categories On compact exceptional objects in derived module Journal of Algebra

(1)

Contents lists available atScienceDirect

Journal of Algebra

www.elsevier.com/locate/jalgebra

On compact exceptional objects in derived module categories

Liping Li¹

CollegeofMathematicsandComputerScience;PerformanceComputingand StochasticInformationProcessing(MinistryofEducation),HunanNormal University,Changsha,Hunan410081,China

a r t i c l e i n f o a bs t r a c t

Article history:

Received13August2014 Availableonline5July2016 CommunicatedbyLeonardL.

Scott, Jr.

Keywords:

Tiltingobjects

Compactexceptionalobjects Derivedmodulecategories Derivedsimple

Recollements

LetAbeabasicandconnectedfinitedimensionalalgebraand D^b(A) betheboundedderivedcategoryoffinitelygenerated left A-modules. Inthispaper weconsider lengthsof tilting objects andindecomposable compact exceptionalobjects in D^b(A), and prove a sufficient condition such that these lengthsareboundedbythenumberofisomorphismclassesof simpleA-modules.Moreover,weshowthatalgebrassatisfying this criterion are bounded derived simple, anddescribe an algorithm to construct a family of algebras satisfying this condition.

1. Introduction

Inalgebraicrepresentationtheory,oneofthemostinterestingandcomplicatedprob- lemsistoclassifyalgebrasuptoderivedequivalence.Explicitly,givenaﬁnitedimensional

E-mailaddress:lipingli@hunnu.edu.cn.

1 TheauthorissupportedbytheNationalNaturalScienceFoundationofChina11541002,theConstruct ProgramoftheKeyDisciplinein HunanProvince,andtheStart-UpFundsofHunanNormalUniversity 830122-0037. Hethanks the referee for carefullychecking the manuscriptand providing many detailed suggestions,whichmakethecurrentformmuchbetter.

(2)

(connectedandbasic) k-algebraA, wewantto characterizeor constructall(connected andbasic)algebraswhoseboundedderivedmodulecategoriesaretriangulatedequivalent to thebounded derivedcategory D^b(A). This bigproject draws theattentionof many people, and quiteafew resultshave been obtainedintwo different approaches. In one direction,peoplehaveclassifiedcertaintypesofalgebraswithspecialproperties;see[1,2, 8,11–13,19,20].Intheotherdirection,severalpropertieshavebeenshowntobeinvariant underderivedequivalence,suchasnumberofisomorphismclassesofsimplemodulesand finitenessof global dimensions([16]), finitenessof finitistic dimensions([30]), finiteness of strong global dimensions ([18]), self-injective property ([3]), etc. However,there are much morequestions unsolved. Forinstance, afinite dimensionallocal algebrais only derivedequivalenttoalgebrasMoritaequivalenttoitself(see[34]),andwewillshowthat path algebras of Kronecker quivers have this property as well (see Example 5.1). But acomplete list of basic algebras which are only derived equivalent to algebras Morita equivalenttothemselves isnotavailableyet.

AccordingtoafundamentalresultofRickard([31,32]),analgebraΓ isderivedequiva- lenttoAifandonlyifthereisatiltingobjectT ∈D^b(A) suchthatΓ isisomorphictothe oppositealgebraofEnd_D^b_(A)(T).Therefore,tiltingobjects,andmoregenerally,compact exceptionalobjectsareofparticularimportance,andhenceareextensivelystudied.For instance,AngeleriHügel,Koenig,andLiu(in[4–6])usethemtoinvestigaterecollements andstratiﬁcationsofderivedcategories; andAl-NofayeeandRickardpointoutin[3,33]

that for a ﬁxed algebra, there are at most countably many basic tilting objects T up to isomorphism and degreeshift, where T is basicif its directsummands are pairwise nonisomorphic.

Inthis paper we mainlyfocus on lengths of objectsin derivedcategories, which are deﬁned as follows. For an arbitrary P^• ∈ K⁻(_AP), the homotopy category of right boundedcomplexesof ﬁnitelygeneratedprojective A-modules,let

a(P^•) = sup{i∈Z|Pⁱ= 0} −inf{i∈Z|Pⁱ = 0},

calledtheamplitudeofP^• ([9]).SinceK⁻(AP) andtherightboundedderivedcategory D⁻(A) areequivalentastriangulatedcategories,forX∈D⁻(A),wedeﬁneitslengthto be²

l(X) = inf{a(P^•) + 1|P^•∈K⁻(AP) is quasi-isomorphic toX}.

Clearly,anobjectX ∈D⁻(A) hasﬁnitelengthifandonlyifX isquasi-isomorphictoa certainP^•∈K^b(AP),orequivalently, X iscompact.

HappelandZachariaprovein[18]thatlengthsofallindecomposableobjectsinD^b(A) areboundedifandonlyifAispiecewisehereditary;thatis,D^b(A) isequivalenttoD^b(H),

2 Notethatourdefinitionoflengthsisslightlydifferentfromthatin[18].Thelengthdefinedherecounts termsbetweenthefirstnonzeroterm(ifitexists)andthelastnonzeroterm,whereasthelengthdefinedin [18]countsthenumberofdifferentialmapsbetweenthefirstnonzeroterm(ifitexists)andthelastnonzero term.Foracompactobject,thedifferenceofthesetwolengthsisexactly1.

