The D -standard and K -standard categories Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

The D-standard and K-standard categories

Xiao-Wu Chen,Yu Ye^∗

KeyLaboratoryof WuWen-TsunMathematics,ChineseAcademyofSciences, SchoolofMathematicalSciences,UniversityofScienceandTechnologyofChina, No.96JinzhaiRoad,Hefei230026,Anhui,PRChina

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

Articlehistory:

Received8October2017 Accepted17May2018 Availableonlinexxxx

CommunicatedbyHenningKrause

MSC:

18E30 16G10 18G35 16E05

Keywords:

Standardequivalence Trianglecenter Trianglefunctor Derivedequivalence

Weintroduce thenotions ofa D-standard abeliancategory andaK-standardadditivecategory.Weprovethatforaﬁnite dimensionalalgebra A,itsmodulecategoryisD-standardif andonlyifanyderivedautoequivalenceonAisstandard,that is, isomorphic to thederived tensor functor by a two-sided tiltingcomplex.Weprovethatifthesubcategoryofprojective A-modules is K-standard, then the module category is D-standard.WeprovidenewexamplesofD-standardmodule categories.

Contents

1. Introduction . . . . 160

2. Trianglefunctorsandcenters . . . . 161

2.1. Trianglefunctorsandextendingnaturaltransformations . . . . 161

2.2. Almostvanishingmorphismsandcenters . . . . 164

3. Pseudo-identitiesandcenters . . . . 166

* Correspondingauthor.

E-mailaddresses:xwchen@mail.ustc.edu.cn(X.-W. Chen),yeyu@ustc.edu.cn(Y. Ye).

URLs:http://home.ustc.edu.cn/~xwchen(X.-W. Chen),http://staﬀ.ustc.edu.cn/~yeyu(Y. Ye).

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3.1. Pseudo-identitiesonboundedhomotopycategories . . . . 166

3.2. Pseudo-identitiesonboundedderivedcategories . . . . 170

3.3. Comparingcenters . . . . 171

4. K-standardadditivecategories . . . . 173

5. D-standardabeliancategoriesandstandardequivalences . . . . 176

5.1. D-standardabeliancategories . . . . 176

5.2. Standardequivalences . . . . 179

6. K-standardnessversusD-standardness . . . . 182

7. Twoexamples . . . . 186

7.1. Thedualnumbers . . . . 187

7.2. Anotherexample . . . . 190

Acknowledgments . . . . 193

References . . . . 193

1. Introduction

Letkbeafield,andletAbeafinitedimensionalk-algebra.WedenotebyA-mod the category of finite dimensional left A-modules and by D^b(A-mod) its bounded derived category. Byaderivedequivalence betweentwo algebras A and B,we meanak-linear triangleequivalenceF:D^b(A-mod)→D^b(B-mod).Itisanopenquestionin [19] whether any derivedequivalenceisstandard,thatis,isomorphictothederivedtensorfunctorby atwo-sidedtiltingcomplex.

We mention thatthere exists a triangle functor betweenthe bounded derived cate- goriesofmodulecategories,whichisnotisomorphictothederivedtensorfunctorbyany complexofbimodules;see [20].

The above open question is answered aﬃrmatively in the following cases: (1) A is hereditary [14]; (2)A is(anti-)Fano [13];(3) Ais triangular [6].Wemention thattheir proofs relyonthework [16] and [1].

In thepresent paper,we takeadiﬀerent approach.Recall from [19] that foragiven derivedequivalenceF,thereisastandardequivalenceFsuchthattheyactidenticallyon objects.Thismotivatesthefollowingnotion:atriangleautoequivalenceGonD^b(A-mod) is called a pseudo-identity, provided that G acts on objects by the identity and that the restriction of G to stalk complexes equals the identity functor. Roughly speaking, a pseudo-identityisverycloseto thegenuineidentityfunctoronD^b(A-mod).Thenany derivedequivalenceF:D^b(A-mod)→D^b(B-mod) allows afactorization F FGwith G a pseudo-identity on D^b(A-mod) and F a standard equivalence; moreover, such a factorization isunique;seeProposition5.8.

We say that the module category A-mod is D-standard if any pseudo-identity on D^b(A-mod) is isomorphic to the identity functor as triangle functors. We prove that A-mod isD-standardifandonlyifanyderivedequivalencefromD^b(A-mod) isstandard;

seeTheorem5.10.Therefore,theopenquestionisequivalenttothefollowingconjecture:

any modulecategoryA-mod isD-standard.

This notion of D-standardness applies to any k-linear abelian category. It is well knownthatforthecategoriesofcoherentsheavesonsmoothprojectivevarieties,derived

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equivalences of Fourier–Mukai type are geometric analogue to standard equivalences.

By [16], any k-linear triangle equivalence between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier–Mukai type. Indeed, the proof therein implies that the abelian category of coherent sheaves on any projective varietyisD-standard;compareProposition5.7.Inacertainsense,thisfactsupportsthe aboveconjecture.

Analogouslytotheaboveconsideration,weintroducethenotionofaK-standard ad- ditivecategory,whereapseudo-identityontheboundedhomotopycategoryisinvolved.

