLemma2.2in [17].)Let0→L→M →N →be anexact sequenceof gradedA-modules.Then:
(1) IfM isgeneratedin degree s,soisN.
(2) IfL andN aregenerated indegree s,soisM.
(3) IfM isgeneratedindegrees,thenLisgeneratedindegreesifandonlyifJM∩L= JL.
Nowwedefinegeneralized Koszulmodules andgeneralizedKoszulalgebras.
Definition 2.3.A graded A-moduleM is called ageneralizedKoszulmoduleif ithasa (minimal)linearprojectiveresolution
. . .−→Pn−→Pn−1−→. . .−→P0−→M−→0
such that Pi is generated in degree i for all i 0. The graded algebra A is called a generalizedKoszulalgebraifA0viewed asanA-moduleisgeneralizedKoszul.
The reader can easily see that M is a generalized Koszul A-module if and only if M is generated in degree 0 and each syzygyΩi(M) is generated indegree i for every i 1. Moreover, from the above projective resolution, we deduce that M0 ∼= P00 and Ωi(M)i∼=Pii areprojectiveA0-modulesforalli1.
Recallforafinite dimensionalalgebraΛ,thefinitisticdimensionfin.dimΛisdefined asthesupremumofprojectivedimensionsofallindecomposableΛ-moduleshavingfinite projectivedimension[4,12].Inparticular,iftheglobaldimensiongl.dimΛisfinite,then fin.dimΛ= gl.dimΛ.Itiswellknownthatfin.dimΛ= 0 ifΛisafinitedimensionallocal algebras or a self-injective algebra. The famous finitistic dimension conjecture asserts thatthefinitistic dimensionofanyfinitedimensionalalgebrais finite.
InthispaperweshowthatmanyimportantresultsoftheclassicalKoszultheoryhold ifweassumethatfin.dimA0= 0= fin.dimAop0 .Itiseasytosee thatthisassumption is equivalenttothefollowing splittingcondition:
(S) Every exact sequence0→P →Q→R →0 of left (right, resp.) A0-modules splits if P andQare left (right,resp.) projectiveA0-modules.
Indeed, from the short exactsequence we deduce thatpdA0R 1. If fin.dimA0 = 0 = fin.dimAop0 , then pdA0R = 0, and hence the exact sequence splits. Conversely, supposethateverysuchexactsequencesplits.If thereissomeA0-moduleM suchthat pdA0M=n1,weconsiderR=Ωn−1(M) anddeducethatpdA0R= 1.Consequently, there is aprojective resolution of R which is non-splitting, contradicting the splitting condition.
Proposition2.4.Let0→L→M→N→beanexactsequenceofgradedA-modulessuch thatLisgeneralizedKoszul.ThenM isgeneralizedKoszulifandonlyifN isgeneralized Koszul.
Proof. ThisisProposition 2.9in[17].Theproofisalmostthesameexceptreplacingthe self-injective propertyof A0 bythesplittingproperty (S).Forthesakeofcompleteness we giveabriefproofhere.
By the second statement of the previouslemma, M is generated indegree 0 ifand onlyifN isgeneratedindegree 0.Considerthefollowingdiagraminwhichallrowsand columns areexact:
0 0 0
0 ΩL M ΩN 0
0 P P⊕Q Q 0
0 L M N 0
0 0 0.
HerePandQaregradedprojectivecoversofLandN respectively.WeclaimM ∼=ΩM.
Indeed,thegivenexactsequenceinduces anexactsequenceofA0-modules:
0−→L0−→M0−→N0−→0.
Observe that L0 is a projective A0-module. If N is generalized Koszul, then N0 is a projective A0-modulesinceN0∼=Q00,andtheabovesequencesplits.IfM isgeneralized Koszul, then M0 is a projective A0-module, and this sequence splits as well by the splitting property (S).In eithercase wehave M0 ∼=L0⊕N0. Thus P⊕Qis agraded projectivecoverofM,andhenceM∼=ΩM isgeneratedindegree 1ifandonlyifΩN is generatedindegree 1byLemma 2.2.ReplaceL,M andN by(ΩL)[−1],(ΩM)[−1] and (ΩN)[−1] (allofthem aregeneralizedKoszul)respectivelyintheshort exactsequence.