(3)

where H isahereditary abeliancategory. Thuswe mayask underwhatconditionsthe lengths ofallindecomposable compactexceptional objectsarebounded. For connected algebras, we show thatthis boundedness property is derived invariant (Theorem 2.2), and it implies the boundedness of lengths of tilting complexes (Proposition 2.1). The following theoremgivesusacriterionguaranteeingtheaboveboundednessproperty.

Theorem 1.1. Let A be a basic and connected ﬁnite dimensional algebra. For a simple A-moduleS,letPS beitsprojectivecoverandletQS bethedirectsumofindecomposable projective A-modules(uptoisomorphism)notisomorphic toPS.Supposethatforevery simple module S, the socle of PS contains a simple summand S_◦ ∼= S satisfying the following conditions:

(1) S_◦ isnotcontainedin theimageof any homomorphismP →PS forP ∈add(QS);

(2) S_◦ iscontainedinthekernel ofany homomorphismPS →P forP ∈add(QS).

Then the lengths of tilting objects and indecomposable compact exceptional objects in D^b(A) are bounded by the number of isomorphism classesof simple A-modules. More- over, everyindecomposableprojectiveA-moduleappearsatpreciselyonedegreeforevery minimaltilting complex.

NotethatthepossibilitythatS=S_◦isexcluded.Indeed,ifithappens,thenPS=S, and hencebythesecond conditionAis adirectsumof twoalgebras,contradicting the assumption thatA isconnected.

By[5]and[6],analgebraAiscalledboundedderivedsimpleifD^b(A) hasnonontrivial recollements bybounded derivedmodule categories of algebras.Clearly, local algebras areboundedderivedsimple.Buttherearemanyotherboundedderivedsimplealgebras ([27,28]).Algebras satisfyingthe conditionsinthe abovetheorem arebounded derived simple.Thatis:

Theorem 1.2. Let A be basic and connected finite dimensional algebra such that every simple A-module satisfies the conditions specified in Theorem 1.1. Then the following statements are equivalent:

(1) K^b(_AP), thehomotopy category of perfect complexes, has a nontrivial torsion pair (T,F)suchthat either T or F isclosed under degreeshift;

(2) Aisatriangularmatrixalgebra,andT (resp.F)coincideswithathicksubcategory generated bya projectiveA-module P (resp. Q).

In this case, Aisbounded derivedsimple.

The conditions inTheorem 1.1 seem artiﬁcial and mysterious, and the reader may havetheintuitionthatalgebrasAsatisfyingthese conditionsarecomplicated.Actually,

(4)

theseconditionsimplythateveryvertexintheordinaryquiverofAhasatleastoneloop.

Moreover,wecanshowthatAhasinfiniteglobaldimension(ifAk)anditsfinitistic dimensionis 0.Therefore, A doesnot haveniceproperties peopleprefer suchas being hereditary, piecewise hereditary, quasi-hereditary, etc. It is reasonable to believe that indecomposableobjects inD^b(A) arecomplicated.However, sinceinthis paperwe are onlyinterestedincompactexceptionalobjects,thebadbehaviorsofAcontrarilyrestrict thesizeofthesespecialobjectsaswellasstratificationsofD^b(A).Furthermore,algebras satisfying these conditions are not rare. Actually, many weakly directed algebras and stringalgebrasareexamples.WewillgiveanexplicitalgorithminSection4toconstruct abigclassof algebrassatisfyingtheseconditions.

Thepaperisorganizedasfollow.InSection2weconsiderlengthsofobjectsinderived module categories, and prove Theorem 1.1. Stratiﬁcation of bounded derived module categoriesisinvestigatedinSection3,whereTheorem 1.2isproved.InSections4and5 wedescribe someapplications,and ask severalquestionsforwhich theanswersarenot cleartous.

Inthispaperweonlyconsiderfinitedimensionalalgebrasoveranalgebraicallyclosed field k, although many results are still true in a much more general framework. All modules, unless specified explicitly, are finitely generated left modules. Composition of maps and morphisms is from right to left. The zeromodule is regardedas atrivial projectiveorfreemodule.ForafixedobjectXinamodulecategoryoraderivedcategory, add(X) istheadditivecategoryconsistingofdirectsummandsoffinitedirectsumsofX, andThick(X) isthesmallesttriangulatedcategory(closed underisomorphisms,degree shift,directsummands,and finitecoproducts) containingX.³ Thedegreeshift functor [−] inderivedcategoriesisas usuallydefined.

2. Lengthsofcompactexceptionalobjects

Throughout this section let A be a basic and connected finite dimensional algebra overan algebraicallyclosed fieldkand letA-modbe thecategory of finitelygenerated left A-modules. Let K^b(AP) be the homotopy category of perfect complexes; that is, complexes of finitely generated projective A-modules such that all but finitely many termsarezero.Thefollowing embeddingandequivalencearewellknown:

K^b(AP)⊆K⁻(AP)∼=D⁻(A).

Moreover, K^b(_AP) can be identified with a full subcategory of D^b(A), and D^b(A) is equivalent to K^−,b(AP), the homotopy category of right bounded chain complexes of finitelygenerated projectiveA-moduleswithfinitehomologies;see [16].

