Almost vanishing morphisms and centers

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.

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 in-decomposable 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,whichisthesetofnatural transforma-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.Thenanynatural transfor-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.

Proposition 2.9.LetT beanon-degenerateblock suchthatEnd_T(X)=kforeach inde-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 X^n+1d

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) =

−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 triv-ial. 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^−nd

−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,

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_X1−n◦F(π_n−1)◦F(f)

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

=−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)is isomor-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

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)=· · · →X^n−2d

n−2

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

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 trunca-tions.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)is isomor-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 comparethetrianglecentersofthehomotopycategoryandthederived cate-gory.

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 −−−−→^(λd)X X

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

Hom(X,−)−−−−−−−−−−→^{Hom((λ1)X,−)} Hom(X,−)→ · · · →Hom(X,−)−−−−−−−−−−→^{Hom((λd)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) is surjec-tive.

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.

Let k be a commutative ring. We will assume that all functors and categories are k-linear.Throughout,A isak-linearadditivecategory, whichis always assumedto be skeletally small.

Definition 4.1.The category A is said to be K-standard (over k), provided that the following holds: given any k-linear triangle autoequivalence (F,ω) : K^b(A) → K^b(A) satisfying F(A)⊆ A and any natural isomorphismθ₀:F|A → Id_A, there is a natural transformation θ: (F,ω)→Id_Kb(A)oftrianglefunctorsextendingθ₀.

The categoryA issaidto be stronglyK-standard (over k),iffurthermore theabove extension θisalwaysunique. 2

We observethattheaboveextensionθ isnecessarilyanisomorphism. Indeed,inthe notation of Lemma2.1, thetriangulated subcategory Iso(θ) contains A.Then we have Iso(θ)=K^b(A).

Lemma 4.2.LetA beasabove. Thenthefollowingstatementsare equivalent.

(1) The categoryA isK-standard.

(2) For any k-linear pseudo-identity (F,ω) on K^b(A), there is a natural isomorphism η: (F,ω)→Id_K^b_(A)of trianglefunctors suchthatη|Aistheidentity.

(3) Any k-linear pseudo-identity(F,ω)on K^b(A)is isomorphic toId_K^b_(A),as triangle functors.

Proof. The implications (1) ⇒ (2) and (2) ⇒ (3) are clear. By Lemma 3.5, we have (3)⇒(2).

For(2)⇒(1),let(F,ω) andθ0beasinDefinition4.1.ByCorollary3.4anditsproof, there is a pseudo-identity(F,ω) onK^b(A) with anatural isomorphismθ: (F,ω)→ (F,ω) such that θ|A = θ0. By assumption, there is an isomorphism η: (F,ω) → Id_K^b_(A)withη|A theidentitytransformation.Take θ=η◦θ.Thenwearedone. 2

The centers of the homotopy category and the underlying additive category play a role forstronglyK-standardcategories.

Lemma 4.3.LetAbeak-linear additivecategory.ThenitisstronglyK-standardif and only if itisK-standard andthehomomorphism res in(3.2) for Aisanisomorphism.

Proof. Forthe“onlyif”part,itsufficestoshowthatthehomomorphism(3.2) isinjective, since we observe that in Section 3 it is always surjective. We claim that each λ in the kernel of (3.2) is zero. Indeed, both 1+λ and 1 are extensions of the identity transformation (Id_Kb(A))|A= Id_A→Id_A.Bytheuniquenessoftheextensions,weinfer that1+λ= 1.

For the “if” part, we take two extensions θ,θ: (F,ω) → Id_K^b_(A) of the given iso-morphism θ0: F|A→Id_A. Asmentionedabove,both θ andθ areisomorphisms.Then

θ◦θ ⁻¹lies inZ(K^b(A)),whoserestrictionto Aistheidentity transformation.Since thehomomorphism(3.2) isinjective,we inferthatθ◦θ ⁻¹ isequalto theidentity and thusθ=θ. 2

WehavethefollowingbasicpropertiesofaK-standardadditivecategory.