We prove that if the category of projective A-modules is K-standard, then A-mod is D-standard;seeTheorem6.1.Thisseemstoshednewlightontheaboveconjecture.

Thepaperisstructuredasfollows.InSection2,werecallbasicfactsontrianglefunc- torsandcenters.Thenotionsofapseudo-identityontheboundedhomotopycategoryof anadditivecategoryandontheboundedderivedcategoryofanabeliancategoryarein- troducedinSection3.InSection4,weintroducethenotionofa(strongly)K-standard additive category, and observe that an Orlov category [1] is strongly K-standard; see Proposition4.6.Analogously,wehavethenotionofa(strongly)D-standardabeliancat- egoryinSection5,whereweobservethatanabeliancategorywithanamplesequence [16]

of objects is strongly D-standard; see Proposition5.7. Weprove Theorem 5.10, which relatestheaboveopenquestiontotheD-standardness.

InSection6,weproveTheorem6.1,whichclaimsthatanabeliancategorywithenough projectivesisD-standardprovidedthatthefullsubcategoryofprojectivesisK-standard.

Inthe ﬁnalsection, we provide two examplesofalgebras, whosemodule categories are D-standard.Inparticular,thealgebraofdual numbersprovides aD-standard,butnot stronglyD-standard,modulecategory;see Theorem7.1.

Throughout thepaper,we assumethatthecategories inconsiderationareskeletally small.Weassumetheaxiomofglobal choice,thatis,theaxiomof choiceforclasses.

2. Trianglefunctorsandcenters

Inthissection,werecallbasicfactsontrianglefunctorsandthecentersoftriangulated categories.

2.1. Triangle functorsandextendingnatural transformations

LetT andTbetriangulatedcategories,whosetranslationfunctorsaredenotedbyΣ andΣ,respectively.Recallthatatrianglefunctor(F,ω) consistsofanadditivefunctor F: T → T and a natural isomorphism ω: FΣ → ΣF such that any exact triangle X →Y →Z →^h Σ(X) inT issenttoanexacttriangleF(X)→F(Y)→F(Z)−−−−−−→^ω^X^◦F(h) ΣF(X) inT.

Thenatural isomorphismω is called the connecting isomorphism forF. When ω is understoodornotimportantinthecontext,wesuppress itand writeF forthetriangle functor (F,ω). Theconnecting isomorphismω is trivial ifFΣ= ΣF and ω = IdFΣ is

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theidentitytransformation.Forexample,theidentityfunctorId_T,asatrianglefunctor, is understoodasthepair(Id_T,IdΣ),whichhasthetrivialconnectingisomorphism.

Foratrianglefunctor(F,ω),wedeﬁnenaturalisomorphismsωⁿ: FΣⁿ →ΣⁿF forall n≥1 asfollows:ω¹=ωand ωⁿ⁺¹= Σⁿω◦ωⁿΣ forn≥1.Weobserveωⁿ⁺¹= Σωⁿ◦ ωΣⁿ.IfbothΣ andΣ areautomorphismsofcategories,wedeﬁnenaturalisomorphisms ω⁻ⁿ:FΣ⁻ⁿ → Σ⁻ⁿF as follows: ω⁻¹ = (Σ⁻¹ω¹Σ⁻¹)⁻¹ and ω⁻ⁿ⁻¹ = Σ⁻ⁿω⁻¹ ◦ ω⁻ⁿΣ⁻¹ forn≥1.Byconvention,ω⁰= Id_F.

For twotriangle functors(F,ω) and (F,ω) fromT to T, a naturaltransformation η: (F,ω)→(F,ω) between trianglefunctors meansanaturaltransformation η:F → F satisfyingω◦ηΣ= Ση◦ω.Thecompositionoftwotrianglefunctors(F,ω) :T → T and (G,γ) :T → T isgiven by(GF,γF ◦Gω) :T → T, whichisoften denotedjust byGF.

Thefollowing factiswell known.

Lemma 2.1.Let η: (F,ω)→ (F,ω) be thenatural transformation as above. Then the full subcategoryIso(η)={X ∈ T |ηX is an isomorphism}ofT isatriangulatedsubcat- egory. 2

WesaythatafullsubcategoryS ofT isgenerating providedthatthesmallesttrian- gulated subcategory containingS is T itself. Thefollowing well-knownresultis known as Beilinson’sLemma;see[9, II.3.4].

Lemma 2.2.Let F: T → T be atriangle functor. Assume that S ⊆ T isa generating subcategory. ThenF isfullyfaithful ifandonly ifF inducesisomorphisms

Hom_T(X,Σⁿ(Y))−→Hom_T(F X, FΣⁿ(Y))

for allX,Y ∈ S andn∈Z. In this case, F is denseif and only if the essentialimage ImF containsagenerating subcategory ofT. 2

LetF:C → Dbe afunctor. Foreachobject CinC,wechooseanobjectF(C) inD and anisomorphismδC:F(C)→F(C).Here,weareusing theaxiomofchoiceforthe classofobjectsinC.WecallthesechosenisomorphismsδC’stheadjustingisomorphisms.

