Silted algebras Advances in Mathematics

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Advances in Mathematics

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Silted algebras

Aslak Bakke Buan,Yu Zhou^∗

DepartmentofMathematicalSciences,NorwegianUniversityofScienceand Technology,7491Trondheim,Norway

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received3August2015 Accepted12July2016

Availableonline30August2016 CommunicatedbyHenningKrause

Keywords:

Siltingtheory Torsionpairs Shodalgebras Derivedcategories

Westudyendomorphismalgebrasof2-termsiltingcomplexes inderivedcategoriesofhereditaryfinitedimensionalalgebras, ormoregenerallyofExt-finitehereditaryabeliancategories.

Modulecategoriesofsuchendomorphismalgebrasareknown to occur as hearts of certain bounded t-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which arealgebras characterized by the propertythat each indecomposablemoduleeitherhasinjectivedimensionatmost one,orithasprojectivedimensionatmostone.

©2016TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

HappelandRingel[11] introducedtilted algebrasinthe earlyeighties.These arethe finite dimensional algebras which occur as endomorphism algebras of tilting modules overhereditaryfinite dimensionalalgebras.

ThenotionwasgeneralizedbyHappel,ReitenandSmalø[12],whointroducedquasi- tilted algebras, which are the algebras occurring as endomorphism algebras of tilting

✩ Thiswork wassupportedby FRINATgrant number 231000,from the NorwegianResearchCouncil.

SupportbytheInstitutMittag-Leffler(Djursholm,Sweden)isgratefullyacknowledged.

* Correspondingauthor.

E-mailaddresses:[email protected](A.B. Buan),[email protected](Y. Zhou).

http://dx.doi.org/10.1016/j.aim.2016.07.004

0001-8708/©2016TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

objectsinhereditaryabelian categorieswithfinitenessconditions.Quasi-tiltedalgebras wereshowntohaveanaturalhomologicalcharacterization.Considerthefollowingprop- ertyonthecategorymodA,offinitedimensionalrightA-modules,forafinitedimensional algebra A: for each indecomposable module X, we have either pdX ≤1 or idX ≤ 1 (that is: the projective or the injective dimensionis at mostone). Let us call this the shod-property(=smallhomologicaldimension).

In [12], theyshowedthatthequasi-tilted algebras areexactlythe algebras ofglobal dimensionatmosttwowiththeshod-property.Theyalsoshowedthatalgebraswiththe shod-propertyhaveglobaldimensionatmostthree.

Then,Coelho andLanzilotta[8]definedshodalgebras,asthealgebras withtheshod- property.Shodalgebrasofglobaldimensionthree,theycalledstrictlyshod.LaterReiten and Skowroński[17] gaveacharacterizationofthe strictlyshodalgebras,intermsof a property oftheAR-quiverQ_AofA:namelytheexistenceofwhattheycalledafaithful double sectioninQA.

Our aim is to show thatshodalgebras admit a verynatural characterization,using the notion of silting complexes, as introduced by Keller and Vossieck [15]. Let P be a complex inthebounded homotopycategory of finitely generated projectiveA-modules K^b(projA). Then P is called silting ifHom_K^b_(projA)(P,P[i]) = 0 for i >0, and if P generatesK^b(projA) asatriangulatedcategory.Furthermore,wesaythatPis2-termif Ponlyhasnon-zerotermsindegree0 and−1.InSection4,wegeneralizethenotionto boundedderivedcategoriesofabeliancategories.NotethatK^b(projA) canbeconsidered as afullsubcategory of theboundedderivedcategory D^b(A),andas equalto D^b(A) if A hasfinite globaldimension.

Definition0.1.LetBbeafinitedimensionalalgebraoverafieldk.WecallBsiltedifthere isafinitedimensionalhereditaryalgebraHanda2-termsiltingcomplexP∈K^b(projH) suchthat B ∼= EndD^b(H)(P). Furthermore, the algebraB is called quasi-silted if B ∼= End_Db(H)(P) for a 2-term silting complex P in the derived category of an Ext-finite hereditary abeliancategoryH.

In[6],westudiedtorsiontheoriesinducedby2-termsiltingcomplexes,andgeneralized classicalresultsofBrenner–Butler[4]andHappel–Ringel[11].Hereweapplytheseresults to provethefollowing mainresult.

Theorem0.2.LetAbeaconnectedfinitedimensionalalgebraoveranalgebraicallyclosed field k.Then

(a) A isastrictly shodor atiltedalgebra if andonlyif itisasilted algebra.

(b) A isashod algebra ifandonly itisaquasi-siltedalgebra.

Thepaperisorganizedasfollows. Wefirstrecall somenotationandfactsconcerning 2-termsiltingcomplexesandinducedtorsionpairs.TheninSection2,weprovepart(a)

ofourmaintheorem.InSection3weprovidealinktotheclassificationofshodalgebras byReitenandSkowroński.Thenwedefine2-termsiltingcomplexesinboundedderived categoriesofExt-finite abeliancategories,inSection4,where alsopart(b)ofourmain theorem isproved. We concludewith providingasmall exampleto illustrate ourmain theorem.See[3]forthedefinitionoftorsionpairsandotherundefinednotionsformodule categories. See [9,12] for the definition of t-structures and other undefined notions for derivedcategories.