Repeating theaboveprocedureweprovetheconclusionbyrecursion. 2
IfM isageneralizedKoszulmodule,itstruncations(withsuitabledegreeshifts)are generalizedKoszulaswell:
Proposition 2.5. Let A be a generalized Koszul algebra and M be a generalized Koszul module. ThenJiM[−i]isalso generalizedKoszulforeach i1.
Proof. This isProposition 2.13 in[17].For theconvenience of the readerwe includea briefproofhere.Considerthefollowingcommutativediagram:
0 ΩM Ω(M0) JM 0
0 P0 id P0 0
0 JM M M0 0
Since M0 is a projective A0-module and A0 is generalized Koszul, Ω(M0)[−1] is also generalizedKoszul. Similarly, ΩM[−1] isgeneralized Koszul since so is M. Therefore, JM[−1] isgeneralizedKoszulbythepreviousproposition.NowreplacingMbyJM[−1]
and using recursion, we conclude that JiM[−i] is a generalized Koszul A-module for everyi1. 2
FromthispropositionweimmediatelydeducethatifAisageneralizedKoszulalgebra, thenitisaprojectiveA0-module.Wenowfocusongradedalgebras withthisproperty.
Proposition2.6.IfAisaprojectiveA0-module,theneverygeneralizedKoszulmoduleM isaprojectiveA0-module.
Proof. Clearly, it suffices to show that Mi is a projective A0-module for each i 0.
SinceM isgeneralizedKoszul, M0 is aprojectiveA0-module. Nowsuppose i1.The minimallinearprojectiveresolutionofM givesrisetoexactsequencesofA0-modules:
0−→Ωs+1(M)i−→Pis−→Ωs(M)i−→0, 0si.
If s = i, we have Ωi+1(M)i = 0 since Ωi+1(M) is generated in degree i+ 1. Thus Ωi(M)i∼=PiiisaprojectiveA0-module.Nowlets=i−1.Weclaimthatthefirstterm Ωi(M)iisaprojectiveA0-module.Indeed,Ωi(M)[−i] isageneralizedKoszulmodule,so (Ωi(M)[−i])0 isaprojective A0-module.ButΩi(M)i ∼= (Ωi(M)[−i])0. Thisproves the claim.SincethefirsttwotermsareprojectiveA0-modules,bythesplittingproperty (S), wededucethatΩi−1(M)iisaprojectiveA0-module.Byrecursion,weconcludethatMi isaprojective A0-moduleforeveryi>0. 2
Thefollowinglemmawill beusedintheproofofTheorem 1.1.
Lemma2.7.LetM beagradedA-modulegeneratedindegree 0.Supposethat bothAand M are projective A0-modules. Then ΩM is generated in degree 1 if and only if every A-modulehomomorphismΩM →A0 extendstoanA-modulehomomorphismJP →A0, whereP isagradedprojective coverof M.
Proof. This isavariedversionofLemma 2.17 in[17].The exactsequence 0→ΩM → P →M →0 induces an exactsequence0→(ΩM)1 →P1→ M1 →0 of A0-modules,
whichsplitssinceM1isaprojectiveA0-module.ApplyingthefunctorHomA0(−,A0) we getanother splitexactsequence
0→HomA0(M1, A0)→HomA0(P1, A0)→HomA0
(ΩM)1, A0
→0.
Note that (ΩM)0 = 0. Therefore, ΩM is generated in degree 1 if and only if ΩM/J(ΩM)∼= (ΩM)1,ifandonlyiftheabovesequenceisisomorphicto
0→HomA0(M1, A0)→HomA0(P1, A0)→HomA0(ΩM/JΩM, A0)→0.