Indeed, the choice makes F into a functor such that δ is a natural isomorphism between F and F. The action of F on a morphism f: C → C is given by F(f) = δ_C◦F(f)◦δ_C⁻¹.ItfollowsthatFpreservestheidentitymorphismandthecomposition of morphisms,thatis,itisindeedafunctor. Bythevery constructionofF,weobserve the naturalityofδ.Inacertainsense,thenew functorF: C → Disadjusted from the given functorF.

Bythefollowingwell-knownlemma, wemightalsoadjusttrianglefunctors.

Lemma 2.3. Let (F,ω) :T → T be a triangle functor. Assume that F:T → T is another functor with a natural isomorphism δ: F → F. Then there is a unique iso-

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morphism ω: FΣ → ΣF such that (F,ω) is a triangle functor and that δ is an isomorphismbetweentrianglefunctors.

Proof. Takeω = Σδ◦ω◦(δΣ)⁻¹.Thestatementsaredirecttoverify. 2

Thefollowingstandardfactwillbeusedlater.Weprovideafullproofforcompleteness.

Lemma 2.4.Let F,G:A → B be two additive functors betweenadditive categories. As- sume that C ⊆ A is a full subcategory such that any object in A is isomorphic to a ﬁnite direct sum of objects in C. Let η: F|C → G|C be a natural transformation. Then there is a unique natural transformation η: F → G extending η. Moreover, if η is an isomorphism,soisη.

Proof. For eachobject A inA, wechoose anisomorphism ξA:A → ⊕i∈ICi with each C_i∈ C andIaﬁniteset.Wemakethechoicesuchthatξ_C istheidentity morphismfor eachobjectCinC.Here,we usetheaxiomofchoicefortheclass ofobjectsinA.

We deﬁne η_A :F(A) →G(A) to be G(ξA)⁻¹◦(⊕i∈IηCi)◦F(ξA). Here, we identify F(⊕i∈ICi) with ⊕i∈IF(Ci), G(⊕i∈ICi) with ⊕i∈IG(Ci). In particular, η_C = ηC for C∈ C byourchoice.

We claim that the morphism η_A is natural in A. For this, we take an arbitrary morphism f: A → A in A. For the object A, we have the chosen isomorphism ξA: A → ⊕j∈JC_j with C_j ∈ C and J is a ﬁnite set. Then we have the following commutativediagram

A

f

ξA

i∈ICi

(fji)_(i,j)∈I×J

A ^ξ^A

j∈JC_j

Here,eachf_ji:C_i→C_j isamorphisminC.Inthefollowingcommutativediagram,the middlesquareusesthenaturalityofη.

F(A)

F(f)

F(ξA)

i∈IF(Ci)

(F(fji))

i∈Iη_Ci

i∈IG(Ci)

(G(fji)) G(ξA)⁻¹

G(A)

G(f)

F(A)^F(ξ^A⁾

j∈JF(C_j)

j∈JηC

j

j∈JG(C_j)^G(ξ^A⁾

−1

G(A) Theoutercommutativediagramprovesthatη_A◦F(f)=G(f)◦η_A,as required.

Wemention thattheabove argumentactually proves thatη_A is independentof the choiceof theisomorphismξA, bytaking A=Aand f = IdA. 2

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Let kbe acommutativering. LetA beak-linearadditivecategory. Foraset Mof morphisms in A, we denote by obj(M) the full subcategory formed by those objects, which are either the domain or the codomain of a morphism in M. We say that M linearly spans A providedthat eachmorphism in obj(M) is ak-linear combination of theidentitymorphismsandcompositionofmorphismsfromM,andthateachobjectin Ais isomorphictoaﬁnite directsumofobjectsinobj(M).

Lemma 2.5. LetM bea spanning set of morphismsin A.Assumethat F:A→ Ais a k-linearendofunctorsuchthatF(f)=f foreachf ∈ M.Thenthereisauniquenatural isomorphism θ:F →Id_A satisfyingθ_S = Id_S foreach objectS fromobj(M).

Proof. TheassumptionimpliesthatF(S)=SforanyobjectS fromobj(M).Moreover, therestrictionofF onobj(M) istheidentityfunctor,sinceitactsonmorphismsbythe identity. ApplyingLemma2.4 toF andId_A,wearedone. 2

2.2. Almost vanishingmorphismsandcenters

Throughout this subsection, k will be a ﬁeld and T will be a k-linear triangulated category, whichisHom-ﬁniteand Krull–Schmidt.

Following [11,Deﬁnition 2.1],a nonzeromorphismw:Z →XinT isalmostvanishing providedthatf◦w= 0 andw◦g= 0 foranynon-sectionf:X →Aandnon-retraction g:B→Z.ThishappensifandonlyifwﬁtsintoanalmostsplittriangleΣ⁻¹X →E→ Z →^w X;see [9,I.4.1].Inparticular,both Z andX areindecomposable.

Proposition 2.6. Assumethat X →^f Y →^g Z →^h ΣX isan exact trianglein T with g = 0 andh= 0suchthatEnd_T(Z)eitherequalskorkIdZ⊕kΔ,wherethemorphismΔ :Z→ Z isalmostvanishing.Thenforanonzeroscalarλ,thetriangleX →^f Y →^g Z−−→^λh Σ(X) is exactif andonly ifλ= 1.

