Comptes Rendus
Mathématique
Bernhard Keller and Henning Krause
Tilting preserves finite global dimension Volume 358, issue 5 (2020), p. 563-570.<https://doi.org/10.5802/crmath.72>
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2020, 358, n5, p. 563-570
https://doi.org/10.5802/crmath.72
Homological Algebra /Algèbre homologique
Tilting preserves finite global dimension
Bernhard Keller∗,aand Henning Krauseb
aUniversité de Paris, UFR de Mathématiques, Institut de Mathématiques de Jussieu–PRG, UMR 7586 du CNRS, Case 7012, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
bFakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.
E-mails:[email protected], [email protected].
Abstract.Given a tilting object of the derived category of an abelian category of finite global dimension, we give (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.
2020 Mathematics Subject Classification.18G80, 18G20.
Manuscript received 2nd February 2020, revised and accepted 11th May 2020.
1. Introduction
Tilting theory [1] allows us to construct derived equivalences in various settings. Prime examples are the derived equivalences between algebras obtained from tilting modules [9] or tilting com- plexes [21] and the derived equivalences between algebras and (non commutative) varieties ob- tained from tilting bundles, cf. for example [2,3,8,11]. An important consequence of the existence of a derived equivalence is the agreement of various subordinate invariants. For instance, the Grothendieck group [21] and Hochschild cohomology [10, 15, 22] are preserved. Another invari- ant is the finiteness of global dimension, to which this note is devoted. It is well-known that finite- ness of global dimension is preserved when two algebras are linked by a tilting module [9, III.3.4]
or a tilting complex [7, 12.5]. Similar facts hold in the geometric examples. It seems natural to unify the algebraic and geometric examples by considering the following general question:
Question. Given a tilting object T in the (bounded) derived category of an abelian categoryA, does finite global dimension ofA imply finite global dimension of the endomorphism ring of T ?
Despite the ubiquity of tilting objects in algebra and geometry, there seems to be no general result in the literature which guarantees that tilting preserves finite global dimension, even when the categoryAis hereditary.1An explanation may be possible confusion about the very definition
∗Corresponding author.
1[19, Theorem 6.1] claims that End(T) has finite global dimension whenA is hereditary, but the proof seems to be incomplete.
564 Bernhard Keller and Henning Krause
of a tilting object. In fact, there are various possible definitions in the literature, and we need to clarify this point.
LetAbe an abelian category. By definition, itsglobal dimensionis the infimum of the integers dsuch that ExtiA(−,−)=0 for alli>d. Denote byD(A) the derived category ofA. Fix an object T∈D(A) and setΛ=End(T). We assume that Hom(T,ΣiT)=0 for alli6=0.
We considertwo settingsforTto be a tilting object, depending on whether the abelian category A is essentially small or not. For the first setting, we focus on the bounded derived category Db(A) of objects with cohomology concentrated in finitely many degrees. Then we define T ∈Db(A) to betiltingifDb(A) equals the thick subcategory generated byT.2For example, if Γis a right coherent ring of finite global dimension andA the abelian category modΓof finitely presented rightΓ-modules, then the objectTofDb(A) is tilting if and only if it is isomorphic to a tilting complex in the sense of [21].
Theorem 1. Let T∈Db(A)be tilting. Suppose thatA is noetherian, that is, each object inA is noetherian. Then the global dimension ofΛis at most2d+t , where d is the global dimension of Aand t the smallest integer such that HiT =0for all i outside an interval of length t . Moreover, RHom(T,−)induces a triangle equivalenceDb(A)−→∼ Db(modΛ)whenΛis right coherent.
For our second setting, assume thatAis a Grothendieck category so thatD(A) has arbitrary (set-indexed) coproducts given by coproducts of complexes. Recall that an objectC ofD(A) is calledcompactif the functor Hom(C,−) commutes with arbitrary coproducts. Each compact object lies inDb(A), cf. Lemma 7. Then we defineT ∈D(A) to betiltingif it is compact and D(A) equals the localizing subcategory generated byT (the closure underΣ±1, extensions and arbitrary coproducts). For example, ifA is the category ModΓof all right modules over a ringΓ, then the tilting objects inD(A) are precisely those isomorphic to tilting complexes in the sense of [21].