WewouldliketothankSteffenOppermannandDongYangfordiscussionsrelatedto thispaper.

1. Backgroundandnotation

In this section we fix notation and recall facts concerning silting theory for 2-term siltingcomplexes.Wereferto[6]fordetails.

Inthis paper, all modulesare right modules. LetA be afinite dimensionalalgebra over a field k. We denote by modA the category of all finitely generated A-modules.

A compositionf gofmorphismsf andg meansfirstgandthenf.Butacompositionab ofarrowsaandbmeansfirstathenb.Underthissetting,wehaveanequivalencefromthe categoryofallfinitedimensionalrepresentationsofaquiverQboundedbyanadmissible idealJ to the category modkQ/J, and acanonical isomorphismA ∼= EndAA. For an A-moduleM,weletaddM denotethefullsubcategoryofalldirectsummandsindirect sumsofcopies ofM,weletFacM denotethefullsubcategoryofallmoduleswhichare factorsofmodulesinaddM,andweletSubMdenotethefullsubcategoryofallmodules whicharesubmodulesofmodulesinaddM.LetradMdenotetheradicalofamoduleM, and let socM denote the socle of M. We let D = Hom_k(−,k) denote the ordinary vector-space duality, we let ν denote the Nakayama functor ν = DHom_A(−,A), and weletτA denote theAuslander–Reiten translationinmodA.Note thattheNakayama functorinduces anequivalence K^b(projA)→K^b(injA),where projA andinjAdenote thefullsubcategories ofprojectives andinjectivesinmodA, respectively.Furthermore, wehaveanisomorphism

Hom_D^b_(A)(X, νY)∼=DHom_D^b_(A)(Y,X)

for X,Y in K^b(projA). Let U,V be full subcategories of a triangulated (or abelian) categoryW.WeletU ∗ V denotethefullsubcategory withobjects occurringas middle termsoftriangles(orshortexactsequences)withleftendtermsinUandrightendterms inV.

Consider a pair of indecomposable A-modules X,Y. If there exists a sequence of non-zero morphisms X = X0

f0

−−→ X1 f1

−−→ X2 f2

−−→ · · · → Xn fn

−−→ Xn+1 = Y with Xi

indecomposablefori= 0,1,· · ·,n+ 1,thenwecallX apredecessorofY,andY iscalled asuccessor ofX.

Let P be a 2-term silting complex in K^b(projA) for afinite dimensional k-algebra A and letB = End_D^b_(A)(P).We alwaysassumeP isbasic,and thatk isalgebraically closed.ThenBisapathalgebramoduloanadmissibleideal.Considerthesubcategories

T(P) ={X ∈modA|Hom_D^b_(A)(P, X[1]) = 0} and

F(P) ={X ∈modA|Hom_D^b_(A)(P, X) = 0}.

Then(T(P),F(P)) isatorsionpairinmodA.Furthermore,thefunctorsHom_D^b_(A)(P,−) restricted to T(P) and Hom_Db(A)(P,−[1]) restricted to F(P) are both fully faith- ful and there is a 2-term silting complex Q in D^b(modB), such that X(P) :=

Hom_D^b_(A)(P,F(P)[1]) = T(Q) and Y(P) := Hom_D^b_(A)(P,T(P)) = F(Q). We will referto thisfact,whichis themain resultof[6],asthesilting theorem.

Note that Hom_D^b_(A)(P,νP) ∼= DHom_D^b_(A)(P,P) is an injective cogenerator for modB.

The following facts from [13] and [6] concerning the torsion pair (T(P),F(P)) are useful.

Proposition 1.1. Withnotation asabove,we have:

(a) (T(P),F(P))= (FacH⁰(P),SubH⁻¹(νP));

(b) themodulesin addH⁰(P)are theExt-projectives inT(P);

(c) foreach X in T(P),thereisan exact sequence 0→L→T0→X →0 with T₀ inaddH⁰(P)andL inT(P);

(d) themodulesin addH⁻¹(νP)are theExt-injectivesinF(P);

(e) foreach Y inF(P),thereisan exactsequence 0→Y →F0→L→0 with F₀ in addH⁻¹(νP)and Lin F(P).

We shallalso need thefollowing factsconcerning B = End_D^b_(A)(P) andthetorsion pair(X(P),Y(P)) inmodB,inthecasewhereA ishereditary.

Proposition 1.2. Let H be a hereditary algebra, let P be a 2-term silting complex in K^b(projH)andlet B= End_Db(H)(P).Thenthefollowinghold.

(a) The torsionpair(X(P),Y(P))issplit.

(b) X(P)isclosed undersuccessors andY(P)isclosed underpredecessors.

(c) Forany X∈ X(P)andY ∈ Y(P),we havepdYB ≤1andidXB ≤1.

(d) Theglobal dimensionofB isatmost3.

(e) Anyalmost splitsequencein modB lies entirelyineither X(P)or Y(P),or elseit is aconnectingsequence

0→F(νP)→F((radP)[1])⊕F(νP/soc(νP))→F(P[1])→0,

forPindecomposableprojectivewithP /∈add(P⊕P[1]),whereF = Hom_D^b_(H)(P,−).

Proof. (a)This followsfrom[6,Lemma 5.5].