Proof. We observe that g is a non-retraction, otherwise h = 0. Similarly, h is a non- section.Assumethatthegiventriangleisexact.Thenwehaveanisomorphismξ:Z →Z makingthefollowing diagramcommute.

X ^f Y ^g Z

ξ

h Σ(X)

X ^f Y ^g Z ^λh Σ(X)

If End_T(Z)=k,we assumethat ξ=μId_Z forsome μ∈k. It follows from themiddle square thatμ= 1, andthusλ= 1 fromtherightsquare.

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Inthesecondcase,weassumethatξ=μIdZ+γΔ forsomeμ,γ∈k.Bythemiddle squareandthefactthatΔ◦g= 0,wehaveμ= 1.Bytherightsquareandthefactthat h◦Δ= 0,weinferthatλ= 1. 2

We denote by indT a complete set of representatives of isomorphism classes of indecomposable objects in T. Here, indT is indeed a set, since T is skeletally small.

Denote by Λ the subset consisting of these objects X with an almost vanishing mor- phismΔ_X:X →X suchthatΔ_X iscentralinEnd_T(X).

Thefollowingisavariantof [11,Lemma 2.2];compare [22,Remark 4.15].

Lemma2.7. For each X ∈Λ,we associate ascalarλX.Thenthere is auniquenatural isomorphismη: Id_T →Id_T such that ηX = IdX+λXΔX forX ∈Λ and ηY = IdY for Y ∈indT \Λ.

Proof. WeviewindT asafullsubcategoryofT.ByLemma2.4,itsuﬃcestoverifythat therestriction ofthe isomorphismη onindT isnatural. Forthis, we takeanarbitrary morphismf:Y →Z with Y,Z∈indT.WeclaimthatηZ◦f =f◦ηY.

IfneitherY norZ liesinΛ,theclaimisclear.IfY liesinΛ andZ doesnotliein Λ, wehavef◦ΔY = 0,sincef isanon-sectionandΔY isalmostvanishing.Thentheclaim follows.ThesameargumentworksforthecaseY ∈Λ andZ /∈Λ.

Fortherest, wemayassume thatboth Y and Z lieinΛ.IfY =Z,then theclaims follows,sinceΔ_Y andthusη_Y arecentralinEnd_T(Y).IfY =Z,wehaveΔ_Z◦f = 0= f◦Δ_Y.Thisimpliestheclaim.Wearedone. 2

LetAbe ak-linearadditivecategory. Wedenote byZ(A) thecenter of A,which is bydeﬁnitionthe set ofnaturaltransformations λ: Id_A →Id_A.To ensurethatZ(A) is indeed aset, we use the assumption that A is skeletally small. Then Z(A) is a com- mutativek-algebra,whoseaddition andmultiplicationareinducedbytheaddition and compositionofnaturaltransformations, respectively.

WedenotebyZ(T) thetrianglecenter ofT,whichisthesetofnaturaltransforma- tionsλ: Id_T →Id_T betweentrianglefunctors, equivalently,the naturaltransformation λsatisﬁes λΣ= Σλ.Then Z(T) isa subalgebraof Z(T). Wemention thatZ(T) is thezerothcomponentofthegradedcenterofT;compare [11,12].

Thefollowingobservationwillbeuseful.

Lemma2.8. Let(F,ω) :T → T beatriangleautoequivalence.Thenanynaturaltransfor- mation(F,ω)→(F,ω)oftrianglefunctorsisof theformF λ forauniquelydetermined λ∈Z(T). 2

Following [8, Section 4],wesaythatT isablock, providedthatT doesnot admita decompositionintotheproductoftwo nonzerotriangulatedsubcategories. Moreover,it is non-degenerate ifthere is anonzero non-invertiblemorphism X →Y betweensome indecomposableobjectsX and Y.

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Proposition 2.9.LetT beanon-degenerateblock suchthatEnd_T(X)=kforeachinde- composable object X.Thenthefollowingstatements hold.

(1) Wehave Z(T)=k=Z(T).

(2) If (Id_T,ω)is atrianglefunctor,thenω= Id_Σ,theidentity transformationonΣ.

Proof. For (1), it suﬃces to show that any natural transformation η: Id_T → Id_T is given by a scalar. By assumption, η_X = λ_XId_X for each indecomposable object X and somescalar λ_X. In viewof Lemma 2.4, it suﬃces to show that λ_X =λ_Y for any indecomposablesX andY.

Weobserve thatλX =λY providedthatthereis anonzero mapX →Y orY →X, usingthenaturalityofη.SinceT isanon-degenerateblock,foranyindecomposablesX and Y,thereis asequenceX =X₀,X₁,· · ·,X_n =Y suchthatHom_T(X_i,X_i+1)= 0 or Hom_T(Xi+1,Xi)= 0; see [8, Proposition 4.2 and Remark 4.7].From this sequence we inferthatλX =λY.

For(2),weobservethatω= Σ(η) forauniqueη∈Z(T).By(1)wemayassumethat η =λ∈k.Takeanonzeronon-invertiblemorphismg:X→Y betweenindecomposables andformanexacttriangleZ →^f X →^g Y →^h Σ(Z).SinceX isindecomposable,weobserve that h= 0. Applyingthe trianglefunctor (Id_T,ω) to this triangle,we obtainan exact triangle

Z −→^f X −→^g Y −→^λh Σ(Z).