Theorem 2. Let T ∈D(A)be tilting. ThenRHom(T,−)induces a triangle equivalenceD(A)−→∼ D(ModΛ)andgl.dimΛ≤2d+t , where d and t are defined as in Theorem 1.
We deduce Theorem 1 from Theorem 2. The proof uses t-structures and the strategy is inspired by [7, 12.5]. For Theorem 2, we compare the canonical t-structure onD(A) with the canonical one onD(ModΛ); this yields the bound for the global dimension ofΛ. For Theorem 1, we embedA into a Grothendieck categoryA¯and employ the fact that a tilting objectT ∈Db(A) identifies with a tilting object in the unbounded derived categoryD( ¯A).
2. t-structures and finite global dimension
LetT be a triangulated category with suspensionΣ:T −→∼ T. A pair (U,V) of full additive subcategories is calledt-structureprovided the following holds [4]:
(1) ΣU ⊆UandΣ−1V ⊆V.
(2) Hom(X,Y)=0 for allX∈UandY ∈V.
(3) For eachX∈T there exists an exact triangleX0→X→X00→ΣX0such thatX0∈U and X00∈V.
2Often the following weaker condition is used: Hom(T,ΣiX)=0 for alli∈ZimpliesX=0. This is not sufficient in our context.
C. R. Mathématique,2020, 358, n5, 563-570
We consider the following example. LetA be an abelian category andT =D(A) its derived category. Forn∈Zset
T≤n:=n X∈T¯
¯
¯HiX=0 for alli>no , and
T>n:=n X∈T¯
¯
¯HiX=0 for alli≤no .
Then we have T≤n =Σ−n T≤0 and T>n =Σ−n T>0for all n ∈Z. For eachX ∈T the truncations in degreenprovide an exact triangle
τ≤nX−→X−→τ>nX−→Σ(τ≤nX)
withτ≤nX ∈ T≤n and τ>nX ∈ T>n. Thus the pair (T≤0,T>0) is a t-structure and called canonical t-structureonD(A). Note that the canonical t-structure restricts to the one onDb(A).
Lemma 3. Let(D≤0,D>0)denote the canonical t-structure onDb(A). Then the global dimension ofA is bounded by d if and only ifHom(X,Y)=0for all X∈D≥0and Y ∈D< −d.
Proof. For objects A,A0 ∈A andi ∈Z we have Exti(A,A0)∼=Hom(A,ΣiA0). Thus the global dimension ofA is bounded byd if and only if for all objectsX,Y ∈Db(A) with cohomology concentrated in a single degree we have Hom(X,Y)=0 when X ∈D≥0and Y ∈D< −d. The assertion of the Lemma 3 follows since forX∈D≥0andY ∈D< −d, the truncations induce finite filtrations
X=τ≥0Xτ≥1Xτ≥2X· · · and
· · ·τ< −d−2Y τ< −d−1Y τ< −dY =Y
such that each subquotient has its cohomology concentrated in a single degreei, withi≥0 for the subquotients ofXandi< −dfor the subquotients ofY. We wish to extend this Lemma 3 from Db(A) toD(A). To this end, we fix a Grothendieck categoryA. Let us recall some basic facts about derived limits and colimits inD(A). We will use derived functors in the sense of Deligne [5, 1.2]. Recall that one of the most pleasant properties of Deligne’s definition is that for an adjoint pair (F,G), if the derived functors exist, they still form an adjoint pair (LF,RG), see for example [14, Section 13]. LetIdenote a small category. We write AI for the Grothendieck category of functorsI→A. The diagonal functor∆:A →AI taking an object to the constant functor has a left adjoint colim and a right adjoint lim. Let us examine their derived functors. Since the functor∆is exact, its left and right derived functors exist and are canonically isomorphic to the induced functorD(A)→D(AI). For generalI, the existence ofLcolim is unclear but ifIis filtered, then colim is exact (by the definition of a Grothendieck category) and so its left derived functor exists and is canonically isomorphic to the induced functorD(AI)→D(A), which we still denote by colim. This implies in particular that arbitrary coproducts exist inD(A) and are computed by coproducts in the category of complexes. For arbitraryI, the categoryAI is still a Grothendieck category. This implies that the right derived functorRlim exists and is computed asRlimX=limiX, whereX→iX is a homotopy injective resolution in the homotopy category ofAI, see for example [12, Theorem 14.3.4]. In particular, this implies that products of arbitrary set-indexed families (Xi)i∈Iof objects ofD(A) exist inD(A) and are computed as products in the category of complexes
Y
i∈I
iXi,
566 Bernhard Keller and Henning Krause
whereXi →iXi is a homotopy injective resolution for eachi∈I. For example, ifXi is homolog- ically left bounded, then foriXi, we may take any strictly left bounded complex with injective components quasi-isomorphic toXi.