(b)This follows from (a), using thefact thatthere areno non-zeromaps from objects inX(P) toobjectsinY(P).

(c) Let Y be anobject in Y(P). By definition,there is an H-moduleM ∈ T(P) such thatHom_D^b_(H)(P,M)=Y. ThenbyProposition 1.1(c),thereisanexactsequence

0→L→T0→M→0

with T0 ∈ addH⁰(P) and L ∈ T(P). Applying Hom_D^b_(H)(P,−), we have an exact sequence

0→Hom_D^b_(H)(P, L)→Hom_D^b_(H)(P, T0)→Hom_D^b_(H)(P, M)→0.

Ontheotherhand,applyingHomH(−,N) foranyN ∈ T(P) wehaveanexactsequence Ext¹_H(T₀, N)→Ext¹_H(L, N)→Ext²_H(M, N)

wherethe firsttermiszero,using Proposition 1.1(b)and thatT₀ is inaddH⁰(P),and wherethelasttermiszerosinceH ishereditary.SoExt¹_H(L,N)= 0 foranyN ∈ T(P), which implies L ∈ addH⁰(P) by Proposition 1.1(b). Since pdH⁰(P)H ≤ 1, we have that H⁰(P) is isomorphic to a direct summand of P. So both Hom_D^b_(H)(P,T₀) and Hom_Db(H)(P,L) areprojective,hence,wehavepdY_B= pd Hom_Db(H)(P,M)_B≤1.The proofforX ∈ X(P) issimilar byusing Proposition 1.1(d,e).

(d)Thisfollowsfrom (c),using[12, Proposition 2.1.1].

(e)Thisfollows from(a)and[6,Proposition 5.6]. 2

Wealsoneedthefollowingobservationwhichfollowsfrom[1,Lemma 2.3].

Lemma 1.3.For aprojective A-module P, wehave that P[1]is adirect summand in P ifand onlyif HomA(P,H⁰(P))= 0.

Werecallthenotionof(strictly)shodalgebras.

Definition1.4.([8])AnalgebraAiscalledshodifforeachindecomposableA-moduleX, either pdXA ≤ 1 or idXA ≤ 1. Also, A is called strictly shod if A is shod and gl.dimA= 3.

Let L be the set of the indecomposableA-modulesY such thatfor allpredecessors X ofY,wehavethatpdX_A≤1 andRbethesetoftheindecomposableA-modulesX such thatforallsuccessors Y of X,we havethatidYA ≤1.We recallsomeequivalent characterizationsofshodalgebras.

Proposition 1.5. ([8])The followingareequivalent foran algebraA:

(a) A isashod algebra;

(b) (addR,add(L\ R))isasplit torsionpairinmodA;

(c) (add(R\ L),addL)isasplit torsionpairinmodA;

(d) thereexistsasplittorsionpair(T,F)inmodAsuchthatpdYA≤1foreachY ∈ F andidX_A≤1foreach X ∈ T.

2. Siltedalgebrasandshodalgebras

NotethatbyProposition 1.2(a,c),anysiltedalgebraisshod.Inthissectionweprove thatafinite dimensionalalgebraoveranalgebraicallyclosed fieldissiltedifandonlyif itis strictlyshodor tilted.Thefollowingwillbe thekeyfactforourproof.

Proposition 2.1. If there is a 2-term silting complex P ∈ K^b(projA) such that (T(P),F(P))satisfies condition(d) inProposition 1.5,thenA isasiltedalgebra.

Theproofofthiswillfollowafteraseriesoflemmas.ForthiswenowfixanalgebraA and assumetheexistenceofa2-termsiltingcomplexP =

P⁻¹−→^p P⁰

∈K^b(projA), such that (T(P),F(P)) satisfies condition (d) inProposition 1.5. Then, in particular, Aisashodalgebra.LetB= End_Db(A)(P)∼=kQ_B/J_B,whereJ_B isanadmissibleideal.

We aim to provethat QB, theGabriel quiver of B, is acyclic.Then we show that the correspondinghereditaryalgebraH =kQBadmitsa2-termsiltingcomplex,suchthatA isisomorphictoitsendomorphismring.ThisisourstrategyforprovingProposition 2.1.

In order to describe QB and JB we need firstto consider two factor algebras of B, namely EndA(H⁰(P)) andEndA(H⁻¹(νP)).

Let P=P_L⊕P_M ⊕P_R, where P_L is thedirect sumof theindecomposable direct summands of P with zero 0th term and PR is the direct sum of the indecomposable directsummandsofPwithzero−1thterm. Weshallneedthefollowing lemma.

Lemma2.2. H⁻¹(P)isprojectiveandH⁻¹(P)[1]∈addPL.Dually,H⁰(νP)isinjective and H⁰(νP)∈addνPR.

Proof. Weprovethefirstpart.Theproofofthesecondpartissimilar.SincePisgiven bythemapP⁻¹−→^p P⁰,we haveanexactsequence

0→H⁻¹(P)→P⁻¹−→^p P⁰−→^cp H⁰(P)→0.

By definition, the map c_p is a projective cover of H⁰(P). Let K = kerc_p. By Propo- sition 1.1(b), H⁰(P) is Ext-projective inT(P).Therefore any directsummand ofK is notinT(P),andsince(T(P),F(P)) issplit,bytheassumption onP, wehavethatK belongstoF(P).SopdK_A≤1 bytheassumption,andhenceH⁻¹(P) isprojective.

