Maximal τ_d-rigid pairs

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Maximal τ

-rigid pairs

Karin M. Jacobsen^a,∗,1, Peter Jørgensen^b

aNorwegianUniversityofScienceandTechnology,DepartmentofMathematical Sciences,Sentralbygg2, Gløshaugen,7491Trondheim,Norway

bSchoolofMathematicsandStatistics,NewcastleUniversity,NewcastleuponTyne NE17RU,UnitedKingdom

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

Articlehistory:

Received30April2019

Availableonline14November2019 CommunicatedbyDavidHernandez

MSC:

16G10 18E10 18E30

Keywords:

d-Abeliancategory (d+ 2)-Angulatedcategory Higherhomologicalalgebra Maximald-rigidobject Maximalτd-rigidpair

Let_T bea2-Calabi–Yautriangulatedcategory,T a cluster tilting object with endomorphism algebra Γ. Consider the functor _T(T,−) : _T → mod Γ. It induces a bijection fromtheisomorphismclassesofclustertiltingobjectstothe isomorphismclassesofsupportτ-tiltingpairs.Thisisdueto Adachi,Iyama,andReiten.

Thenotionof(d+2)-angulatedcategoriesisahigheranalogue oftriangulatedcategories.Weshowahigheranalogueofthe aboveresult,basedonthenotionofmaximalτ_d-rigidpairs.

©2019PublishedbyElsevierInc.

0. Introduction

In triangulated categories, the notions of cluster tilting objects (introduced in [4, p. 583]) and maximal rigid objects have recently been extensively investigated. They

* Correspondingauthor.

E-mailaddresses:[email protected](K.M. Jacobsen),[email protected] (P. Jørgensen).

URL:http://www.staff.ncl.ac.uk/peter.jorgensen(P. Jørgensen).

1 Currentaddress:FakultätfürMathematik,UniversitätBielefeld,33501Bielefeld,Germany.

https://doi.org/10.1016/j.jalgebra.2019.10.046 0021-8693/©2019PublishedbyElsevierInc.

frequentlycoincide,by[22,thm.2.6],andtheyarecloselylinkedtothenotionofsupport τ-tiltingpairsinabeliancategories (introducedin[1, def.0.3]).Indeed,there isoftena bijection betweenthecluster tiltingobjects inatriangulated categoryand thesupport τ-tiltingpairsinasuitable(abelian)modulecategory, see[1, thm.4.1].

This paperinvestigatestheanalogoustheoryin(d+ 2)-angulatedand d-abeliancat- egories, whichare the main objectsof higher homologicalalgebra,see [8, def.2.1] and [15,def.3.1].Severalkeypropertiesfromtheclassiccasedonotcarryover.Forexample, cluster tilting objects are maximal d-rigid, but the converse is rarely true. Moreover, the higher analogueof support τ-rigid pairs permita bijection to the maximal d-rigid objects, butnottotheclustertiltingobjects.

For furtherreading inhigherhomological algebraanumber ofreferences havebeen included inthe bibliography,see [3], [6], [7], [8], [9], [10], [11],[12],[13],[14], [15], [16], [17],[18],[19],[20],[21].

Let k be an algebraically closed field, d 1 an integer, T a k-linear Hom-finite (d+ 2)-angulated category with split idempotents, see [8, def. 2.1].Assume thatT is 2d-Calabi–Yau,see[21,def.5.2],andletΣ^d denote thed-suspensionfunctorof_T. Clustertiltingandmaximald-rigidobjects.AnobjectX ∈T isd-rigid ifExt^d_T(X,X)= 0.Werecallthreeimportantdefinitions.

Definition 0.1 ([21, def. 5.3]).Anobject X ∈T is Oppermann–Thomas cluster tilting in T if:

(i) X isd-rigid.

(ii) ForanyY ∈T there existsa(d+ 2)-angle

Xd → · · · →X0→Y →Σ^dXd

withX_i∈addX forall0≤i≤d.