ByProposition2.6,weinferthatλ= 1. Thenwearedone. 2 3. Pseudo-identitiesandcenters

Inthissection,westudytriangleendofunctorsontheboundedhomotopycategoryof an additive category and on thebounded derived category of an abelian category. We introduce the notion of a pseudo-identityendofunctor on them. Their triangle centers are studied.

3.1. Pseudo-identitiesonboundedhomotopy categories

LetAbeanadditivecategory.WedenotebyK^b(A) thehomotopycategoryofbounded complexes inA.A boundedcomplexX isvisualizedas follows

· · · −→X^{n d}

n

−→X Xⁿ⁺¹^d

n+1

−→X Xⁿ⁺²−→ · · ·

where Xⁿ= 0 foronlyfinitelymanyn’sandthedifferentialssatisfydⁿ⁺¹_X ◦dⁿ_X= 0. The translation functor Σ oncomplexes is defined suchthat Σ(X)ⁿ =Xⁿ⁺¹ and dⁿ_Σ(X) =

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−dⁿ⁺¹_X .For achainmap f:X →Y, thetranslated chainmap Σ(f) : Σ(X)→ Σ(Y) is givenbyΣ(f)ⁿ =fⁿ⁺¹ foreachn∈Z.

Anadditivefunctor G: A→ B induces atriangle functor K^b(G) :K^b(A)→K^b(B), which acts componentwise on complexes and whose connecting isomorphism is trivial. Similarly, a natural transformation η: G → G induces a natural transformation K^b(η) :K^b(G)→K^b(G) betweentrianglefunctors.

ForanobjectA inA,wedenote byA thecorresponding stalkcomplexconcentrated ondegreezero.In thisway, weviewA asafull subcategory ofK^b(A).ForA∈ A and n∈Z,thecorresponding stalkcomplexΣⁿ(A) isconcentratedondegree−n.

ForacomplexX and n∈Z,weconsider thebrutaltruncation σ_≥−nX =· · · →0→ X⁻ⁿ^d

−n

→X X¹⁻ⁿ→ · · ·,whichisasubcomplexofX.Thereisaprojectionπ_n:σ_≥−nX → Σⁿ(X⁻ⁿ),andthusanexacttriangleinK^b(A)

Σⁿ⁻¹(X⁻ⁿ)−→^f σ_≥1−nX −→ⁱⁿ σ_≥−nX −→^πⁿ Σⁿ(X⁻ⁿ), (3.1) where in is the inclusionmap and f is given bythe minus diﬀerential −d⁻_Xⁿ: X⁻ⁿ → X¹⁻ⁿ.Usingthesetriangles,oneobservesthatAisageneratingsubcategory ofK^b(A).

Lemma 3.1. LetF:K^b(A) →K^b(A) be atriangle functor satisfying F(A)⊆ A. Then thefollowingstatements hold.

(1) F isfullyfaithfulif andonlyif soistherestriction F|A:A→ A. (2) If therestriction F|A: A→ Aisan equivalence,so isF.

(3) Assume thatA has splitidempotents. IfF isan equivalence,so isF|A.

Proof. The “only if” part of (1) is trivial. For the “if” part, we observe that Hom_Kb(A)(X,Σⁿ(Y)) = 0 for X,Y ∈ A and n = 0. Since A is a generating subcat- egoryofK^b(A),weapplyLemma2.2toobtainthatF isfullyfaithful.

For(2),weobservethatifF|Aisanequivalence,theessentialimageImF containsA, a generatingsubcategory of K^b(A).Inviewof thesecond statementofLemma2.2, we inferthatF isdense.

For (3), we recall the following well-known observation: a bounded complex Y is isomorphic to some object in A if and only if Hom_K^b₍_A₎(Y,Σⁿ(A)) = 0 = Hom_K^b_(A)(Σⁿ(A),Y) foreachA∈ A andn= 0.

It suﬃces to prove that for any complex X, if F(X) is isomorphic to some object inA,so isX.ForeachA∈ Aandn= 0, wehave

Hom_Kb(A)(X,Σⁿ(A))Hom_Kb(A)(F(X), FΣⁿ(A)) Hom_K^b_(A)(F(X),Σⁿ(F A)) = 0,

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where the ﬁrst isomorphism uses thefully-faithfulness of F and the last equality uses thefactthatF(A)∈ A.Similarly,wehaveHom_K^b_(A)(Σⁿ(A),X)= 0.Thenwearedone bytheaboveobservation. 2

Thefollowingresultisanalogousto [18,Proposition 7.1],whereacompletelydiﬀerent argument isused.

Proposition 3.2. Let (F,ω) :K^b(A) → K^b(A) be a triangle autoequivalence satisfying F(A)⊆ A. Assume that therestriction F|A is isomorphic to theidentity functor Id_A. Then foreachcomplex X ∈K^b(A),F(X) isisomorphicto X.

Proof. Assume that φ: F|A → Id_A is the given isomorphism. Using the translation functor andtheconnectingisomorphismω,itsuﬃcesto provethestatementunderthe assumption thatXⁱ= 0 fori>0.