Wenow provethat H⁻¹(P)[1] is inaddP_L. ByLemma 1.3, it is sufficientto prove thatHomA(H⁻¹(P),H⁰(P))= 0.ApplyingHom_D^b_(A)(−,H⁰(P)) tothetriangle

H⁻¹(P)[1]→P→H⁰(P)→H⁻¹(P)[2], wehaveanexactsequence

Hom_D^b_(A)(P, H⁰(P)[1])→Hom_D^b_(A)(H⁻¹(P)[1], H⁰(P)[1])

→Hom_Db(A)(H⁰(P), H⁰(P)[2]),

where the first term is zero by H⁰(P) ∈ T(P) and the third term is zero by idH⁰(P)A≤1.ThenwehaveHomA(H⁻¹(P),H⁰(P))= 0. 2

By this we get the following relations between the algebras EndA(H⁰(P)), End_A(H⁻¹(νP)) andEnd_D^b_(A)(P).

Lemma2.3. The functorH⁰(−)gives asurjectivehomomorphismof algebras H⁰(−) : End_D^b_(A)(P)→EndA(H⁰(P))

whose kernel is the space consisting of morphisms which factor through addPL. The functorH⁻¹(ν−)gives asurjectivehomomorphismof algebras

H⁻¹(ν−) : End_Db(A)(P)→End_A(H⁻¹(νP))

whose kernelisthespaceconsisting ofmorphisms whichfactorthroughaddP_R. Proof. Since H⁰(−) is a k-linear functor, it gives a homomorphism of the algebras.

UsingthatPisaprojectivepresentationofH⁰(P),weobtainthatthishomomorphism is surjective. By H⁰(P_L) = 0, we have that H⁰(f) = 0 for any morphism f which factorsthrough P_L. Nowwe assume thatH⁰(f)= 0 for a chainmap f = (f⁻¹ f⁰) ∈ End_D^b_(A)(P).Consideringthefollowingdiagram:

P^{−1 p}

f⁻¹

P⁰

f⁰

H⁰(P)

H⁰(f)

0

P^{−1 p} P⁰ H⁰(P) 0

wehavethatf⁰factorsthroughp.Sowemayassumethatf⁰= 0 uptohomotopy.Inthis case, f⁻¹ factors through H⁻¹(P). Due to Lemma 2.2, we haveH⁻¹(P)[1]∈ addPL. Hencef factorsthroughaddPL.Thus,theproofofthefirststatementiscomplete.The second statementisdualtothefirstone. 2

Thefollowing willbecrucial forprovingthatQB isacyclic.

Lemma 2.4.Both EndA(H⁰(P))andEndA(H⁻¹(νP))arehereditaryalgebras.

Proof. WeprovethestatementforB0= EndA(H⁰(P)).TheproofforEndA(H⁻¹(νP)) issimilar. Tocompletetheproof,itissufficienttoprovethatpdSB0 ≤1 foranysimple B0-module S. Since, by Lemma 2.3, the algebraB0 is afactor algebra of B, we have that modB₀ is afull subcategory of modB closed underboth submodules and factor modules.Therefore,thetorsionpair(X(P),Y(P)) inmodB,givesrisetoatorsionpair (X(P)∩modB₀,Y(P)∩modB₀) inmodB₀.Itisstraightforwardtoverifythatwehave Hom_D^b_(A)(P,H⁰(P))∼= HomA(H⁰(P),H⁰(P)),so Hom_D^b_(A)(P,H⁰(P)) isaprojective generatorofmodB0.

Let S be asimple B0-moduleand Hom_D^b_(A)(P,T0)→^pS S →0 bea projective cover ofS,whereT0∈addH⁰(P).WehavethatHom_D^b_(A)(P,T0) isinY(P),andsinceY(P) is closed under submodules, also the kernel of p_S is in Y(P). Hence there is an L in T(P),suchthatthere isanexactsequence

0→Hom_D^b_(A)(P, L)→Hom_D^b_(A)(P, T₀)→S→0. (2.1) WeclaimthatExt¹_A(L,T(P))= 0.Indeed,sinceSissimple,eitherS ∈ X(P)∩modB0

orS ∈ Y(P)∩modB0.IfS∈ Y(P)∩modB0,then,bydefinition,thereisanX ∈ T(P) such thatS = Hom_Db(A)(P,X). Bythesiltingtheorem,the exactsequence(2.1) gives risetoanexactsequence

0→L→T₀→X →0 (2.2)

in modA. By the assumption on T(P), we have that Ext²_A(−,T(P)) = 0. In particular Ext²_A(X,T(P)) = 0, and hence it follows from the sequence (2.2) that Ext¹_A(L,T(P)) = 0.

Now consider the case with S ∈ X(P)∩modB₀. Then, by definition, there is a Y ∈ F(P) suchthatS= Hom_D^b_(A)(P,Y[1]).Sowehaveatriangle(see[6,Theorem 2.3]) inD^b(A):

Y →L→T0→Y[1].