3 WesayXclassicallygeneratesThick(X) inthissituation.

(5)

A complexP^•∈K^b(AP) isminimal ifithasnosummandsofthefollowingform:

. . . 0 P ^id P 0 . . . ,

where P ∈add(_AA).ItiseasytoseethatP^• isminimalifandonlyifeverydifferential map di : Pⁱ → Pⁱ⁺¹ sends Pⁱ into the radical of Pⁱ⁺¹. Anobject X is compact (that is, the functor Hom_D^b_(A)(X,−) commutes with small coproducts) if and only if it is quasi-isomorphic to a minimal perfect complex P^• ∈ K^b(_AP) (which is unique up to isomorphism), and if and only if its length is finite. To calculate its length, we first choose a minimal perfectcomplex P^• ∈K^b(AP) quasi-isomorphic to X,and letr and s be the degreesof the first and thelast nonzero terms inthis complex.Then l(X)= l(P^•)=s−r+ 1.

AnobjectX ∈D^b(A) isexceptional (orrigid)if

Hom_D^b_(A)(X, X[n])∼= HomK⁻(AP)(P^•, P^•[n]) = 0

whenever n = 0. Anobject X ∈ D^b(A) is tilting ifit is quasi-isomorphicto acertain exceptional P^• ∈ K^b(AP) (called a tilting complex) such that Thick(P^•) = K^b(AP).

Clearly,atiltingobjectisadirectsumofﬁnitelymanyindecomposablecompactexcep- tionalobjects.Ifalldirectsummandsarepairwisenonisomorphic,wesaythatthetilting objectis basic.

Proposition 2.1. Let A be abasic and connected algebra. If the lengths of allindecom- posable compactexceptionalobjects arebounded, soare thelengthsof tiltingobjects.

Proof. LetT ∈D^b(A) beatiltingobject.Withoutlossofgeneralitywecanassumethat T is atilting complex (i.e., T ∈ K^b(AP))and is basicsince atilting complex and the corresponding basictilting complexobtainedbytaking onedirectsummand from each isomorphism class havethesame length.Write T =⊕ⁿi=1T_i, where n isthe numberof isomorphismclassesofsimpleA-modulesandeachTiisindecomposable.Clearlywecan assumethateveryindecomposablesummandisminimal. Ifr,s∈ZsatisfyT^r= 0=T^s and Tⁱ = 0 for i > s or i < r, the length of T is s−r+ 1. Since the length of each indecomposablesummand isboundedbyaﬁxednumberm,l(T)nmisalsobounded if we can show thatthere is nogap among these indecomposable summands. Thatis, for any rj s, we haveT^j = 0.Butthis is clear.Indeed, ifit is nottrue, then we usethis gaptodecomposeT =T⊕T suchthat

Hom_K^b₍_A_P)(T, T) = 0 = Hom_K^b₍_A_P)(T, T).

Consequently, A is derivedequivalent to End_Kb(AP)(T)^op⊕End_Kb(AP)(T)^op. This is impossible sinceconnectednessisinvariantunderderivedequivalences. 2

(6)

By [18], boundedness of lengths of all indecomposable compact objects is equiva- lenttothepiecewisehereditaryproperty,whichisinvariantunderderivedequivalences.

Similarly,boundedness oflengthsof allindecomposable compactexceptionalobjects is invariantunderderivedequivalences,too.

Theorem2.2. LetA andB be two connectedbasic ﬁnite dimensional algebras. Suppose that A and B are derived equivalent. If lengths of indecomposable compact exceptional objects inD^b(A)are bounded,then B has thesameproperty aswell.

Proof. Without loss of generality weonly consider minimal objects inhomotopy cate- goriesof perfectcomplexes. Assumethatl(X)m forallindecomposable exceptional objectsX ∈K^b(AP). SinceD^b(B) is derivedequivalentto D^b(A),by [31],there exists a tilting complex T ∈ K^b(BP) such that End_K^b₍_B_P₎(T)^op ∼= A. Moreover, T in- duces a triangulated equivalence F : K^b(AP) → K^b(BP) such that F(A) = T. Let G:K^b(_BP)→K^b(_AP) bethequasi-inverseofF.

NowletY beaminimal, indecomposableexceptionalobjectinK^b(_BP).ThenG(Y) isanindecomposableexceptionalobjectinK^b(_AP),whichhasatmostmnonzeroterms sincebyourassumptionG(Y)∈K^b(AP) isminimalandbythegivenconditionlengths ofindecomposableexceptionalobjectsinK^b(AP) areboundedbym.

Note that F G(Y) is constructed as follows (Proposition 2.10 in [31]). First, for all i∈Z,nonzeroG(Y)ⁱ(whichareprojectiveA-modules)arereplacedbyobjectsinadd(T) to obtainabigradedcomplexY^∗∗ over add(B). Thenwe takethetotalcomplexY^∗ of Y^∗∗ anddeﬁne F G(Y)=Y^∗. Clearly,Y ∼=F G(Y)=Y^∗,andl(Y^∗)m+l(T),where l(T) isthelengthofT.Thatis,lengthsofindecomposableexceptionalobjectsinK^b(_BP) areboundedbym+l(T). 2

Thefollowinglemmasandtheircorollarieswill beusedfrequently.