Here weusethefactthatM1,P1 and(ΩM)1areprojectiveA0-modules.Buttheabove sequence isisomorphicto
0→HomA(JM, A0)→HomA(JP, A0)→HomA(ΩM, A0)→0
since JM and JP are generated indegree 1. Therefore, ΩM is generated in degree 1 if and only if every (non-graded) A-module homomorphism ΩM → A0 extends to a (non-graded) A-modulehomomorphismJP →A0. 2
LetM beagradedA-moduleandΓ = Ext∗A(A0,A0).ThenExt∗A(M,A0) isagraded Γ-module.Nowwerestateand proveTheorem 1.1.
Theorem 2.8. Let A =
i0Ai be a locally finite graded algebra with fin.dimA0 = fin.dimAop0 = 0.If A isaprojective A0-module, thenagraded moduleM is generalized Koszulif andonly ifitisaprojective A0-moduleandforalli0,
Ext1A(A0, A0)·ExtiA(M, A0) = Exti+1A (M, A0).
Inparticular,thegradedΓ = Ext∗A(A0,A0)-moduleExt∗A(M,A0)isgeneratedindegree 0.
Proof. This is a varied version of Theorem 2.16 in[17]. Since the proof is almost the same,weonlygiveasketch.Pleasereferto[17] fordetails.
The only if part. Let M be ageneralized Koszul A-module. Without loss of gener- ality we can suppose thatM is indecomposable. ByProposition 2.6 M is aprojective A0-module.Asintheoriginalproof,itsufficestoshowthatthegivenidentityistruefor i= 1,i.e.,
Ext1A(M, A0) = Ext1A(A0, A0)·HomA(M, A0).
The proof of this identity is completely the same as the original proof. We omit the details.
Theifpart. Asintheoriginalproof, weonlyneed toshow thatΩM isgenerated in degree 1. Bytheprevious lemma, itsuffices to showthateach (non-graded)A-module
homomorphism g : ΩM →A0 extends to JP0, where P0 is a graded projective cover ofM.Theproofofthis factiscompletely thesameas theoriginal proof.
Toprovethesecond statement,we applyrecursionto theidentity Ext1A(A0, A0)·ExtiA(M, A0) = Exti+1A (M, A0), anddeducethat
Exti+1A (M, A0) = Ext1A(A0, A0)·ExtiA(M, A0)
= Ext1A(A0, A0)·. . .·Ext1A(A0, A0)·HomA(M, A0)
⊆Exti+1A (A0, A0)·HomA(M, A0).
Theotherinclusionis obvious.Therefore,Ext∗A(M,A0) isgeneratedindegree 0. 2 Animmediate corollaryoftheabovetheorem is:
Corollary2.9.Thegradedalgebra AisgeneralizedKoszulifandonlyifAisaprojective A0-moduleandΓ = Ext∗A(A0,A0)isgenerated indegrees0and 1.
NowwecanproveageneralizedKoszulduality.
Theorem2.10.LetA=
i0Ai bealocallyfinitegradedalgebra suchthat fin.dimA0= fin.dimAop0 = 0.IfAisageneralizedKoszulalgebra,thenE = Ext∗A(−,A0)givesadual- itybetweenthecategoryof generalizedKoszulA-modulesandthecategoryofgeneralized KoszulΓ = Ext∗A(A0,A0)-modules.Thatis,ifM isageneralizedKoszulA-module,then E(M) is ageneralized Koszul Γ-module, and EΓEM = Ext∗Γ(EM,Γ) ∼=M as graded A-modules.
Proof. SinceA0isageneralizedKoszulmoduleandM isaprojectiveA0-module,M0 is generalizedKoszulas well.ByProposition 2.5, JM[−1] isalsogeneralizedKoszul.Fur- thermore,wehavethefollowingshort exactsequenceofgeneralizedKoszulmodules:
0−→ΩM[−1]−→Ω(M0)[−1]−→JM[−1]−→0.
AsintheproofofProposition 2.4,thissequence inducesexactsequencesofgeneralized Koszulmodulesrecursively:
0−→Ωi(M)[−i]−→Ωi(M0)[−i]−→Ωi−1
JM[−1]
[1−i]−→0.
Take a fixed sequence for a certain i > 0. It gives a splitting exact sequence of A0-modules:
0−→Ωi(M)i−→Ωi(M0)i−→Ωi−1
JM[−1]
i−1−→0.