Since(T(P),F(P)) issplit,itfollowsthatExt¹(Y,T(P))= 0.ThereforeExt¹(L,T(P))

= 0.Thus,wehavefinishedtheproofoftheclaimthatExt¹_A(L,T(P))= 0.Thisimplies thatL∈addH⁰(P) byProposition1.1(b).Hencetheexactsequence(2.1)isaprojective resolutionofS,whichshowsthatpdS_B0 ≤1. 2

WefirstgiveapreliminarydescriptionofQB andJB forB= End_D^b_(A)(P).Thiswill beimprovedinLemma 2.9.

LetV?be theset ofthe verticesof QB corresponding to thedirectsummandsof P?

andlete?bethesumofprimitiveorthogonalidempotentsev withv∈V?,where?=L, M orR.LetQ^LM_B (resp.Q^MR_B )be thefullsubquiverofQB consistingoftheverticesin V_L∪V_M (resp.V_M ∪V_R)and thearrowsbetweenthem.ForanyalgebraΛ,weuseQ_Λ todenote its(Gabriel) quiver.

Lemma2.5. Withtheabove notation,thefollowinghold.

(a) Each path in QB from VL to VR, or from VR to VL, is in JB.In particular, there are noarrows from V_L toV_R andnoarrows fromV_R toV_L.

(b) The surjective algebra morphisms in Lemma 2.3 induce isomorphisms of quivers Q^MR_B ∼=Q_EndA(H⁰_(P)) andQ^LM_B ∼=Q_EndA(H−1(νP)).

(c) The algebra B is monomial, andthe ideal JB is generated by thepaths from VL to V_R and thepaths from V_R toV_L.

Proof. The first assertion in (a) follows from Hom_D^b_(A)(PR,PL) = 0 and Hom_D^b_(A)(PL,PR)= 0.Then thesecond assertionfollowssinceJB isadmissible.

The surjective algebra morphisms in Lemma 2.3 induce algebra-isomorphisms End_A(H⁰(P)) ∼= B/Be_LB and End_A(H⁻¹(νP)) ∼= B/Be_RB. The (Gabriel) quiver ofB/Be?BisobtainedfromQB byremovingtheverticesinV?andthearrowsadjacent tothese vertices,where?=L,R.Then(b)follows.

Moreover, it follows from the above algebra-isomorphisms that J_B/J_Be_LJ_B = 0 = J_B/J_Be_RJ_B since, by Lemma 2.4, End_A(H⁰(P)) and End_A(H⁻¹(νP)) are hereditary.

Sofor anyminimal relation

iλifi with λi ∈knonzero andfi apathinQB,eachfi

factorsthroughVL andVR.Hence,by(a),theassertionsin(c)follow. 2

Beforewe canshow thatQB is acyclicwe needsomemore propertiesofthe torsion pair(X(P),Y(P)).

Lemma2.6. Withtheabove notation,thefollowinghold.

(a) For any X in modA, we have that the B-module Hom_Db(A)(P,X[1]) is in X(P), and thatHom_D^b_(A)(P,X)is inY(P).

(b) We have that the projective B-module Hom_D^b_(A)(P,PL) is in X(P) and that the injective B-moduleHom_D^b_(A)(P,νPR) isinY(P).

(c) LetCM betheprojectivecoverofaB-moduleM.IfCM isinadd Hom_D^b_(A)(P,PL), thenM isinX(P).

(d) LetEM betheinjectiveenvelopeofaB-moduleM.IfEM isinadd Hom_D^b_(A)(P,νPR), thenM isinY(P).

Proof. Part(a)iscontainedin[6,Lemma 3.6].Part(b)followsdirectlyfrom (a).Then parts (c) and (d) follow from the factsthatX(P) is closed under factormodules, and Y(P) isclosed undersubmodules. 2

Lemma 2.7.We haveExt²_B(X(P),Y(P))= 0.

Proof. Since (T(P),F(P)) is split byassumption, wehavethatPis atilting complex by[6,Proposition 5.7].Then wehave

Ext²_B(X(P),Y(P))∼= HomD^b(A)(F(P)[1],T(P)[2])∼= Ext¹_A(F(P),T(P)) = 0. 2 Lemma 2.8.There areno pathsfrom VL toVR inQB.

Proof. Assumethere isapath pfrom v₁ inV_L to v₂ inV_R, andassume itcontains no proper subpathfromV_L to V_R. Then, byLemma 2.5(c),itfollowsthatpisaminimal relation. LetS_i fori = 1,2 be thesimple B-module corresponding to vertex v_i. Then Ext²_B(S₁,S₂)= 0,by[5,Proposition 3.4] (notethatwe useright modulesbuttheyuse left modules).

ByLemma 2.6 (c,d), wehavethatS1 isinX(P) andS2 is inY(P).Now theclaim follows fromLemma 2.7. 2

Summarizing, wehavethefollowingdescriptionofQ_B and J_B.

Lemma 2.9.The quiver QB is acyclic and the ideal JB is generated by the paths from V_R toV_L.

Proof. ByLemma 2.4 and2.5(b),therearenocyclesinthesubquivers Q^LM_B andQ^MR_B . On theother hand,there areno pathsfrom VL toVR by Lemma 2.8.Hence, thereare nocycles inQB. Itfollowsfrom Lemma 2.5(c)and Lemma 2.8thatJB isgenerated by thepathsfrom VR to VL. 2

The followingfacts, see[1, Theorem 0.5andProposition 1.1], willbe crucial.Recall that a torsion pair (T,F) in mod Λ is called functorially finite if both T and F are functorially finitesubcategories.