Definition 0.2.AnobjectX∈T isd-self-perpendicularinT if addX ={Y ∈T |Ext^d_T(X, Y) = 0}. Definition 0.3.AnobjectX∈T ismaximal d-rigidin _T if

addX={Y ∈T |Ext^d_T(X⊕Y, X⊕Y) = 0}. Ourfirstmain resultis:

TheoremA.X isOppermann–Thomascluster tilting⇒X isd-self-perpendicular⇒X is maximald-rigid.

Weprove this inTheorem 1.1. Of equalimportance is thatthe implications cannot bereversedingeneral,seeRemark1.2.Inparticular, whend2,theclassofmaximal d-rigidobjects istypically strictly largerthanthe classof Oppermann–Thomascluster tiltingobjects,incontrasttotheclassiccased= 1 wherethetwoclassesusuallycoincide, see[22,thm.2.6].

Maximalτd-rigidpairs.LetT ∈T beanOppermann–Thomasclustertiltingobjectand letΓ= End_T(T).Recallthefollowingresult.

Theorem0.4 ([14,thm.0.6]). Consider theessentialimage _D of thefunctor_T(T,−): T →mod Γ.Then D isad-clustertiltingsubcategoryofmod Γ.Thereisacommutative diagram,asshownbelow,wheretheverticalarrowisthequotientfunctorandthediagonal arrowisan equivalence ofcategories:

T

T/add Σ^dT.

D

(−)

T(T ,−)

∼

ThecategoryDisad-abeliancategoryby[15,thm.3.16].Ithasad-Auslander–Reiten translationτ_d,whichisahigheranalogueoftheclassicAuslander–Reitentranslationτ, see[12,sec. 1.4.1].AmoduleM ∈D iscalledτ_d-rigid ifHom_Γ(M,τ_dM)= 0.

Remark0.5.Theclassicadd-proj-correspondenceholds,as_T(T,−) restrictstoanequiv- alenceaddT →proj Γ.ThefunctoralsorestrictstoanequivalenceaddST →inj Γ.[14, lem.2.1]

It isnaturalto ask if D permitsahigheranalogueof theτ-tiltingtheory of [1]. We willnotanswerthisquestion,butwillinsteadintroducethefollowingdefinitionsinspired byit.

Definition0.6.A pair(M,P) withM ∈D and P ∈proj Γ iscalledaτ_d-rigid pairin _D ifM isτd-rigidand HomΓ(P,M)= 0.

Definition 0.7.A pair(M,P) withM ∈D and P ∈proj Γ is called amaximal τd-rigid pairin D ifitsatisfies:

(i) IfN ∈D then

N∈addM ⇔

⎧⎪

⎨

⎪⎩

HomΓ(M, τdN) = 0, Hom_Γ(N, τ_dM) = 0, HomΓ(P, N) = 0.

(ii) IfQ∈proj Γ,then

Q∈addP ⇔Hom_Γ(Q, M) = 0.

A maximalτd-rigid pairisaτd-rigidpair.

Oursecond mainresultis:

Theorem B.If eachindecomposable objectof _T isd-rigid,thenthere isabijection isomorphism classes of

maximald-rigid objects in_T

→

isomorphism classes of maximal τ_d-rigid pairs in_D

.

We prove this inSection 3. If d = 1, then (M,P) is amaximal τ1-rigid pairif and only ifit isasupportτ-tiltingpairinthesense of[1,def.0.3(b)],see [1, def.0.3,prop.

2.3, andcor.2.13].HenceTheoremBisahigheranalogueofthebijection

isomorphism classes of cluster tilting object in_T

→

isomorphism classes of supportτ-tilting pairs in mod Γ

which exists by[1, thm. 4.1] when T is triangulated, i.e. inthe cased = 1. However, when d2, wedonot thinkofmaximal τd-rigid pairsas supportτd-tiltingpairs. The reasonisthatbyTheoremB,maximalτd-rigidpairsarelinkedtomaximald-rigidobjects inhigherangulated categories. Asremarkedabove,this classis typicallystrictly larger than theclass ofOppermann–Thomasclustertiltingobjectswhend2.

Note that[19] makes anapproachtohighersupporttiltingtheory.