Weclaimthatforeachn≥0,there isanisomorphism an:F(σ_≥−nX)−→σ_≥−nX

satisfying πn◦an = Σⁿ(φ_X−n)◦ω_Xⁿ−n◦F(πn). Theclaim will be provedby induction onn.

We takea₀ =φ_X⁰. Weassumethattheisomorphisma_n−1is alreadygiven forsome n≥1.Considertheexacttriangle(3.1).We claimthattheleft squareinthefollowing diagram commutes.

FΣⁿ⁻¹(X⁻ⁿ) ^F(f)

Σⁿ⁻¹(φ_X−n)◦ωⁿ⁻¹_X−n

F(σ_≥1−nX)

a_n−1 F(in)

F(σ_≥−nX)

ω_Σn−1 (X−n)◦F(πn)

ΣFΣⁿ⁻¹(X⁻ⁿ)

Σⁿ(φ_X−n)◦Σ(ω_X−nⁿ⁻¹)

Σⁿ⁻¹(X⁻ⁿ) ^f σ_≥1−nX ⁱⁿ σ_≥−nX ^πⁿ Σⁿ(X⁻ⁿ) Indeed,thefollowing mapinducedbyπn−1:σ_≥1−nX →Σⁿ⁻¹(X¹⁻ⁿ) isinjective

Hom_Kb(A)(FΣⁿ⁻¹(X⁻ⁿ), σ_≥1−nX)−→Hom_Kb(A)(FΣⁿ⁻¹(X⁻ⁿ),Σⁿ⁻¹(X¹⁻ⁿ)).

Hence,fortheclaim,itsuﬃcesto prove

πn−1◦an−1◦F(f) =πn−1◦f◦Σⁿ⁻¹(φ_X−n)◦ω_Xⁿ⁻−n¹. Bytheinductionhypothesis,theﬁrstequality inthefollowingidentity holds:

π_n−1◦a_n−1◦F(f) = Σⁿ⁻¹(φ_X1−n)◦ωⁿ⁻¹_X_1−n◦F(π_n−1)◦F(f)

=−Σⁿ⁻¹(φ_X1−n)◦ω_Xⁿ⁻1−n¹ ◦FΣⁿ⁻¹(d⁻_Xⁿ)

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=−Σⁿ⁻¹(d⁻_Xⁿ)◦Σⁿ⁻¹(φ_X−n)◦ω_Xⁿ⁻−n¹

=πn−1◦f◦Σⁿ⁻¹(φ_X−n)◦ω_Xⁿ⁻¹₋n.

Here,the second and fourth equalities use the factthat πn−1◦f =−Σⁿ⁻¹(d⁻ⁿ_X ), and thethirdusesthenaturalityofωⁿ⁻¹and φ.

Thanks to the above diagram between exact triangles, the required isomorphism a_n:F(σ_≥−nX)→σ_≥−nX follows from theaxiom(TR3) inthe triangulatedstructure ofK^b(A). 2

Inspiredby theaboveresult, itseemsto be of interestto havethefollowing notion.

Foreach n∈Z,we denote byΣⁿ(A) the full subcategory ofK^b(A) consistingofstalk complexesconcentratedondegree−n.WeidentifyΣ⁰(A) withA.

Deﬁnition 3.3. A triangle functor (F,ω) :K^b(A) → K^b(A) is called a pseudo-identity, providedthatF(X)=X foreachboundedcomplexX andthatitsrestrictionF|Σⁿ(A)

tothesubcategoryΣⁿ(A) equals theidentity functoronΣⁿ(A),foreachn∈Z. 2 Thediﬀerencebetweenapseudo-identityandthegenuineidentity functoronK^b(A) liesintheiractiononmorphismsandtheirconnectingisomorphisms.

Corollary3.4.Let(F,ω) :K^b(A)→K^b(A)beatrianglefunctor.Then(F,ω)isisomor- phic to a pseudo-identity if and only if F is an autoequivalence satisfying F(A) ⊆ A suchthat therestrictionF|A isisomorphic totheidentityfunctor.

Proof. ByLemma3.1,a pseudo-identityisanautoequivalence.Thenwehavethe“only if”part.

Forthe“if” part,we assumethatγ:F|A→Id_A isthegivenisomorphism.Weapply Proposition 3.2 and choose for each complex X an isomorphism δX: F(X) → X = F(X) suchthatforeachobject A∈ A andn∈Z, δ_Σⁿ_(A):F(ΣⁿA)→F(ΣⁿA) equals Σⁿ(γ_A)◦ω_Aⁿ; here, we refer to Subsection 2.1 for the notation ωⁿ. Using δ_X’s as the adjustingisomorphismsandLemma2.3,weobtainapseudo-identity(F,ω) onK^b(A), whichisisomorphicto(F,ω) astrianglefunctors. 2

Lemma 3.5. Let (F,ω) :K^b(A) → K^b(A) be a pseudo-identity. Assume that (F,ω) is isomorphic totheidentity functorId_K^b_(A),astrianglefunctors.Thenthere isanatural isomorphism θ: (F,ω) → Id_K^b₍_A₎ of triangle functors, whose restriction to A is the identity transformation.