Lemma 2.3. Let S, PS, S_◦ and QS be as deﬁned in Theorem 1.1 and suppose that the twoconditionsinithold. Theneveryhomomorphismα:PS →P,whereP isprojective, iseither asplitmonomorphismor hasS_◦ inthekernel.

Proof. Write P = P⊕Q with P ∈ add(PS) and Q ∈ add(QS). If α is not asplit monomorphism,thenS_◦isinthekerneloftheinducedmapPS →P.ButS_◦alsoliesin thekerneloftheinducedmapPS→Q bytheassumption.Theconclusionfollows. 2 LetP^•∈K^b(AP).WesaythatanindecomposableprojectiveA-moduleP appearsat degree iifPⁱ hasasummand isomorphictoP.Thefollowinglemmais crucialtoprove Theorem 1.1.

Lemma2.4.LetS,P_S,S_◦andQ_S beasdeﬁnedinTheorem 1.1andsupposethatthetwo conditions inithold. If P^• and Q^• areobjects in K^b(AP)suchthat r isthelastdegree ofP^• andstheﬁrstdegree ofQ^• wherePS appears,thenHom_K^b₍_A_P)(P^•[r],Q^•[s])= 0.

(7)

Proof. Byassumption,wehavethefollowingdecompositions:P^r∼=PS⊕P,andP^s∼= PS⊕Q.Now weconstructachainmapas follows:

. . . ^d^s−3 P^s−2 ^d^s−2

0

P^s−1 ^d^s−1

0

PS⊕P ^d^s

˜ α hs

P^s+1 ^d^s+1

0 hs+1

. . .

. . . ^d^r−3 Q^r⁻² ^d^r−2 Q^r⁻¹ ^d^r−1 PS⊕Q ^d^r Q^r+1 ^d^r+1 . . . ,

where α˜=

α 0

0 0

, andαmaps thetopofPS toS_◦ inthesocle ofPS.

We check that αd˜ s−1 = 0 and drα˜= 0. Indeed, sinceP^• isminimal, ds−1(P^s−1) is containedintheradicalofP^s.Butα˜mapstheradicalofP^sto0,soαd˜ _s−1= 0.Besides, by deﬁnition, α(P˜ ^s)= S_◦ ⊆PS ⊆ Q^r. Since Q^r is the last degree where PS appears, the map P_S →Q^r+1 induced byd_r cannot be asplit monomorphism.Bythe previous lemma, S_◦ iscontainedinthekernel ofthisinducedmap.Consequently,drα˜= 0.

The chainmap isnot nullhomotopic. Otherwise,there must exist maps hs: P^s → Q^r−1 and h_s+1 : P^s+1 → Q^r such that α˜ = h_sd_r−1+h_s+1d_r. In particular, α is the composite ofthefollowingmaps:

PS

⎡

⎣hsι dsι

⎤

⎦

Q^r⁻¹⊕P^s+1

pd_r−1 ph_s+1

PS ,

where ι : PS → P^s and p : Q^r → PS are the inclusion and projection respectively.

Butthis isimpossible.Indeed,sincer andsaretheﬁrstandthelastdegreeswhereP_S appears, Q^r⁻¹⊕P^s+1 hasno summand isomorphicto PS.Therefore, by condition(1) inTheorem 1.1,S_◦⊆P_S isnotintheimageof[pd_r−1 ph_s+1],andhenceisnotinthe image of αbytheabovefactorization. Thiscontradicts thedeﬁnitionα,and ourclaim is proved. 2

Severalimmediate corollariesofthislemmaare:

Corollary2.5.LetS,P_S,Q_S,andS_◦ beasdeﬁnedinTheorem 1.1,andsupposethatthe two conditions in ithold. LetP^• ∈K^b(AP) be a minimal exceptional object.Then PS

appearsin atmostone degreeof P^•.

Proof. Suppose that PS appears in at least two degrees of P^•. Let r and s be the ﬁrst degree and the last degree where PS appears. By the previous lemma, Hom_Kb(AP)(P^•,P^•[s−r])= 0, where s−r= 0. This contradicts theassumption that P^• isexceptional. 2

(8)

Corollary2.6. LetAandPS be asinLemma 2.4.If P^•,Q^•∈K^b(AP) aretwominimal objects satisfying Hom_K^b₍_A_P)(P^•,Q^•[n]) = 0 for all n ∈ Z, then PS cannot appear in both P^• andQ^•.

Proof. SupposethatPS appearsinbothP^• andQ^•.Applyingdegreeshiftifnecessary, we can assume thatboth the last degree of P^• where PS appearsin P^• and the ﬁrst degreeofQ^•whereP_SappearsinQ^•are0.ThusHom_Kb(AP)(P^•,Q^•)= 0 byLemma 2.4, contradictingtheassumption. 2

Nowwerestateand proveTheorem 1.1.

Theorem 2.7. Let A be a basic and connected ﬁnite dimensional algebra, and suppose that all simple A-modules satisfy the conditions in Theorem 1.1. Then the lengths of tiltingobjectsandindecomposablecompactexceptionalobjects D^b(A)areboundedbythe number of isomorphism classes of simple A-modules. Moreover, every indecomposable projectiveA-module appearsatpreciselyone degreeforevery minimaltilting complex.