Thispaperisorganisedasfollows:Section1provesTheoremA,Section2investigates thepreciserelationbetweenHom spacesin T and D,Section3provesTheoremB,and Section4givesanexample.

Setup0.8. Throughoutthepaperweusethefollowingnotation:

k: Analgebraically closedfield.

D: ThedualityfunctorHom_k(−,k).

T: Ak-linear,Hom-finite,(d+ 2)-angulatedcategorywithsplit idempotents.Weas- sume that _T is 2d-Calabi–Yau, that is _T(X,Y) ∼= DT(Y,Σ^2dX) naturally in X,Y ∈T.

Σ^d: Thed-suspensionfunctoronT.

T: AnOppermann–ThomasclustertiltingobjectinT.

(−): ThecanonicalfunctorT →T/add Σ^dT,whosetargetisthenaivequotientcate- goryofT modulothemorphismswhichfactorthroughanobjectinadd Σ^dT.

Γ: TheendomorphismringEnd_T(T).

νΓ: TheNakayamafunctoronmod Γ.

τd: Thed-Auslander–Reitentranslationonmod Γ.

D: Theessentialimageofthefunctor T(T,−):T →mod Γ.

1. ProofofTheoremA

Theorem1.1. LetX∈T be given.

(i) There are implications

X is Oppermann–Thomas cluster tilting

⇓

X isd-self-perpendicular

⇓

X is maximal d-rigid

⇓ X isd-rigid.

(ii) Ifeach indecomposableobjectin T isd-rigid, then

X isd-self-perpendicular ⇔X is maximal d-rigid.

Proof. (i), thefirst implication:Suppose X is Oppermann–Thomas cluster tilting. We mustprovetheequalityinDefinition0.2,andtheinclusion⊆isclear.Fortheinclusion

⊇, suppose Ext^d_T(X,Y) = 0. Then each morphism X₀ → Σ^dY with X₀ ∈ addX is zero.This applies in particular to the(d+ 2)-angleX_d → · · · → X₀ → Σ^dY →Σ^dX_d withX_i ∈addX,whichexists sinceX isOppermann–Thomascluster tilting.Butthen themorphism Σ^dY → Σ^dX_d is asplit monomorphism, and applying Σ^−d givesasplit monomorphismY →X_d provingY ∈addX.

(i), the second implication: Suppose that X is d-self-perpendicular. We must prove theequalityinDefinition0.3,andtheinclusion⊆isclear.Fortheinclusion⊇,suppose Ext^d_T(X⊕Y,X⊕Y)= 0.Theninparticular,Ext^d_T(X,Y)= 0,whenceY ∈addX.

(i),thethirdimplication:Thisisclear.

(ii):Supposethateachindecomposable objectinT isd-rigid.Because ofpart(i),it isenoughtoprovetheimplication ⇐in(ii), sosuppose thatX ismaximal d-rigid. We mustprovetheequalityinDefinition0.2,and⊆isclear.

Fortheinclusion⊇, observethat{Y ∈T |Ext^d_T(X,Y)= 0}isclosedunderdirect sumsand summands by additivityof Ext. Henceit is enoughto suppose that Y is an

indecomposable objectinthissetand proveY ∈addX.However,Ext^d_T(X,Y)= 0 im- pliesExt^d_T(Y,X)= 0 because T is2d-Calabi–Yau,andExt^d_T(Y,Y)= 0 byassumption.

Finally, X isd-rigidbypart(i),so Ext^d_T(X,X)= 0.Combining these equalitiesshows Ext^d_T(X⊕Y,X⊕Y)= 0, andY ∈addX follows. 2

Remark1.2.TheimplicationsinTheorem1.1(i)cannotbe reversedingeneral:

– An example of a d-self-perpendicular object X which is not Oppermann–Thomas cluster tilting is given in Section 4. In fact, the objects in the last three rows of Fig.4aresuchexamples.Theexamplewasoriginally givenin[21, p.1735].

– An example of a maximal d-rigid object which is not d-self-perpendicular can be obtainedbycombiningproposition 2.6andcorollary 2.7in[5].These resultsgivea maximal1-rigidobjectwhichisnotclustertilting,butinthetriangulatedsettingof [5],clustertiltingisequivalent to1-self-perpendicular,see[5,bottomofp.963].