Proof. Takeanaturalisomorphismδ: (F,ω)→Id_Kb(A).Therestrictionofδto Aisan invertibleelement μinZ(A).Setθ=K^b(μ⁻¹)◦δ.Then wearedone. 2

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3.2. Pseudo-identitiesonboundedderived categories

Throughout this subsection, A is an abelian category. We denote by D^b(A) the bounded derived category. We identify A as the full subcategory of D^b(A) formed by stalk complexesconcentratedondegreezero.

An exact functor G: A → B between abelian categories induces a triangle functor D^b(G) : D^b(A) → D^b(B), which acts componentwise on complexes and has a trivial connecting isomorphism.Similarly,a natural transformation μ:G→G between exact functors induces a natural transformation D^b(μ) :D^b(G) → D^b(G) between triangle functors.

For abounded complex X and n ∈ Z, we denote by Hⁿ(X) the n-th cohomology.

We recallthegood truncations τ_≤n(X)=· · · →Xⁿ⁻²^d

n−2

X→ Xⁿ⁻¹ →Kerdⁿ_X →0→ · · · and τ_≥n(X) =· · · → 0→ Cokdⁿ_X⁻¹ → Xⁿ⁺¹ ^d

n+1

→X Xⁿ⁺² → · · ·. This gives riseto the truncationfunctorsτ_≤nandτ_≥nonD^b(A).ThereisafunctorialisomorphismHⁿ(X) Σⁿτ_≥nτ_≤n(X).

The following observation seems to be known; compare Lemma 3.1 and [7, Corol- lary 2.5].

Lemma 3.6. LetF:D^b(A)→D^b(A)be atrianglefunctor satisfying F(A)⊆ A.The F is anequivalence ifand onlyif soistherestrictionF|A:A→ A.

Proof. Forthe“if”part,weobservethatF induces anisomorphism Extⁿ_A(X, Y)−→Extⁿ_A(F(X), F(Y))

foranyX,Y ∈ Aandeachn∈Z.Here,weidentifyforeachn≥1,theExtⁿ_A-groupswith then-thYodenaextensiongroups,wherethelatterarepreservedbytheautoequivalence F|A.SinceAisageneratingsubcategoryofD^b(A),weinferbyLemma2.2thatF isan equivalence.

Forthe“onlyif”part,weonlyneedtoprovethedensenessofF|A.Itsuﬃcestoclaim thatifF(X) isisomorphictosomeobjectinA,so isX.

We observe that a complex X is isomorphic to some object in A if and only if Hⁿ(X) = 0 for n = 0. By the assumption that F(A) ⊆ A, we infer that F com- muteswiththetruncationfunctorsτ_≤nandτ_≥n.Consequently,itcommuteswithtaking cohomologies. More precisely, foreach boundedcomplex X and eachn∈Z, there is a naturalisomorphism

F|A(Hⁿ(X))−→^∼ Hⁿ(F(X)).

Since F|Aisfullyfaithful,theclaimfollowsimmediately. 2

WehavethefollowinganalogueofProposition3.2;compare [18,Proposition 7.1].

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Proposition 3.7. Let F: D^b(A) → D^b(A) be a triangle autoequivalence satisfying F(A) ⊆ A. Assume that the restriction F|A is isomorphic to the identity functor Id_A. Thenforeach complex X∈D^b(A),F(X) isisomorphic toX.

Proof. The sameargument of Proposition3.2 works,where we still usebrutal truncations.It suﬃces toobserve thattheprojectionπn−1:σ_≥1−nX →Σⁿ⁻¹(X¹⁻ⁿ) induces aninjectivemap

Hom_D^b_(A)(FΣⁿ⁻¹(X⁻ⁿ), σ_≥1−nX)−→Hom_D^b_(A)(FΣⁿ⁻¹(X⁻ⁿ),Σⁿ⁻¹(X¹⁻ⁿ)), sincewehave

Hom_D^b_(A)(FΣⁿ⁻¹(X⁻ⁿ), σ_≥2−nX)Hom_D^b_(A)(Σⁿ⁻¹(X⁻ⁿ), σ_≥2−nX) = 0.

Weomitthedetails. 2

The following deﬁnition and corollary are analogous to the ones for the homotopy category. Recall that for each n ∈ Z, Σⁿ(A) denotes the full subcategory of D^b(A) consisting of stalk complexes concentrated on degree−n. As usual, we identify Σ⁰(A) withA.

Deﬁnition3.8.A trianglefunctor(F,ω) :D^b(A)→D^b(A) isapseudo-identity provided thatF(X)=X foreachboundedcomplexX and thatitsrestrictionF|Σⁿ(A)toΣⁿ(A) equalstheidentity functoronΣⁿ(A) foreachn∈Z. 2

ThefollowingresultisaconsequenceofLemma3.6andProposition3.7.

Corollary3.9.Let(F,ω) :D^b(A)→D^b(A)beatrianglefunctor.Then(F,ω)isisomor- phic to a pseudo-identity if and only if F is an autoequivalence satisfying F(A) ⊆ A suchthat therestrictionF|A isisomorphic totheidentityfunctor. 2

Thefollowingisanalogousto Lemma3.5.