Proof. Take anindecomposableexceptionalobject P^•∈K^b(AP).Without loss ofgen- erality we can assume that P^• is minimal. Let r,s ∈ Z such thatP^r and P^s are the ﬁrstandlast nonzerotermsinP^•.Thenl(P^•)=s−r+ 1 byourdeﬁnition.SinceP^• is indecomposable,forrts,P^t= 0.However,byCorollary 2.5,everyindecomposable projective A-module (up to isomorphism)appears at no more than one degreein P^•. Therefore,P^• canhaveatmostnnonzeroterms,wherenisthenumberofisomorphism classesofsimpleA-modules.Thatis,l(P^•)=s−r+ 1n.

LetT ∈K^b(AP) beaminimaltiltingcomplex.ByCorollary 2.5,everyindecomposable projective A-module appears at no more than onedegree of T. But sinceThick(T) = K^b(_AP),everyindecomposable projectiveA-modulemust appearatsomedegree ofT. Otherwise,supposethatP_e∼=Aedoesnotappear,wheree∈Aisaprimitiveidempotent.

Then

Thick(T)⊆Thick(A(1−e))=K^b(_AP).

This is impossible. Therefore, every indecomposable projective A-module appears at preciselyonedegreeof T.

ObservethatT mustbe connected asintheproof ofLemma 2.1.Thatis,ifr,s∈Z satisfy T^r = 0 = T^s and Tⁱ = 0 for i > s or i < r, then T^j = 0 for r j s.

Therefore, the lengthof T must be bounded bythe number of isomorphismclasses of simpleA-modulesbythesecond statement. 2

Inthefollowingexampleweuse theaboveresultsto classifythederivedequivalence classes of aparticular algebra A. A similar examplehas been considered in[7, Exam- ple 5.13],andinSection4wewillconsider amoregeneralconstruction.

(9)

Example2.8.LetAbethepathalgebraofthefollowingquiverwithrelationsδ²=ρ²= ρα=αδ= 0.

x

δ α

y ^ρ

Uptoisomorphism,thisalgebrahastwoindecomposableprojectivemodulesasfollows:

Px= x

x y, Py= y y, both ofwhichsatisfytheconditionsinTheorem 1.1.

ByCorollary 2.5,minimalindecomposableexceptionalobjectsinK^b(_AP) havelengths at most2.Moreover,ifl(P^•)= 2,thenthere existsacertain i∈ZsuchthatPⁱ∼=P_y^⊕^a and Pⁱ⁺¹ ∼=P_x^⊕^b witha,b1.We cancheck thatP^• is indecomposableif andonly if a=b= 1. Inconclusion, upto degreeshiftand quasi-isomorphisms,K^b(AP) only has three indecomposableexceptionalobjects:stalkcomplexesP_x andP_y,and

X := Py

d Px ,

where dmapsthetopof Py ontothesimplesummandSy inthesocleofPx.

UsingTheorem 2.7,thereadercancheckthatuptodegreeshiftandquasi-isomorphism K^b(_AP) has three basictilting complexes: T₁ =P_x⊕P_y, T₂ =P_x[−1]⊕X, and T₃ = P_y⊕X.Clearly,End_Kb(AP)(T₁)^op∼=A.

By computation, B^op = End_K^b₍_A_P)(T2) is isomorphic to the path algebra of the following quiverwithrelationsβαβ = 0=δ²,δα=βδ= 0:

x

α

y δ β

.

Similarly, by computation,C^op = End_Kb(AP)(T₃) isisomorphic to the path algebraof thefollowing quiverwithrelationsαβα= 0=δ²,δα=βδ= 0:

x

α

y

β

δ.

WeconcludethatuptoMoritaequivalence,Aisderivedtothreealgebras:A,B,andC as theylieindiﬀerentMoritaequivalenceclasses.

A careful observation tells us that B ∼= Côp. This is reasonable since A ∼= Aôp. Moreover, B hasatiltingcomplexT =Bex[1]⊕M whereM ∼=Bey/Bex.It iseasyto check thatEndB(T)ôp∼=A.

(10)

Proposition 2.9. Let A be a basic and connected ﬁnite dimensional algebra, and sup- posethatallsimpleA-modulessatisfy theconditionsinTheorem 1.1.Thentheﬁnitistic dimensionofA is0.

RecallthatthefinitisticdimensionofAisthesupremumofprojectivedimensionsof indecomposableobjectsinA-modwithfiniteprojectivedimension.Thefinitisticdimen- sionofAequals 0ifandonlyifallfinitelygenerated A-moduleshavingfiniteprojective dimensionareprojective.

Proof. Supposethattheconclusioniswrong.Thenwe canﬁndsomeM∈A-mod such thattheprojectivedimensionofMisnwithn>0.Takeaminimalprojectiveresolution ofM as follows

0→Pⁿ→Pⁿ⁻¹→. . .→P⁰→0.

Byourassumption,Pⁿ = 0,andthemapdn :Pⁿ→Pⁿ⁻¹must beinjective.

Take a nonzero indecomposable summand P_S of Pⁿ and denote by ι the inclusion P_S → Pⁿ. Since the projective resolution is minimal, ι cannot be a split monomorphism.ByLemma 2.3,ιandhenced_n cannotbeinjective.Theconclusionfollowsbythe contradiction. 2

3. Stratiﬁcationofboundedderivedmodulecategories

In this section we consider stratification of bounded derived module categories of algebrassatisfyingtheconditionspecifiedinTheorem 1.1,showingthattheyarebounded derivedsimple.Thefollowingdefinitionistakenfrom [10,Section2,ChapterI].