– Finally,anexampleofad-rigidobjectwhichisnotmaximald-rigidisthezeroobject, assoonasT hasanon-zerod-rigidobject.

WeendthesectionbyobservingthatTheorem1.1(ii)canbeappliedtoanimportant class ofcategories.

Proposition 1.3. Let Λ be a d-representation finite algebra, OΛ the (d+ 2)-angulated cluster category associatedtoΛin [21,thm.5.2].Theneach X ∈OΛ satisfies

X isd-self-perpendicular ⇔X is maximald-rigid.

Proof. Each indecomposable in OΛ is d-rigid by [21, Lemma 5.41], so the equivalence follows fromTheorem 1.1(ii). 2

2. Adimensionformula forExt^d_T

Recall from Setup 0.8 that T is a fixed Oppermann–Thomas cluster tilting object in _T, and that _T is 2d-Calabi–Yau, that is, _T(X,Y) ∼= DT(Y,Σ^2dX) naturally in X,Y ∈T.

Lemma 2.1.There isanaturalisomorphism ν_ΓT(T, T)∼=T

T,Σ^2d(T) forT ∈addT.

Proof. Bythe2d-Calabi-Yau propertywehave T

T,Σ^2d(T) ∼= DT(T, T).

By[14,Lemma2.2(i)],

DT(T, T)∼= DHomΓ

T(T, T),T(T, T) = DHomΓ

T(T, T),Γ .

Finally,bydefinitionwehave DHom_Γ

T(T, T),Γ =ν_ΓT(T, T), see[2,def.III.2.8]. 2

Lemma2.2.If X ∈T has nonon-zerodirectsummandsinadd Σ^dT,thenthereexistsa (d+ 2)-angle

T_d→ · · · →T₀→X →Σ^dT_d

in T withthefollowingproperties:EachTiisinaddT,andapplyingthefunctor T(T,−) gives acomplex

T(T, T_d)→ · · · →T(T, T₀)→T(T, X)→0

whichisthestart oftheaugmented minimalprojectiveresolutionof T(T,X).

Proof. Given X,there existsa(d+ 2)-angle

Σ^−dX→T_d → · · · →T₀→X

with eachTi inaddT byDefinition 0.1. Since X hasno non-zero directsummands in add Σ^dT, the first morphism in the (d+ 2)-angle is in the radical of T. Bydropping trivialsummandsoftheformT ^∼−→⁼ T,wecanassumethatsoaretheothermorphisms exceptthelastmorphism.

By[8,prop.2.5(a)],applyingthefunctor T(T,−) givesanexactsequence T(T,Σ^−dX)→T(T, T_d)→ · · · →T(T, T₀)→T(T, X)→T(T,Σ^dT_d) = 0.

ByTheorem 0.4, applying the functor T(T,−) is,up to isomorphism, justto apply a quotient functor, and this preserves radical morphisms. So in the exactsequence each morphism,exceptpossibly T(T,T0)→T(T,X),isintheradicalofmod Γ.Thisproves theclaimofthelemma. 2

Lemma 2.3. If X ∈ T has no non-zero direct summands in add Σ^dT, then there is a naturalisomorphism

τdT(T, X)∼=T(T,Σ^dX).

Proof. As X has no non-zero directsummands inadd Σ^dT, we can consider the (d+ 2)-angle from Lemma 2.2. Apply T(T,−) to get the following part of an augmented minimal projectiveresolutioninmod Γ:

T(T, Td)→ · · · →T(T, T0)→T(T, X)→0.

UsingtheNakayamafunctorandLemma2.1wegetthefollowingcommutativediagram.