ApplyingHomA0(−,A0) toitandusingthefollowingisomorphismforagradedA-module N generated indegreei
HomA(N, A0)∼= HomA(Ni, A0)∼= HomA0(Ni, A0), we get:
0→HomA
Ωi−1
JM[−1]
, A0
→HomA
Ωi(M0), A0
→HomA
ΩiM, A0
→0,
whichisisomorphicto 0→Exti−1A
JM[−1], A0
→ExtiA(M0, A0)→ExtiA(M, A0)→0.
Now lettheindex ivaryandputthesesequencestogether. Wehave:
0−→E
JM[−1]
[1]−→E(M0)−→EM −→0.
Let usfocus onthis sequence. WeclaimΩ(EM)∼=E(JM[−1])[1].Indeed,sinceM0
is a projective A0-module, E(M0) is a projective Γ-module. But JM[−1] is general- ized Koszul, so E(JM[−1]) is generated in degree 0 by the previous theorem. Thus E(JM[−1])[1] isgenerated indegree 1,andE(M0) isagradedprojectivecoverofEM. Thisprovestheclaim.Consequently,Ω(EM) isgeneratedindegree 1.Moreover,replac- ingM byJM[−1] (whichisalsogeneralizedKoszul)andusing theclaimedidentity, we havethat
Ω2(EM) =Ω E
JM[−1]
[1]
=Ω E
JM[−1]
[1]
=E
J2M[−2]
[2]
isgeneratedindegree2.Byrecursion,Ωi(EM)∼=E(JiM[−i])[i] isgeneratedindegreei for all i 0. Thus EM is a generalized Koszul Γ-module (note that Γ0 ∼= Aop0 and fin.dimΓ0= fin.dimAop0 = 0). Inparticular forM=AA,
EA= Ext∗A(A, A0) = HomA(A, A0) =Γ0 is ageneralizedKoszulΓ-module.
Since Ωi(EM) isgeneratedindegreei, Ωi(EM)i∼=E
JiM[−i]
[i]i∼=E
JiM[−i]
0
= HomA
JiM[−i], A0
∼= HomA(Mi, A0).
Wealsohave
HomΓ
Ωi(EM), Γ0
∼= HomΓ0
Ωi(EM)i, Γ0
∼= HomΓ0
HomA(Mi, A0), Γ0
∼= HomΓ0
HomA0(Mi, A0), Γ0
∼=Mi.
ThelastisomorphismholdsbecauseMiisaprojectiveA0-moduleandΓ0∼=Aop0 .There- fore,weget
ExtiΓ(EM, Γ0)∼= HomΓ
Ωi(EM), Γ0
∼=Mi
foreveryi0.Addingthemtogether, EΓE(M)∼=∞
i=0Mi∼=M.
Now wehave EΓ(E(A))=EΓ(Γ0)∼=A. Moreover, Γ isa graded algebrasuchthat Γ0∼=Aop0 hasfinitisticdimension0andisgeneralizedKoszulasaΓ-module.Therefore, wecanexchangeAandΓ intheabovereasoningandgetEEΓ(N)∼=N foranarbitrary KoszulΓ-moduleN.
LetLbeanothergeneralizedKoszulA-module.SinceL,M,EL,EM areallgenerated indegree 0 andareprojectiveA0-modules,wehave
homΓ(EL, EM)∼= HomΓ0
(EL)0,(EM)0
∼= HomΓ0
HomA(L, A0),HomA(M, A0)
∼= HomAop0
HomA0(L0, A0),HomA0(M0, A0)
∼= HomA0(M0, L0)∼= homA(M, L).
Consequently,E isadualitybetweenthecategory ofgeneralizedKoszulA-modulesand thecategoryofgeneralizedKoszulΓ-modules.3 2
Itis well knownthatalocally finite gradedalgebraA isclassicalKoszulifand only ifthe oppositealgebra Aop is also classicalKoszul. Unfortunately, this resultdoes not holdinthegeneralizedtheory.Here isanexample.