Lemma 4.4. Let A be a K-standard additive category. Then the following statements hold.

(1) Let (F,ω) : K^b(A) → K^b(A) be a triangle autoequivalence with F(A) ⊆ A. If A has split idempotents, thenthere isan isomorphism(F,ω)−→^∼ K^b(F|A)of triangle functors.

(2) AssumefurtherthatAisstronglyK-standard.LetF1,F2:A→ Abetwo autoequiv-alences, whichareisomorphic. Thenany natural transformationK^b(F₁)→K^b(F₂) of triangle functors is of the form K^b(η) for a unique natural transformation η:F1→F2.

Proof. (1)WehaveobservedinLemma3.1(3)thatF|A:A→ Aisanautoequivalence.

We fix its quasi-inverse G. Consider the triangle autoequivalence K^b(G)F, whose re-striction to A is isomorphic to the identity functor. By the K-standard property, we inferthatK^b(G)F is isomorphicto theidentityfunctor. Consequently,we havethatF isisomorphictoK^b(F|A).

(2) We fix a natural isomorphism δ: F₂ → F₁. Take any natural transformation θ:K^b(F1)→K^b(F2) oftrianglefunctors andset η=θ|A tobe itsrestrictionto A.By Lemma2.8thereareγ,γ∈Z(K^b(A)) satisfyingK^b(F₁)γ=K^b(δ)◦θandK^b(F₁)γ= K^b(δ)◦K^b(η).We observe thatthe restrictionsof γ and γ to A coincide.Lemma 4.3 impliesthat thehomomorphism (3.2) isinjective. It follows thatγ =γ, whichproves thatθ=K^b(η). 2

An additive category A is split provided that it has split idempotents and every morphism f: X → Y admits a factorization f = v◦u with u a retraction and v a section.

The following observation provides a trivial example for strongly K-standard cate-gories.

Lemma4.5. LetAbea splitcategory.ThenA isstronglyK-standard.

Proof. By assumption, we observe that any complex X in K^b(A) is isomorphic to a directsumofstalk complexes.Let(F,ω) andθ0 be asinDefinition4.1. Weset

θ_Σⁿ_(A)= Σⁿ((θ0)A)◦ω_Aⁿ:F(ΣⁿA)−→Σⁿ(A)

for any A ∈ Aand n ∈ Z. Bythe additivity,θX:F(X) →X is defined forany com-plexX;compareLemma2.4.Thisyieldstherequiredextensionofθ0,whichisobviously unique. 2

ForaKrull–SchmidtcategoryA,wedenotebyindAacompletesetofrepresentatives of isomorphismclassesofindecomposableobjects.

The following notion is slightly generalized from [1]; see also [6]. A Krull–Schmidt category A is called an Orlov category provided that the endomorphism ring of each indecomposableobjectisadivisionringandthatthereisadegreefunctiondeg : indA→ Z satisfyingHom_A(S,S)= 0 forany non-isomorphicS,S∈indAwithdegS≤degS.

Thefollowing basicresultisdueto [1, Section 4].

Proposition 4.6. LetA bean Orlovcategory.Then AisstronglyK-standard.

Proof. Let (F,ω) and θ₀ be as in Definition 4.1. Then F|A is automatically homoge-neous in the sense of [1, Definition 4.1]. Then theexistence of the extension θ follows from [1, Theorem 4.7],whose uniquenessfollows from thecommutativediagram (4.10) and Lemma 4.5(2)in [1].

In particular, the homomorphism (3.2) for A is an isomorphism. This can also be deduced from[6,Proposition 2.2(ii)]. 2

Example 4.7.Let k be acommutative artinian ring, and let A be an Artin k-algebra.

Denote by A-proj the category of finitely generated projective left A-modules. Then

Denote by A-proj the category of finitely generated projective left A-modules. Then

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