Deﬁnition 3.1.A pairof strict full subcategories(T,F) of atriangulatedcategory C is calledatorsionpairifthefollowing conditionshold:

(1) C(T,F)= 0 foranyT ∈ T andF ∈ F.

(2) T ∈ T impliesT[1]∈ T;andF ∈ F impliesF[−1]∈ F.

(3) Forany X ∈ C, thereisatriangle(whichisuniqueuptoisomorphism) T_X X F_X T_X[1] .

NotethatT istheleft perpendicular category of F,and F istheright perpendicular category ofT (see[15]).

Thefollowingkeyobservationiscrucialto provethemainresultofthis section.

Lemma3.2. LetA bea ﬁnitedimensional algebra. Leteandf betwoorthogonal idem- potents such that e+f = 1. If there existsa torsion pair (T,F) of K^b(AP) suchthat T ⊆Thick(Ae),F ⊆Thick(Af),thenT = Thick(Ae)and F= Thick(Af).

(11)

Proof. We only need to show the other inclusions. Take an arbitrary object U ∈ Thick(Af).Since(T,F) isatorsionpairofK^b(AP),thereisacanonicaltriangle

V U W V[1]

with V ∈ T ⊆Thick(Ae) andW ∈ F ⊆Thick(Af). However,sincebothU and W are containedinThick(Af),soisV.Therefore,V ∈Thick(Ae)∩Thick(Af).Weclaimthat V isquasi-isomorphicto 0.Ifthis holds,thenU ∼=W ∈ F,so Thick(Af)⊆ F.

Indeed,sinceV ∈Thick(Ae),wegetaminimalrepresentationofV P^•: . . .→0→P^r→. . .→P^s→0→. . .

suchthatallPⁱarecontainedinadd(Ae).Similarly,sinceV ∈Thick(Af),wegetanother minimal representationofV

Q^•: . . .→0→Q^l→. . .→Q^t→0→. . . suchthatallQⁱ arecontainedinadd(Af).

Now let us regard V and its two representations as objects in D^b(A). If V = 0, then there is a simple module X and a certain n ∈ Z with Hom_D^b_(A)(V,X[n]) = 0.

Consequently, we have both Hom_D^b_(A)(P^•,X[n]) = 0 and Hom_D^b_(A)(Q^•,X[n]) = 0.

However,thisimpossiblesincetermsinP^• andQ^• havenoisomorphicindecomposable summands.Therefore,V isquasi-isomorphicto0,andourclaimisproved.

We haveshownF ⊇Thick(Af).ThefactthatT ⊇Thick(Ae) canbe shownby the sameargument.This ﬁnishestheproof. 2

Animmediatecorollary is:

Corollary 3.3. LetT,F,e,f be asin thepreviousproposition. ThenA is isomorphic to thetriangular matrixalgebra (see[26])

eAe 0

f Ae f Af

.

Proof. SinceAe∈ T,Af∈ F, and(T,F) isatorsionpairofK^b(AP),oneshouldhave eAf∼= HomA(Ae, Af)∼= HomK^b(AP)(Ae, Af) = 0. 2

Thefollowing propositionimpliestheﬁrstpartofTheorem 1.2.

Proposition 3.4. LetA beabasic andconnectedﬁnite dimensionalalgebra, andsuppose that allsimple A-modules satisfy the conditions in Theorem 1.1.If (T,F) is a torsion pair of K^b(AP) such that either T or F is closed under degree shift,then itcoincides with atorsionpair(Thick(P),Thick(Q))inducedbytwoprojectiveA-modulesP andQ.

(12)

Proof. LetE beacompleteset ofprimitiveorthogonalidempotentsofA,anddeﬁne E1={∈E|Aappears in a certain minimalX ∈ T };

E2={∈E|Aappears in a certain minimalY ∈ F}. Deﬁne e =

∈E1 and f =

∈E2. Since (T,F) is a torsion pair of K^b(_AP) and eitherT orF isclosedundershifts,ononehandHom_K^b₍_A_P)(X,Y[n])= 0 foralln∈Z, X ∈ T and Y ∈ F,so E1∩E2 =∅byCorollary 2.6; ontheother hand,E1∪E2 =E sinceobjectsinT and F classicallygenerated K^b(AP). Consequently,e+f = 1. Thus byLemma 3.2T = Thick(Ae) andF = Thick(Af). 2

Now we deﬁne recollementsof triangulated categories; for moredetails, see [4–7,14, 21,29].

Deﬁnition3.5. A recollementof atriangulatedcategory C bytriangulatedcategories D andE isexpressed diagrammaticallyas follows

D ⁱ^∗ C ^j^∗

i^∗

i^!

E

j_∗ j!

with six exact, additivetriangulated functors i^∗,i_∗,i^!,j^!,j^∗,j_∗ satisfying the following conditions:

(1) (i^∗,i_∗,i^!),and(j!,j^∗,j_∗) bothareadjointtriples;

(2) i_∗,j_! andj_∗ arefullyfaithful;

(3) i^!j_∗= 0;

(4) foreachX ∈ C,thereare triangles

i_∗i^!(X) X j_∗j^∗(X) i_∗i^!(X)[1] , j_!j^∗(X) X i_∗i^∗(X) j_!j^∗(X)[1].