Example 2.11. Let A be the path algebra of the following quiver with relations δ2 = αδ= 0.Put A0=1x,1y,δandA1=α.
x
δ α
y
Then the (graded) left indecomposable projective modules and right indecomposable projectivemodulesaredescribedas follows,where theindices meanthedegree.
3 WewouldliketotakethisopportunitytocorrectasmallmistakeintheproofofTheorem 4.1in[17].
Attheendofthatproofwewritedown
HomAop0
HomA0(L0, A0),HomA0(M0, A0)∼= HomA0(L0, M0),
whichiswrongsinceHomA0(−,A0) isacontravariantfunctorfromA0-mod toAop0 -mod.Accordingly,the functorE= Ext∗A(−,A0) isactuallycontravariantratherthancovariant.
LPx= x0
x0 y1
, LPy =y0; RPx= x0
x0
, RPy= y0
x1
.
It isnothardtocheck thatA isageneralizedKoszulalgebra,butAopisnot.Actually, it isbecausethatA isaprojectiveas aleftA0-module,butAop isnotaleftprojective Aop0 ∼=A0-module.
3. Arelationbetweenthegeneralizedtheoryandtheclassicaltheory
In this section we describea correspondencebetween the generalizedKoszul theory wejustdevelopedandtheclassicaltheory.Asbefore,letA=
i0Ai bealocallyfinite graded algebra generated in degrees 0 and 1. At this moment we do not assume the conditionthatfin.dimA0= fin.dimAop0 = 0.
LetrbetheradicalofA0,andR=ArAbethetwo-sidedidealgeneratedbyr.Wethen definethequotientgradedalgebraA¯=A/R=
i0Ai/Ri.Clearly,A¯isalocallyfinite gradedalgebraforwhichthegradingisinducedfromthatofA,andRs=s
i=0AirAs−i. Note that A¯0 = A0/r is a semisimple algebra. Given an arbitrary graded A-module M =
i∈ZMi such thatMi = 0 fori 0, define M¯ =M/RM =
i∈ZMi/(RM)i. Then M¯ isagraded A-module¯ andM¯ ∼= ¯A⊗AM.
Weuseanexampletoshowourconstruction.
Example 3.1. Let A be the path algebra of the following quiver with relations: δ2 = θ2= 0,θα=αδ.Put A0=1x,1y,δ,θandA1=α,θα.
x
δ α
y θ
Thestructures ofgradedindecomposableprojectiveA-modulesare:
Px=
x0
x0 y1
y1
Py = y0
y0
.
Wefindr=δ,θ,R=δ,θ,θα.ThenthequotientalgebraA¯isthepathalgebraofthe following quiverwithanaturalgrading:
x−→α y.
LetM= radPx=δ,α,αδwhichisagradedA-module.Ithasthefollowingstructure and isnotgenerated indegree 0:
M= x0 y1
y1
.
Then M¯0 = M0/rM0 = δ¯ ∼= Sx, the simple A-module¯ corresponding to x; M¯1 = M1/(rM1+A1rM0)= α¯ ∼=Sy[1]. Therefore, M¯ ∼=Sx⊕Sy[1] isa directsumof two simpleA-modules,¯ andisnotgeneratedindegree 0either.
Thefollowingpropositioniscrucialto provethemainresultinthissection.
Proposition3.2. AgradedA-module M isgenerated indegree 0if andonlyif thecorre- spondinggradedA-module¯ M¯ isgenerated indegree 0.
Proof. IfM isgeneratedindegree 0,thenAiM0=Miforalli0.Byourconstruction, itisclearthatA¯iM¯0= ¯Mi. Thatis,M¯ is generatedindegree 0.
Conversely,suppose thatM¯ isgenerated indegree 0.We wanttoshow AiM0 =Mi for i 0. We use induction to prove this identity. Clearly, it holds for i = 0. So we supposethatitistrueforall0i< nandconsider Mn.