0 τdT(T, X) νΓT(T, Td) · · · νΓT(T, T0)

0 T(T,Σ^dX) T(T,Σ^2dTd) · · · T(T,Σ^2dT0)

∼ ∼

Thetopsequenceisexactbythedefinitionofτd,see[12,sec.1.4.1].Thebottomsequence isexactbecauseitisobtainedbyapplyingHom_T(T,−) toa(d+ 2)-anglein_T,see [8, prop. 2.5(a)].Thefirsttermofthebottomsequence isactually T(T,Σ^dT0),butthisis zero.Since wehaved≥1,thediagramimplies

τdT(T, X)∼=T(T,Σ^dX). 2

Wewrite [addT](X,Y)={f ∈T(X,Y)|f factors through an object of addT}. Lemma 2.4.There isanaturalisomorphism

D[addT](X, Y)∼= Hom_{T/add Σ}^d_T(Y ,Σ^2dX) forX,Y ∈T.

Proof. Picka(d+ 2)-anglein_T:

Td→. . .→T0→Y →Σ^dTd,

with Ti ∈addT.Use T(X,−) toobtainthemorphismΨ:T(X,T0)→T(X,Y).This isahomomorphismofk-vectorspaces,hencewecantalkabouttheimageofΨ.Wefirst note thatany morphismf intheimage ofΨ must factorthroughaddT.Now suppose f ∈T(X,Y) factorsthroughT ∈addT.Wehavethefollowingcommutativediagram, where thelowerrowisapartofthe(d+ 2)-angleabove:

· · · T0 Y Σ^dTd.

· · · T T 0

X

1T

f

Thedashedarrow existsbycompletingthecommutativesquareto amorphism of(d+ 2)-angles.Weconcludethatf ∈Im Ψ.Hence

Im Ψ = [addT](X, Y).

Wenowreturntothelongexactsequence

· · · →T(X, T0)−→^Ψ T(X, Y)→T(X,Σ^dTd)→ · · ·.

Usingthe dualityfunctorD andSerre dualitywe getthefollowing diagram withexact rows:

DT(X,Σ^dTd) DT(X, Y) DT(X, T0)

T(Σ^dTd,Σ^2dX) T(Y,Σ^2dX) T(T0,Σ^2dX)

DΨ

α β

∼ ∼ ∼

[add Σ^dT](Y,Σ^2dX) T(Y,Σ^2dX)/[add Σ^dT](Y,Σ^2dX)

α β

Analogous to the above discussion, the space [add Σ^dT](Y,Σ^2dX) is the image of the map α.Hence αisthe kernel ofβ and DΨ (by isomorphism).The morphismβ is by definitionthecokernel of α,and T(Y,Σ^2dX)/[add Σ^dT](Y,Σ^2dX) is thusthe image of DΨ.Thuswehave

D[addT](X, Y)∼= D Im Ψ∼= Im DΨ∼=T(Y,Σ^2dX)/[add Σ^dT](Y,Σ^2dX)

∼= Hom_T/add Σ^dT(Y ,Σ^2dX). 2

Lemma2.5. SupposeX,Y ∈T.Thenwe haveashortexact sequence

0→DHom_{T/add Σ}^d_T(Y ,Σ^dX)→Ext^d_T(X, Y)→Hom_{T/add Σ}^d_T(X,Σ^dY)→0.

Proof. Bythedefinitionofthequotientfunctorwehaveashort exactsequence 0→[add Σ^dT](X,Σ^dY)→T(X,Σ^dY)→Hom_{T/add Σ}^d_T(X,Σ^dY)→0.

Wehave[add Σ^dT](X,Σ^dY)∼= [addT](Σ^−dX,Y).ByLemma2.4wehave

[addT](Σ^−dX, Y)∼= DHom_{T/add Σ}^d_T(Y ,Σ^2dΣ^−dX)∼= DHom_{T/add Σ}^d_T(Y ,Σ^dX).

Wealsoknow that_T(X,Σ^dY)∼= Ext^d_T(X,Y),so theconclusionfollows. 2

Lemma 2.6. Suppose X,Y ∈ T have no non-zero direct summands in add Σ^dT. Then we haveashortexact sequence

0→DHom_Γ

T(T, Y), τ_dT(T, X) →Ext^d_T(X, Y)

→Hom_Γ

T(T, X), τ_dT(T, Y) →0.