Thefollowingtheorem,provedin[5],givesarelationbetweenrecollementsofderived modulecategoriesandexceptionalcompactobjects.

Theorem 3.6. [5, Corollary 2.5] Let Γ be a ﬁnite dimensional algebra. If the bounded derived category D^b(Γ) is a recollement of D^b(B) and D^b(C), then there are objects T₁,T₂∈D^b(Γ)satisfying:

(1) Both T1 andT2 arecompactandexceptional;

(2) Hom_D^b_(Γ)(T1,T2[n])= 0 foralln∈Z;

(13)

(3) T1⊕T2 generates D^b(Γ) astriangulated category.⁴

Moreover, wehave End_D^b_(Γ)(T1)^op∼=C andEnd_D^b_(Γ)(T2)^op∼=B.

According to[27],analgebraΓ issaidto be boundedderived simple ifD^b(Γ) has no nontrivial recollementsbybounded derivedmodulecategoriesofalgebras.

Werestate thesecond partof Theorem 1.2.

Theorem3.7.LetAbeabasicandconnectedﬁnitedimensionalalgebra,andsupposethat all simpleA-modulessatisfy the conditionsin Theorem 1.1.Then A isboundedderived simple.

Proof. This is clearly true if A is a local algebra. Suppose that A is not local and is not bounded derivedsimple. Therefore,by [7,Theorem 5.12], A isnot K^b(AP)-simple either.Thatis,thehomotopycategoryK^b(AP) hasanontrivial recollementas follows

K^b(_SP) ⁱ^∗ K^b(_AP) ^j

∗

i^∗

i^!

K^b(_RP)

j_∗ j!

,

where R and S are ﬁnite dimensional algebras. It is clear thati_∗S, j!R, and j_∗R are all compact. Without loss of generality we can assume that they are minimal. Since K^b(SP) = Thick(S) and i_∗ is afullembedding, wehaveImi_∗= Thick(i_∗S). Similarly, Imj_! = Thick(j_!R),andImj_∗= Thick(j_∗R).

Let E be acomplete set ofprimitive orthogonal idempotentsof A, andlet X (resp.

Y andZ)beaminimalobjectinK^b(_AP) isomorphictoj_!R(resp.i_∗S andj_∗R).Deﬁne E1={∈E |Aappears inX},

E2={∈E|Aappears inY}, E3={∈E|A appears inZ}. Correspondingly,let

e=

∈E1

, f =

∈E2

, λ=

∈E3

.

By[14,Lemma2.6],(Thick(j_!R),Thick(i_∗S)) isatorsionpairofK^b(_AP).ByPropo- sition 3.2 anditsproof,weknow

4 HerewesayT1⊕T2 generatesD^b(Γ) ifforanynonzeroobjectX ∈ D^b(Γ),there isacertainn∈Z suchthatHom_Db(Γ)(T1⊕T2,X[n])= 0.ThisholdsifThick(T1⊕T2)=K^b(AP);i.e.,T1⊕T2 classically generatesK^b(AP).

(14)

e+f = 1, Thick(j!R) = Thick(Ae), Thick(i_∗S) = Thick(Af).

Therefore,eAf∼= HomA(Ae,Af)= 0.

By[14,Lemma2.6],(Thick(i_∗S),Thick(j_∗R)) isatorsionpairofK^b(AP),too.Again, byProposition 3.2anditsproof,weknow

f +λ= 1, Thick(j_∗R) = Thick(Aλ).

Consequently,

λ=e, Thick(Aλ) = Thick(Ae) ⇒ f Ae∼= HomA(Af, Ae) = 0.

ButthisimpliesthatA∼=eAe⊕f AfsinceeAf= 0=f Ae,contradictingtheassumption thatA isconnected.Theconclusionfollows. 2

However, the algebras A in the above theorem in general are not derived sim- ple. Indeed, the algebra A in Example 2.8 is bounded derived simple by the above theorem, but it is not derived simple. Actually, for every triangular matrix algebra A = (eAe,f Af,f Ae), as we pointed out in [26] D(A) always has the following non- trivialrecollement

D(f Af) ⁱ^∗ D(A) ^j

∗

i^∗

i^!

D(eAe)

j_∗ j!

,

wherethefunctorsarespeciﬁed asinProposition3.6 of[26].

4. Weaklydirectedalgebras

In this section we apply the results in the previous sections to a special class of algebras. As before, we only consider connected and basic ﬁnite dimensional algebras overanalgebraicallyclosedﬁeldk.

Deﬁnition 4.1. A ﬁnite dimensional algebra A is said to be weakly directed if there is a complete set of primitive orthogonal idempotents E = {e_i}ⁿi=1 such that Hom_A(Ae_j,Ae_i)∼=e_jAe_i= 0 onlyifji.

Sincethefieldisalgebraicallyclosed,Aisweaklydirectedifandonlyiforientedcycles intheordinaryquiverareallloops.Examplesofweaklydirectedalgebrasincludequotient algebras of finite dimensional hereditary algebras, local algebras, category algebras of skeletalfiniteEIcategories([23]),extensionalgebrasofstandardmodulesofstandardly stratifiedalgebras ([24]).