Lemma 3.10.Let (F,ω) :D^b(A) → D^b(A) be a pseudo-identity. Assume that (F,ω) is isomorphic totheidentity functorId_D^b₍_A₎,astrianglefunctors.Thenthere isanatural isomorphism θ: (F,ω) → Id_Db(A) of triangle functors, whose restriction to A is the identity transformation. 2

3.3. Comparing centers

Wewill comparethetrianglecentersofthehomotopycategoryandthederivedcate- gory.

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LetP beanadditivecategory.There isaringhomomorphism

res : Z(K^b(P))−→Z(P), λ→λ|P (3.2) sending λtoitsrestrictiononP.Itissurjective.Indeed,thereisanothercanonicalring homomorphism

ind :Z(P)−→Z(K^b(P)), μ→K^b(μ),

which sends μ: Id_P →Id_P to K^b(μ) : Id_Kb(P)→Id_Kb(P).More precisely,the actionof K^b(μ) oncomplexes iscomponentwise byμ. Sincethecomposition res◦ind equals the identity, thehomomorphism(3.2) issurjective.

Thefollowingnotationisneeded.ForaclassSofobjectsinatriangulatedcategoryT, we denote bySthe smallestfull additivesubcategory containingS and closed under taking directsummands,Σ andΣ⁻¹.Fortwo classesX andY ofobjects,wedenote by X Y the classformed bythose objectsZ, whichﬁt into anexact triangleX →Z → Y →Σ(X) forsomeX ∈ X andY ∈ Y.Weset S1 =SandSd+1 =SdS1 ford≥1.

Thefollowing lemmaisimplicitin [21,Lemma 4.11].

Lemma 3.11. LetA→^a B →^b C be twomorphisms in T suchthat Hom_T(a,−) vanishes on X andHom_T(b,−)vanishes onY.ThenHom_T(b◦a,−)vanishes onX Y. Proof. AssumethatX →^u Z→^v Y →Σ(X) isanexacttrianglewith X∈ X and Y ∈ Y. Take anymorphismf:C→Z.Thenv◦f◦b= 0.Itfollowsthatf◦b=u◦g forsome morphism g:B→X.Using g◦a= 0,weinferthatf ◦b◦a= 0. 2

The second statement of the following result is analogous to [12, Proposition 2.9];

compare [21,Remark 4.12].

Proposition3.12.Keepthenotationasabove.ThenthekernelN ofthemapresin(3.2) lies in theJacobsonradicalof Z(K^b(P)).

If K^b(P)=Pd forsomed≥1,we haveN^d= 0.

Proof. Letλ∈ N.Thenres(1+λ)= 1.InthenotationofLemma2.1,thetriangulated subcategory Iso(1+λ) contains P. It forces thatIso(1+λ)=K^b(P),thatis, 1+λis invertible.Consequently,theidealN liesintheJacobsonradical.

For thesecondstatement, wetakeλi ∈ N for1≤i≤d.It suﬃcestoclaim thatfor eachcomplexX,thecomposition

X−−−−→^(λ¹⁾^X X −→ · · · −→X −−−−→^(λ^d⁾^X X

is zero. Thissequence of morphisms induces asequenceof naturaltransformations be- tweentheHomfunctorsonK^b(P)

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Hom(X,−)−−−−−−−−−−→^Hom((λ¹⁾^X^,−) Hom(X,−)→ · · · →Hom(X,−)−−−−−−−−−−→^Hom((λ^d⁾^X^,−) Hom(X,−).

We observe that each of these natural transformations vanishes on P. Indeed, for an objectAinP andanymorphismf:X →A,f◦(λ_i)_X = (λ_i)_A◦f = 0.ByLemma3.11 thecompositionvanishesonPd, whichisequaltoK^b(P).An applicationofYoneda’s Lemmayields therequiredclaim. 2

LetAbeanabeliancategory.Then thereisaringhomomorphism

res : Z(D^b(A))−→Z(A), λ→λ|A (3.3) sending λ to its restriction on A. By a similar argument as above, there is another canonicalringhomomorphism

ind :Z(A)−→Z(D^b(A)), μ→D^b(μ),

satisfyingthatres◦ind isequaltotheidentity.Thenthehomomorphism(3.3) issurjective.

Thefollowingresultisprovedbythesameargumentas Proposition3.12.

Proposition 3.13.Let A be an abelian category. Then the kernel M of the map res in (3.3) liesin theJacobsonradical of Z(D^b(A)).

IfD^b(A)=Ad forsome d≥1,wehave M^d= 0. 2

LetA be anabelian category with enoughprojective objects. Denote byP thefull subcategoryformedbyprojectiveobjects.WeviewK^b(P) asatriangulatedsubcategory ofD^b(A).

Weconsiderthefollowingcommutativediagramofringhomomorphisms,where“res”

denotesthecorrespondingrestrictionofnaturaltransformations Z(D^b(A))

res∼

res

Z(K^b(P))

res

Z(A)

res∼

Z(P)

(3.4)

It is well knownthat thelower row map is an isomorphism.By [10, Theorem 2.5] the upperoneisalsoanisomorphism.Consequently,wemayidentifythekernelsofthetwo verticalhomomorphisms.

4. K-standardadditivecategories

In this section, we introduce the notions of a K-standard additive category and a stronglyK-standardadditivecategory.