Take v ∈ Mn and consider its image v¯ in M¯n = Mn/n
i=0AirMn−i. Since M¯ is generated in degree 0, we can find some ai ∈ An and vi ∈ M0, 1 i r, such that
¯ v=r
i=1¯aiv¯i.This means v−
r
i=1
aivi∈
n
i=0
AirMn−i=rMn+
n
i=1
AirMn−i=rMn+
n
i=1
AirAn−iM0,
where the last identity follows from the induction hypothesis. Butit is clear AnM0 ⊇
n
i=1AirAn−iM0, so v−r
i=1aivi ∈rMn+AnM0. Consequently, v ∈ rMn+AnM0. Sincev∈Mn isarbitrary,wehaveMn⊆rMn+AnM0.ApplyingNakayama’slemmato these A0-modules, weconclude thatMn =AnM0 as well. Theconclusion thenfollows frominduction. 2
Lemma3.3.LetM beagradedA-modulegeneratedindegree 0.IfP isagradeprojective coverof M,thenP¯isagradedprojective coverof M.¯
Proof. Clearly,P¯isagradedprojectivemodule.BothP¯andM¯ aregeneratedindegree 0 bythepreviousproposition.ToshowthatP¯0isagradedprojectivecoverofM¯0,itsuffices toshowthatP¯0isaprojectivecoverofM¯0 asA¯0-modules.Butthisisclearlytruesince P¯0=P0/rP0∼=M0/rM0∼= ¯M0. 2
In general, A1r = rA1. However, if this is true, then Rs = rAs. Indeed, Rs =
s
i=0AirAs−i. Using the factthat A is generated by A0 and A1, and the abovecom- mutativerelation,wecanshowRs=rAiAs−i=rAs.Therefore,A¯=
i0Ai/rAi,and foreverylocallyfinite gradedA-moduleM=
i∈ZMi suchthatMi = 0 fori0, we have
M¯s=Ms/(RM)s=Ms/
i0
RiMs−i=Ms/
i0
rAiMs−i =Ms/rMs,
here
i0RiMs−iisafinitesumsincebyourassumptionMs−i= 0 fors−i0. More- over, theprocedureofsendingM toM¯ preserves exactsequencesofgradedA-modules whichareprojective regardedasA0-modules.
Lemma 3.4. Let 0→L→M →N →0be a shortexact sequence of gradedA-modules such that all terms are projective A0-modules. If rA1 = A1r, then the corresponding sequence 0→L¯→M¯ →N¯→0isalso exact.
Proof. By the above observation, for s ∈ Z, we have L¯s = Ls/rLs, and similar iden- tities hold for M and N. The given exactsequence induces a short exact sequence of A0-modules 0→Li → Mi →Ni →0.Since all terms areprojective A0-modules, this sequence splits,andgivesasplit shortexactsequence0→rLi →rMi→rNi →0.Tak- ingquotients,wegetanexactsequenceofA¯0-modules0→L¯i →M¯i →N¯i→0.Letthe index i varyand takedirectsum.Thenwe getanexactsequence ofgraded A-modules¯ 0→L¯→M¯ →N¯→0 asclaimed. 2
TheconditionthatalltermsareprojectiveA0-modulescannotbedropped,asshown bythefollowing example.
Example 3.5.Let A=A0=k[t]/(t2) and S be thesimplemoduleandconsider ashort exact sequence of graded A-modules 0→S → A→ S → 0. Wehave A¯∼= k. Butthe corresponding sequence 0 → S¯ → A¯ → S¯ → 0 is not exact. Actually, the first map S¯→A¯is0sincetheimage ofS iscontainedinrA0.
Now wecanprovethemainresultofthissection.
Theorem 3.6. Let A =
i0Ai be a locally finite graded algebra and M be a graded A-module.Supposethat bothAandM areprojectiveA0-modules,andrA1=A1r.Then M isgeneralizedKoszulif andonly ifthecorrespondinggradedA-module¯ M¯ isclassical Koszul. In particular, A is a generalized Koszul algebra if and only if A¯ is a classical Koszulalgebra.