Thefollowinglemmaisdescribed in[25].

(15)

Lemma 4.2. Let A be a weakly directed algebra. Then every simple A-module can be identiﬁed with a simple eAe-module for some e ∈ E. Moreover, for any M ∈ A-mod and f ∈E,theleftf Af-modulef M is alsoan A-module.

Proof. Theﬁrst statement is containedin Proposition2.2 in[25]. Toshow the second one,notethatbytheweaklydirectedstructureofA,allarrowsbetweendiﬀerentvertices generate a two-sided ideal J of A, and ⊕ⁿi=1eiAei ∼= A/J. As a summand, f Af is a quotient algebraofA.Thesecondstatementfollowsfromthis observation. 2

Bythislemma,theA-modulef M andthef Af-modulef M havethesamecomposi- tionfactors,and hencethesameLoewylength.

ThepropositionbelowgivesapracticalmethodtochecktheconditionsinTheorem 1.1 forweaklydirectedalgebras.

Lemma 4.3. Let A be a weakly directed algebra and e∈ E. Then the simple A-module Se ∼=Ae/radAe satisﬁes theconditions in Theorem 1.1 if and only if the socle of Ae contains a simple summand S_e ∼=Ae/radAe such that for any x∈eA(1−e)we have S_ex= 0.

ByLemma 4.2,Ae/radAe∼=eAe/radeAe,andS_e⊆eAe.Therefore,therightaction of x∈eA(1−e) onSe isinducedfromthemultiplicationofeAebyeA(1−e) from the right side.

Proof. Weﬁrstprovetheifpartbychecking conditionsinTheorem 1.1.Note thatthe traceofQ_S =A(1−e) isA(1−e)Ae,whichisnotsupportedonesinceeA(1−e)Ae= 0 by theweaklydirectedstructure.ButSe⊆eAeisnotcontainedin(1−e)Ae,socondition (1) inTheorem 1.1isalsotrue.

Let β : Ae → P be an A-module homomorphism, where P ∈ add(A(1−e)). Since Hom_A(Ae,A(1−e)) ∼= eA(1−e), β corresponds to an element x = β(e) ∈ eP, so β(Ae)=Aex.Bytheassumption,Sex= 0.Therefore, β(Se)= 0.Inotherwords, Seis containedinthekernelofβ, andhencecondition(2)inTheorem 1.1holds.

Conversely,supposethatconditionsinTheorem 1.1hold.Thenbythepreviouspara- graph, for any x ∈ eA(1−e), we can ﬁnd a module homomorphism β ∈ Hom_A(Ae, A(1−e)) such thatβ(Ae)=Aex. Inparticular, Sex=β(Se)= 0 since byLemma 2.3 Seisinthekernelofβ.This ﬁnishestheproof. 2

Remark 4.4. Since for any e ∈ E, eA(1−e) is a ﬁnite dimensional vector space, we can takeaﬁnite basis{x_i}ⁿi=1 ofeA(1−e). Moreover,we canchooseaminimal set of elements{v_j}^mj=1 ⊆eAewhichgeneratesthesocleofeAe.Therefore,inpracticeweonly need toverifythatvjxi= 0 for all1inand1jm.

Considerthefollowingexample:

(16)

Example4.5.LetAbethepathalgebraofthefollowingquiverwithrelationsδ²=θ²= ρ²= 0, θρ=ρθ= 0,αδ=βδ= 0,andθα=ρβ = 0.

x

α

β

δ y

θ ρ

Indecomposableprojectivemodulesare:

Px= x x y y

y y

, Py = y y y.

ItisroutinetocheckthatPxsatisﬁesallconditionsinTheorem 1.1,buttheindecom- posableprojectivemoduleP_y failscondition(2).Indeed,thereisashortexactsequence

0 Py Px M 0 ,

whereM= x

x y.ThisisbecausethesocleofP_yisspannedbyρandθ,butθβ= 0 andρα= 0.

Weend this section byintroducing a way to construct weaklydirected algebras for whichallsimplemodulessatisfytheconditionsinTheorem 1.1.LetQ= (Q0,Q1) be a ﬁniteconnectedquiverwithoutorientedcycles,whereQ₀andQ₁arethevertexset and the arrow set respectively.To each v ∈ Q0 we assign an integer mv 2,and to each arrowα:v→wwe assignanintegerlα1 such thatlα<min{mv,mw}.

Nowaddaloopt_v toeachvertex v∈Q₀to getanotherquiverQ.˜ Deﬁne R=kQ/˜ t^m_v^v; t_wα−αt_v;αt^l_v^α|v∈Q₀; Q₁α:v→w.

Thisisaﬁnitedimensionalalgebracontainingatwo-sidedidealJgeneratedbyallarrows inQ₁.LetI⊆J² be anarbitrarytwo-sidedideal,anddeﬁne A=R/I.

Hereisanexampleexplainingtheaboveconstruction.

Example4.6.LetAbe thepathalgebraofthefollowingquiver withrelations (1) δ³=ρ³=θ³= 0;

(2) αδ =ρα,αδ²= 0;

(3) βρ=θβ,βρ²= 0;

(4) βα= 0.