Proposition 2.10.Let(T,F) be a torsion pairin mod Λ, fora finite dimensional alge- bra Λ.

(a) (T,F) is functorially finite if and only if there is a 2-term silting complex C in K^b(proj Λ),suchthat(T,F)= (T(C),F(C)).

(b) (T,F)isfunctoriallyfiniteifandonlyifT = FacU,whereU isExt-projectiveinT. Weneedonemoreobservationbeforefinishing theproof ofProposition 2.1.

Lemma2.11.Wehave X(P)⊂mod EndA(H⁻¹(νP))andY(P)⊂mod EndA(H⁰(P)).

Proof. Weonlyprovethefirstinclusion,thesecondcanbeproveddually.Bythesecond partofLemma 2.3,weonlyneedtoprovethatforanyelement γinEnd_Db(A)(P) which factorsthroughaddPR, wehaveXγ= 0 forany X inX(P).This holdssinceX(P)= Hom_D^b_(A)(P,F(P)[1]) andclearlyHom_D^b_(A)(PR,M[1])= 0 for anyA-moduleM. 2 Proof ofProposition 2.1. Let P be a 2-term silting complex in K^b(projA) such that theinducedtorsionpair(T(P),F(P)) satisfiescondition (d)inProposition 1.5 andlet B= End_D^b_(A)(P)=kQB/JB.ByLemma 2.9,thequiverQBisacyclicandtheidealJB

isgenerated bythe pathsfrom thevertices inV_R to thevertices inV_L. LetH =kQ_B. Wethen haveaninducedembedding modB ⊂modH. Weclaim that(X(P),Y(P)) is alsoatorsion pairinmodH.

ForanyH-moduleM,weonlyneedtoprovethatMisinX(P)∗ Y(P).Wefirstnote thatM eLH isinX(P) byLemma 2.6(c).Considertheshortexactsequence

0→M eLH→M→M/M eLH →0. (2.3) BythedescriptionofJB itisclearthatMJB⊂M eLH,andhencethatN =M/M eLH is in modB. Now, using that (X(P),Y(P)) is atorsion pair in modB, we have N ∈ X(P)∗ Y(P).SinceX(P) isclosedunderextensionsinmodB,byLemma 2.11,itisalso anextension-closedsubcategoryofmod EndA(H⁻¹(νP)).ByLemma 2.4,Lemma 2.5(b) and the definition of H, we have End_A(H⁻¹(νP)) ∼= H/He_RH. Hence X(P) is also closedunderextensionsinmodH.Usingthesequence (2.3),wehavethatM ∈ X(P)∗ X(P)∗ Y(P)=X(P)∗ Y(P).Therefore, (X(P),Y(P)) isalsoatorsionpairinmodH.

WealsoclaimthatforanyX ∈ X(P) andY ∈ Y(P),wehaveafunctorialisomorphism Ext¹_H(X,Y) ∼= Ext¹_B(X,Y). For this, it is sufficient to prove that for any short exact sequenceinmodH:

0→Y →E→X →0,

wehavethatE∈modB. Foranyvertex vr inVR, wehaveXevr = 0,byLemma 2.11, andsimilarly Y evl= 0 forany vertexvlinVL.Foranarbitrarypath evrpevl inJB, we thenhavethatX(e_vrpe_vl)= 0,so E(e_vrpe_vl)⊂Y e_vl= 0.HenceEJ_B = 0,and E isin modB.

The torsion pair (X(P),Y(P)) is a functorially finite torsion pair in modB, by [6, Corollary 3.9], and it then follows from Proposition 2.10(b) thatit is also functorially finite inmodH.

Hence, by Proposition 2.10(a), it follows that there is a 2-term silting complex R in K^b(projH) such that (T(R),F(R)) = (X(P),Y(P)). Since H is hereditary, the torsionpair(X(R),Y(R)) issplitbyProposition 1.2(a).Bythesiltingtheorem,wehave X(R) F(R)=Y(P) T(P) andsimilarlyY(R) F(P).

So we have split torsion pairs (T(P),F(P)) in modA, and (X(R),Y(R)) in mod End_Db(H)(R). We claim that we actually have modA mod End_Db(H)(R). For this, we need in addition a functorial isomorphism HomEnd_Db(H)(R)(Y(R),X(R)) ∼= HomA(F(P),T(P)).Indeed,wehave

Hom_A(F(P),T(P))∼= Hom_D^b_(A)(F(P)[1][−1],T(P))

∼= Ext¹_B(X(P),Y(P))

∼= Ext¹_H(X(P),Y(P))

= Ext¹_H(T(R),F(R))

∼= HomD^b(H)(T(R),F(R)[1])

∼= HomEnd_Db(H)(R)(Y(R),X(R)).

It nowfollowsthatmodAmod End_D^b_(H)(R).HenceA∼= EndD^b(H)(R),andwehave provedthatAissilted. 2

Thefollowinglemmagivesasufficientconditionforafinitedimensionalalgebratobe tilted.