Proof. Let
. . .−→P2−→P1−→P0−→M−→0 (3.1) be a minimal projective resolution of M. Note that all terms in this resolution and all syzygies are projective A0-modules. By Lemmas 3.3 and 3.4, M¯ has the following minimal projectiveresolution
. . .−→P2−→P1−→P0−→M −→0. (3.2) Moreover, this resolutionis linearifand onlyiftheresolution (3.1)islinearby Propo- sition 3.2. That is, M is generalized Koszul if and only if M¯ is classical Koszul. This
provesthefirststatement.ApplyingittothegradedA-moduleA0wededucethesecond statementimmediately. 2
If fin.dimA0 = fin.dimAop0 = 0, i.e., A0 hasthe splitting property (S), we havethe followingcorollary:
Corollary 3.7. Let A =
i0Ai be a locally finite graded algebra with fin.dimA0 = fin.dimAop0 = 0,and supposerA1=A1r.Then:
(1) A isa generalized Koszulalgebra if and only if it isa projective A0-module andA¯ is aclassicalKoszulalgebra.
(2) Suppose that A is a projective A0-module. A graded A-module M is generalized Koszul if and only if it is a projective A0-module and the corresponding graded A-module¯ M¯ isclassicalKoszul.
Proof. If A is a generalized Koszul algebra, then applying Proposition 2.5 to AA we concludethatitisaprojectiveA0-module.Moreover,A¯isaclassicalKoszulalgebraby theprevious theorem.The conversestatementalso follows from the previoustheorem.
Thisprovesthefirststatement.
If A is a projective A0-module and M is generalized Koszul, by Proposition 2.6M is a projective A0-module. Moreover, M¯ is a classical Koszul module by the previous theorem.Theconversestatementalsofollowsfromtheprevioustheorem. 2
WecannotdroptheconditionthatAisaprojectiveA0-moduleintheabovetheorem, asshownbythefollowing example.
Example 3.8. Let A be the path algebra of the following quiver with relations: δ2 = θ2= 0, θα=αδ= 0.Put A0=1x,1y,δ,θandA1=α.
x
δ α
y θ
Thestructuresofgradedindecomposableprojective A-modulesare:
Px= x0
x0 y1 Py= y0
y0.
We find r = δ,θ. Then the quotient algebra A¯ is the path algebra of the following quiver:
x−→α y.
Let Δx = Px/Sy = δ,1x which is a graded A-module concentrated in degree 0.
The first syzygy Ω(Δx) ∼= Sy[1] is generated in degree 1, but the second syzygy
Ω2(Δx) ∼=Sy[1] isnot generated indegree 2. Therefore,Δx is notgeneralizedKoszul.
However, Δ¯x ∼= ¯Sx is obviously a classical Koszul A-module.¯ Moreover, we can check thatAisnotageneralizedKoszulalgebra,butA¯isaclassicalKoszulalgebra.
From theconclusions ofProposition 2.6 and Theorem 3.6, thereader mayhave the feeling that the generalized Koszul algebra A might be a tensor product Λ⊗k A¯ (or A¯⊗kΛ)ofaclassicalKoszulalgebraA¯andafinitedimensionalalgebraΛwithfinitistic dimension 0 concentrated in degree 0. This is not true as explained in the following example.Therefore,ourgeneralizedKoszultheoryworksinsituationsmuchmoregeneral than thisspecialcase.
Example 3.9. Let A be the path algebra of the following quiver with relations δ2 = αδ = 0. Put A0 =1x,1y,δ and A1 =α. This is ageneralized Koszul algebra,and A1r= 0=rA1.Thusconclusionof Theorem 3.6canapplytothisexample.
x
δ α
y
Thequotientgraded algebraA¯isspanned by1x,1y,α,andA¯0=1x,1yissemisimple.
Clearly, AΛ⊗kA¯(andAA¯⊗kΛ)foranyfinitedimensionalalgebraΛ.
4. Applicationstodirectedcategories
Inthissectionwedescribesomeapplications ofthegeneralizedKoszultheorytocer- taincategories.Fortheconvenienceofthereader,letusgivesomebackgroundknowledge.