Proof. Considerthe short exactsequence from Lemma 2.5. ByTheorem 0.4 we know that

DHom_{T/add Σ}^d_T(Y ,Σ^dX)∼= DHomΓ

T(T, Y),_T(T,Σ^dX) .

Applying Lemma2.3wehave DHomΓ

T(T, Y),T(T,Σ^dX) ∼= DHomΓ

T(T, Y), τdT(T, X) .

SimilarlywecanshowHom_{T/add Σ}dT(X,Σ^dY)∼= HomΓ

T(T,X),τ_dT(T,Y) . 2

Themap definednextwilleventuallyinducetheequivalence ofTheoremB.

Definition 2.7. Foreach X ∈T, pickanisomorphism X ∼=X⊕X suchthatX has nonon-zerodirectsummandsinadd Σ^dT andX∈add Σ^dT.Let

Δ(X) =

T(T, X),T(T,Σ^−dX) .

This isapairofΓ-moduleswhere T(T,X) isinD andT(T,Σ^−dX) isinproj Γ.

Proposition 2.8. Given X,Y ∈T,set(M,P)= Δ(X) and(N,Q)= Δ(Y),where Δ is themapin Definition 2.7.Then

dimkExt^d_T(X, Y) = dimkHomΓ(M, τdN) + dimkHomΓ(N, τdM) + dimkHomΓ(P, N) + dimkHomΓ(Q, M).

Proof. ByadditivityofExt we have Ext^d_T(X, Y)∼= Ext^d_T(X⊕X, Y⊕Y)

∼= Ext^d_T(X, Y)⊕Ext^d_T(X, Y)⊕Ext^d_T(X, Y)⊕Ext^d_T(X, Y).

AsT isd-rigid,weseethatExt^d_T(X,Y)= 0,and hencewehave

dim Ext^d_T(X, Y) = dim Ext^d_T(X, Y) + dim Ext^d_T(X, Y) + dim Ext^d_T(X, Y). (2.1) FromLemma2.6wehavetheshort exactsequence:

0→DHom_Γ

T(T, Y), τ_dT(T, X) →Ext^d_T(X, Y)

→Hom_Γ

T(T, X), τ_dT(T, Y) →0, whichmeansthat

dim Ext^d_T(X, Y) = dim_kHom_Γ

T(T, X), τ_dT(T, Y) + dim_kHom_Γ

T(T, Y), τ_dT(T, X)

= dimkHomΓ(M, τdN) + dimkHomΓ(N, τdM). (2.2) Weseethat

Ext^d_T(X, Y)∼=T(X,Σ^dY)∼=T(Σ^−dX, Y)∼= HomΓ

T(T,Σ^−dX),T(T, Y)

∼= HomΓ(P, N).

Thethirdisomorphismfollowsfrom[14,Lemma2.2(i)] andthefactthatΣ^−dX∈addT. Similarly,

Ext^d_T(X, Y)∼= DExt^d_T(Y, X)∼= DHomΓ(Q, M).

Thuswehave

dim Ext^d_T(X, Y) = dim_kHom_Γ(P, N) (2.3) dim Ext^d_T(X, Y) = dim_kHom_Γ(Q, M). (2.4) Substituting(2.2),(2.3),and(2.4) into(2.1) givestheresult. 2

Asaconsequencewehave:

Corollary2.9. Given X,Y ∈T,set(M,P)= Δ(X)and(N,Q)= Δ(Y).Then Ext^d_T(X, Y) = 0⇔

HomΓ(M, τdN) = HomΓ(N, τdM) = HomΓ(P, N) = HomΓ(Q, M) = 0.

3. ProofofTheorem B

Thefollowing resultsusethemapΔ fromDefinition2.7.

Lemma3.1.GivenX,Y ∈T,set(M,P)= Δ(X)and(N,Q)= Δ(Y).ThenY ∈addX if andonly ifN ∈addM andQ∈addP.

Proof. Let X ∼=X⊕X be thedecomposition from Definition2.7, where X has no non-zero directsummands from add Σ^dT while X is in add Σ^dT. We have (M,P) = T(T,X),T(T,Σ^−dX) .Similarly,(N,Q)=

T(T,Y),T(T,Σ^−dY) .