Lemma2.12. LetAbeafinitedimensionalalgebra andPa2-termsilting complex,such thattheconditionofProposition 2.1holds.If,inaddition,T(P)containsalltheinjective A-modules,orF(P)contains alltheprojectiveA-modules,then Aisatilted algebra.

Proof. Assume T(P) contains all the injectiveA-modules. Then, we have νPL[−1] ∈ T(P). So 0= Hom_D^b_(A)(P,νPL[−1][1]) = Hom_D^b_(A)(P,νPL) ∼=DHom_D^b_(A)(PL,P).

In particular, Hom_D^b_(A)(P_L,P_L) = 0. Hence P_L = 0. Then |H⁰(P)| = |P| = |A|. Since H⁰(P) ∈ T(P), we have idH⁰(P)_A ≤ 1 by assumption. It is clear that Ext¹_A(H⁰(P),H⁰(P)) = 0. So H⁰(P) is a cotilting A-module. By Lemma 2.4, the al- gebraEnd_A(H⁰(P)) ishereditary.Therefore,thealgebraAistilted.Theothercasecan be proveddually. 2

Now weprovethemain resultinthissection.

Theorem 2.13.Let A be a connected finite dimensional algebra over an algebraically closed fieldk.Thenthefollowingareequivalent:

(a) Aisasilted algebra;

(b) there isasplit functorially finitetorsion pair(T,F)in modA suchthat idAX ≤1 forany X ∈ T and pd_AY ≤1forany Y ∈ F;

(c) Ais ashodalgebra with (add(R\ L),addL)functorially finite;

(d) Aisatilted algebra orastrictly shodalgebra.

Inthis case, theglobaldimensionof A isatmost3.

Proof. (a)⇒(b): This follows from combining Proposition 1.2(a,c) and Proposition 2.10(a).

(b)⇒(d):Assuming(b),itfollowsfromProposition 2.10(a)thatthereisa2-termsilting complex P ∈ K^b(projA) such that (T,F) = (T(P),F(P)). If F(P) contains all the projectiveA-modules,thenA isatiltedalgebrabyLemma 2.12.Ifthereisaprojective A-moduleinT(P),sinceT(P) iscontainedinaddR,thenRcontainsanExt-projective module.By[12,Theorem II.3.3],ifAis quasi-tilted,thenAis tilted.Thustheproof is complete.

(d)⇒(c):See[7,Theorem 3.6].

(c)⇒(b):This istrivial.

(b)⇒(a):Thisfollows fromcombiningPropositions 2.1 and2.10.

Finally,recall thatitwas provedinProposition 1.2(e),thatthe globaldimension of asiltedalgebraisatmost 3. 2

Notethatwehavenowprovedpart(a)ofTheorem 0.2.

3. Doublesections

In [17], Reiten and Skowroński characterized strictly shod algebras as strict double tilted algebras, which are algebras whose AR-quiver contains a strict faithful double sectionwithcertainconditions.LetB beanalgebrawhichistiltedorstrictly shod.By the previous section, we know that B ∼= End_D^b_(H)(P) for a2-term silting complex P inthebounded derivedcategory ofsomehereditary algebraH.In thissection, we will usethis fact to givean alternativeproof of whyB hasa faithfuldouble section Δ,by identifyingthemodulesinΔ asimagesofsomeinjectiveorprojectiveA-modulesunder thefunctorsHom_D^b_(H)(P,−) orHom_D^b_(H)(P,−[1]).Furthermore, wehavethatΔ isa sectionwhenB istilted, whileitisastrict doublesectionwhenB isstrictlyshod.The constructionofthedoublesectionΔ inasiltedalgebraisananalogueoftheconstruction ofasection inatiltedalgebra.Forthelatter,wereferto[3,Section VIII.3].

WerecallsomedefinitionsconcerningAR-quivers.ForanalgebraΛ,denotebyΓΛthe AR-quiverofΛ andbyτΛ=DTr andτ_Λ⁻¹= TrDtheAR-translationsin ΓΛ.AτΛ-orbit of amodule M ∈mod Λ isthe collection {τ_Λ^mM |m ∈Z}. A path x₀ →x₁ → · · · → xs−1 → xs in ΓΛ is called sectional if there is no i with 1 ≤ i ≤ s−1 such that

x_i−1=τΛxi+1 andis calledalmostsectional ifthereisexactlyoneiwith1≤i≤s−1 suchthatx_i−1=τΛxi+1.

Let C be aconnectedcomponentofΓΛ.Aconnectedfull subquiverΔ of C iscalled adoublesection in C ifthefollowingconditionshold:

– Δ isacyclic,i.e. thereisnoorientedcyclesinΔ;

– Δ is convex, i.e.for eachpath x₁ →x₂ → · · · →x_s in C with x₁,x_s ∈Δ we have x_i∈Δ forall1≤i≤s;

– Foreachτ_Λ-orbit_O in_C wehave1≤ |Δ∩O|≤2;

– If _O is aτ_Λ-orbitin_C and |Δ∩O| = 2 then Δ∩O ={X,τ_ΛX}for someX ∈C and there are sectionalpaths I → · · · → τ_ΛX and X → · · · → P, with I injective andP projective.