Bythedefinitionin[16,17],adirectedcategoryAisak-linearcategorysuchthatthere isapartialorderonObAsatisfyingtheconditionthatxywheneverA(x,y)= 0.In thissectionalldirectedcategoriesAaresupposedtohavethefollowingconditions:Ais skeletal andhasonlyfinitelymanyobjects;Aislocallyfinite,i.e.,dimkA(x,y)<∞for allx,y∈ObA;theendomorphismalgebraofeveryobjectisalocalalgebra.Wealsosup- posethatAisgradedandA0=
x∈ObAA(x,x).Notethatthespaceofallmorphisms in A forms a graded algebraA whose multiplication is determined by composition of morphisms. Wecallittheassociatedalgebra ofA.
In [17] we described another close relation between the generalized theory and the classical theoryfor directedcategories. Theexplicit correspondenceis describedas fol- lows. LetAbe agradeddirectedcategory withrespect toapartial order.Wedefine B tobe thegraded subcategoryofAformed byreplacingtheendomorphismalgebraof every object by k·1, the span of the identity endomorphism. That is, ObB = ObA; B(x,y)=A(x,y) if x=y andB(x,x)=k·1x.Let A andB be theassociated graded algebras ofAandB respectively.Thenwehave
i1Ai =
i1Bi. NotethatB0 isa semisimple algebra,so theclassicaltheory canbe applied.Onthe otherhand, A0 as a direct sumof several finite dimensional local algebras has finitistic dimension 0,so we canusethegeneralizedtheory.
Theorem4.1. LetAandB bedefined asabove.
(1) Suppose that A is a generalized Koszul algebra. If M is a generalized Koszul A-module,then the restricted moduleM ↓AB is classical Koszul. In particular, B is aclassical Koszulalgebra.
(2) Suppose that B is a classical Koszulalgebra. If M is a gradedA-modulesatisfying that Ωi(M)i isaprojectiveA0-moduleforeach i0andM↓AB isclassical Koszul, then M isgeneralized Koszul.
Proof. ThesetwostatementsarepreciselyTheorems5.13and 5.14in[17].Intheoriginal proofswedidnotassumethatA0 isself-injective,seeRemark 5.15. 2
Theorem4.2. LetAandB bedefined asabove.
(1) A isa generalizedKoszul algebra if and only if itis aprojective A0-module and B is aclassicalKoszulalgebra.
(2) Suppose thatA isageneralized Koszulalgebra.Thena gradedA-moduleM isgen- eralized Koszul if and only if it is a projective A0-module and M ↓AB is classical Koszul.
Proof. This isTheorem 5.16in[17],butwe dropthe unnecessarycondition thatA0 is self-injective.
(1) If A is a generalized Koszul algebra, then it is a projective A0-module, see the paragraphafterProposition 2.5.By(1)oftheprevioustheorem,B isaclassicalKoszul algebra.Conversely, ifB is aclassicalKoszul algebra,then A0 ↓AB is aclassicalKoszul B-modulesinceitisaprojective B0-module.Thusby(2)oftheprevioustheorem,Ais ageneralizedKoszulalgebraifwecanshowthatΩi(A0)i isaprojective A0-modulefor eachi0.Weproveastrongerstatement,thatis,Ωi(A0) isaprojectiveA0-modulefor eachi0.
Clearly,Ω0(A0)=A0 isaprojectiveA0-module.Considertheexactsequence 0−→Ωi+1(A0)−→Pi −→Ωi(A0)−→0.
Bytheinductionhypothesis,Ωi(A0) isaprojectiveA0-module.Thustheabovesequence splits as A0-modules. But Pi is a projective A0-module since we assume that A is a projectiveA0-module,so isΩi+1(A0).Thisproves (1).
(2). Since A is a generalized Koszul algebra, it is a projective A0-module. If M is generalized Koszul, then it is a projective A0-module (Proposition 2.6) and M ↓AB is classicalKoszul(by(1)ofTheorem4.1).Conversely,ifM↓ABisclassicalKoszul,toprove thatM is generalizedKoszul,by(2) ofTheorem 4.1it sufficesto showthatΩi(M)i is aprojectiveA0-moduleforeveryi0.Thiscanbeprovedbyasimilarinductionaswe justdid. 2