The condition Q ∈addP is equivalent to Y ∈ addX by theadd-proj-correspon- dence, (see Remark 0.5). The condition N ∈ addM is equivalent to Y ∈ addX by Theorem 0.4becauseX,Y havenonon-zerodirectsummandsinadd Σ^dT. Theresult follows. 2

Lemma 3.2.The category _T is skeletallysmall.The mapΔ inducesabijection

δ: isoT →isoD×iso proj Γ, (3.1) where isodenotesthesetof isomorphismclassesof askeletally smallcategory.

Proof. Let Iso denote the class of isomorphisms of a category. For a skeletally small category _C wehavethatIso_C = iso_C.Notethatsinceamodulecategoryoveraringis skeletally small,wehavethatD,proj Γ⊆mod Γ areskeletally small.

It isclearthatΔ inducesawell-definedmapoftheform δ : Iso_T →iso_D×iso proj Γ.

To see thatδ is injective,arguelike the proofof Lemma3.1,replacing membership of add withisomorphism.

ItfollowsthatT isskeletallysmall.Wecanthusreplaceδwiththemapδfrom(3.1).

To see thatδ is surjective, let (M,P) be a pair with M ∈ D and P ∈ proj Γ. By Theorem 0.4 thereis anobjectX ∈T withno non-zerodirectsummandsinadd Σ^dT such that M ∼= T(T,X). By the add-proj correspondence, see Remark 0.5, there is an object X ∈ add Σ^dT such that P ∼= T(T,Σ^−dX). Setting X = X ⊕X gives (M,P)∼= Δ(X). 2

Lemma3.3.IfX ∈T isd-self-perpendicular,then(M,P)= Δ(X)isamaximalτ_d-rigid pair.

Proof. LetN ∈D and Q∈proj Γ begiven. ByLemma3.2, thereis anobject Y ∈T suchthat(N,Q)∼= Δ(Y).Then

N ∈addM andQ∈addP

⇔Y ∈addX

⇔Ext^d_T(X, Y) = 0

⇔Hom_Γ(M, τ_dN) = Hom_Γ(N, τ_dM) = Hom_Γ(P, N) = Hom_Γ(Q, M) = 0, wheretheequivalences,respectively,arebyLemma3.1,Definition0.2,andCorollary2.9.

TheconditionsofDefinition0.7arerecoveredbysettingQ= 0 respectivelyN = 0. 2 Lemma3.4. LetX ∈T begiven. If(M,P)= Δ(X)isamaximal τ_d-rigid pair, thenX isd-self-perpendicular.

Proof. LetY ∈T begiven andset(N,Q)∼= Δ(Y).Then Ext^d_T(X, Y) = 0

⇔HomΓ(M, τdN) = HomΓ(N, τdM) = HomΓ(P, N) = HomΓ(Q, M) = 0

⇔N∈addM and Q∈addP

⇔Y ∈addX,

wheretheequivalences,respectively,arebyCorollary2.9,Definition0.7,andLemma3.1.

2 Theorem3.5.RecallthatthemapΔfromDefinition2.7inducesthebijectionδ: isoT → isoD×iso proj Γfrom Lemma3.2.

(i) δ restrictsto abijection

isomorphism classes of d-rigid objects inT

→

isomorphism classes of τd-rigid pairs in D

.

(ii) δ restrictsfurther toabijection

isomorphism classes of d-self-perpendicular objects inT

→

isomorphism classes of maximalτd-rigid pairs inD

.

Proof. (i):ConsiderX ∈T andset (M,P)= Δ(X).Then

Ext^d_T(X, X) = 0⇔HomΓ(M, τdM) = 0 and HomΓ(P, M) = 0 byCorollary2.9,so theresultfollows.

(ii):SeeLemmas3.3and3.4. 2

Proof of TheoremB(from the introduction). CombineTheorems3.5(ii)and1.1(ii). 2

1357 1358 1368

1468

2468

2469

2479

2579

3579

Fig. 1.The AR quiver of the 5-angulated categoryT.