AdoublesectionΔ in C iscalledstrictifthereexistsaτΛ-orbit O in C with|Δ∩O|= 2 and iscalled asection ifforanyτΛ-orbit O in C wehave|Δ∩O|= 1.Adoublesection is calledfaithful,ifthedirectsumofthecorrespondingmodulesisfaithful.

Now let B be a connected silted algebra, that is, B is connected and there is a hereditary algebra H and a 2-term silting complex P ∈ K^b(projH) such that B = End_Db(H)(P).LetF(−)= Hom_Db(H)(P,−) :D^b(H)→modB.

Let _P be acomplete set of non-isomorphic indecomposable projective H-modules.

Let_Pl bethesubsetof_P consisting ofP withP ∈addPandlet_Prbe thesubsetof P consistingofP withP[1]∈addP.Itisclearthat_Pl∩Pr=∅.

Lemma 3.1.With theabove notation,thefollowinghold.

(a) For any P ∈P, wehave F(P[1])∈ X(P) andF(νP)∈ Y(P);F(P[1])= 0 if and only ifP ∈Pl;F(νP)= 0if andonly ifP ∈Pr.

(b) ForanyP ∈P\(Pl∪Pr),wehavethatboth ofF(νP)andF(P[1])areindecom- posable andthereisan AR-sequence

0→F(νP)→F(νP/soc(νP))⊕F((radP)[1])→F(P[1])→0.

In particular,inthis case, F(νP) isnotinjective andF(P[1])isnot projective.

(c) An indecomposable B-module in X(P) is projective if and only if it is isomorphic to F(P[1]) for P ∈ Pr.In this case, there is a right minimalalmost split mapin modB

F(νP/soc(νP))⊕F((radP)[1])→F(P[1]).

(d) An indecomposableB-module in Y(P)isinjective if andonly if itis isomorphic to F(νP)forP ∈Pl.Inthis case, thereisaleft minimalalmostsplit mapinmodB

F(νP)→F(νP/soc(νP))⊕F((radP)[1]).

Proof. Statement(a)followsfromLemma 2.6andthedefinitionsofPlandPrand(b) follows from Proposition 1.2(f). We will prove(c). Statement (d) canbe proved simi- larly.For any P ∈Pr,we haveP[1]∈addP. SoF(P[1]) is projective. Now weprove thatall indecomposable projective modules inX(P) areof this from.Let F(M[1]) be anindecomposableprojectiveB-moduleinX(P) withM∈ F(P).Applyingthefunctor F(−) to the projective cover P_M −−→^pM M of M in modH, we obtain anepimorphism F(P_M[1])→F(M[1]) sinceF(kerp_M[2])= 0.SinceF(M[1]) isprojective, this epimor- phismis split.HenceF(M[1]) isadirectsummand ofF(P_M[1]). Therefore,by(a) and (b),F(M[1]) hasto havetheform F(P[1]) forsomeP ∈Pr.

By[9,Chap.4],thereis anAR-triangle

νP →νP/S⊕(radP)[1]→P[1]→(νP)[1]

inD^b(H), where S = soc(νP). Applying the functor F to this triangle, we obtain an exactsequence

0→F(νP/S)⊕F(radP[1])→F(P[1])−→^u F((νP)[1])

inmodB.Notethatthelast mapuinthisexactsequencefactorsthroughF(S[1]).For each indecomposable summand P of P, if P is of the form P[1] forsome P ∈ Pr, then Hom_D^b_(H)(P,S[1]) is 1-dimensional for P ∼= P, and 0-dimensional for P P. If P is not of such form, then H⁻¹(P) = 0 and hence Hom_D^b_(H)(P,νP[1]) ∼= DHom_D^b_(H)(P[1],P) = 0. So the image of u is the simple top of F(P[1]). Hence F(νP/S)⊕F(radP[1])→F(P[1]) isarightminimalalmost splitmap. 2

Let P_r be the subset of P consisting of modules from which there are nonzero morphisms to modules inPr which do notfactor through modules inPl. Dually, let P_lbethesubsetof Pconsistingofmodulestowhichtherearenonzeromorphismsfrom modulesin Pl whichdonotfactorthroughmodulesin Pr.

Theorem3.2.LetH beafinite dimensionalhereditaryalgebraandPbea2-termsilting complexinK^b(projH)suchthatB = End_D^b_(H)(P)isconnected.Thenthefullsubquiver Δ of ΓB formed by F(P[1]) forP ∈ P_r and by F(νP) for P ∈ P_l∪(P\P_r), is a faithful doublesection in acomponent ΨP of ΓB.Moreover, Δ is asection if andonly ifB istilted,while Δisastrict doublesectionif andonly ifB isstrictlyshod.

Proof. We first note that, since A is hereditary, then for a projective P with P/

radP ∼= S, we have that radP is projective and νP/S is injective. By Lemma 3.1, Pr is the set of modules P ∈ P such that there is a path in Γ_B from F(P[1]) to F(P[1]) forsomeP∈Pr, and_Pl isthesetofP ∈P suchthatthereis apathfrom F(νP) toF(νP) forsomeP∈Pl.

Let Z1 → Z2 → · · · → Zs be a path in ΓB with Z1,Zs ∈ Δ. To prove that Δ is convex,itisclearlysufficientto provethatZ2is inΔ,andproceedbyinduction.