4. Anexample

In this section we letd= 3 and T =OA³₂. This isthe 5-angulated (higher)cluster categoryoftypeA2,see[21,def.5.2,sec.6,andsec.8].Theindecomposableobjectscan be identifiedwith theelements oftheset

I³₉={1357,1358,1368,1468,2468,2469,2479,2579,3579},

see[21,sec.8].TheARquiverof_T isshowninFig.1.By[21,thm.5.5andsec.8],the object

T = 1357⊕1358⊕1368⊕1468 is Oppermann–Thomasclustertilting.

IfX,Y ∈T areindecomposableobjects,then T(X, Y) =

k ifY isX or its immediate successor in the AR quiver, 0 otherwise,

see [21,prop.6.1and def.6.9].ItfollowsthatΓ= End_T(T)=kQ/I,where Q= 1→2→3→4

andIistheidealgeneratedbyallcompositionsoftwoconsecutivearrows.Theactionof thefunctor_T(T,−):_T →mod Γ on indecomposableobjectsisshowninFig.2,where P(q) andI(q) denotetheindecomposableprojectiveandinjectivemodulesassociatedto thevertex q∈Q.Note thattheessentialimageof T(T,−) is

X 1357 1358 1368 1468 2468 2469 2479 2579 3579

T(T , X) P(4) P(3) P(2) P(1) I(1) 0 0 0 0

Fig. 2.The action of the functorT(T ,−) :T →mod Γ.

X

◦

◦

◦

Y1

Y2

◦

◦

◦

Fig. 3.The functor Ext³_T(X,−) is non-zero onY1andY2. It is zero on every other indecomposable object.

Maximal 3-rigid objectX Maximalτ3-rigid pair Δ(X) 1357⊕1358⊕1368⊕1468 (Γ,0)

1358⊕1368⊕1468⊕2468 (DΓ,0) 1368⊕1468⊕2468⊕2469

P(2)⊕P(1)⊕I(1), P(4) 1468⊕2468⊕2469⊕2479

P(1)⊕I(1), P(4)⊕P(3) 2468⊕2469⊕2479⊕2579

I(1), P(4)⊕P(3)⊕P(2) 2469⊕2479⊕2579⊕3579 (0,Γ)

2479⊕2579⊕3579⊕1357

P(4), P(3)⊕P(2)⊕P(1) 2579⊕3579⊕1357⊕1358

P(4)⊕P(3), P(2)⊕P(1) 3579⊕1357⊕1358⊕1368

P(4)⊕P(3)⊕P(2), P(1) 1357⊕1468⊕2479

P(4)⊕P(1), P(3) 1358⊕2468⊕2579

P(3)⊕I(1), P(2) 1368⊕2469⊕3579

P(2), P(4)⊕P(1)

Fig. 4.Theseareallthebasicmaximal3-rigidobjectsofT andtheircorrespondingmaximalτ3-rigidpairs inD.

D= add{P(4), P(3), P(2), P(1), I(1)}.

Thisisa3-clustertiltingsubcategoryofmod Γ andhenceitis3-abelian.

The3-suspension functorΣ³ acts ontheAR quiver bymovingfour stepsclockwise.

Combined withour knowledgeofHom, this shows thatifX is afixed indecomposable objectin_T,thentheindecomposableobjectsY withExt³_T(X,Y)= 0 arepreciselythe twoobjectsfurthest fromX intheARquiver,seeFig.3.

Based on this, we can compute all basic 3-self-perpendicular objects in T, and by Proposition 1.3 they coincide with the basic maximal 3-rigid objects in T. For each suchobject X,there is amaximal τ3-rigid pairΔ(X)=

T(T,X),T(T,Σ⁻³X) by TheoremB.SeeFig.4.NotethatthefirstnineobjectsinFig.4areOppermann–Thomas clustertilting,butthethreelast objectsarenot.

Acknowledgment

This work was supported by EPSRC grant EP/P016014/1 “Higher Dimensional Homological Algebra”. Karin M. Jacobsen is grateful for the hospitality of Newcastle University duringhervisitinOctober2018.

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