Lemma 4. For each complex X∈D(A)its truncations induce exact triangles Σ−1X `
p≥0τ≤pX `
p≥0τ≤pX X
and
Rlimq≤0τ≥qX Q
q≤0τ≥qX Q
q≤0τ≥qX Σ¡
Rlimq≤0τ≥qX¢ .
Moreover, we have X −→∼ Rlimτ≥qX when the injective dimension of each HnX admits a global bound not depending on n and HnX=0for nÀ0.
Remark 5. As explained above, the coproducts in the first triangle may be computed in the category of complexes, whereas the products in the second triangle are products in the derived category or equivalently derived products computed using homotopy injective resolutions. The triangles exist by [16, Proposition A.5 (3)] and its dual but we give a direct proof for the special case at hand below.
Proof. For the first triangle we observe that the colimit of theτ≤pXin the category of complexes can be computed degreewise. This gives an exact sequence
0 `
p≥0τ≤pX `
p≥0τ≤pX X 0
of complexes and therefore an exact triangle inD(A), as in the assertion of the Lemma 4.
For the second triangle we need to construct a K-injective (homotopy injective) resolution of (τ≥qX) in the category of complexes of inverse systems. For eachq<0, choose an injective resolutionHqX→Jq. Then choose a K-injective resolutionτ≥0X→I0and, forq<0, recursively define morphismsεq:Iq+1→Σq+1Jqsuch that we have morphisms of triangles inD(A)
ΣqHqX τ≥qX τ≥q+1X Σq+1HqX
ΣqJq Iq Iq+1 εq Σq+1Jq
where the vertical morphisms are quasi-isomorphisms and ΣIq is the cone over a lift to a morphism of complexes ofεq. The system (Iq) is then quasi-isomorphic to (τ≥qX) and K-injective in the homotopy category of complexes of inverse systems. Thus, it may be used to compute the right derived limit of (τ≥qX). We obtain a degreewise split exact sequence of complexes
0 limIq Q
q≤0
Iq Q
q≤0
Iq 0
and therefore an exact triangle inD(A), as in the assertion of the Lemma 4, with Rlimτ≥qX ∼=RlimIq ∼=limIq.
Now suppose that the injective dimension ofHqX admits a global bound, sayd, and we may assume thatHqX =0 for allq>0. To show the isomorphism X −→∼ Rlimτ≥qX we modify the above construction of a K-injective resolution of (τ≥qX) as follows. For eachq≤0, choose an injective resolutionHqX→Jq, where the components ofJqvanish in all degrees strictly greater thand. We putI0=J0and, forq<0, recursively define morphismsεq:Iq+1→Σq+1Jqas before.
Again, the system (Iq) may be used to compute the right derived limit of (τ≥qX). Since theJqare
C. R. Mathématique,2020, 358, n5, 563-570
uniformly right bounded, the system (Iq) becomes stationary in each degree. This yields inD(A) the required isomorphism
X∼=limIq∼=Rlim(τ≥qX).
Lemma 6. Let (D≤0,D>0) denote the canonical t-structure onD(A) and suppose the global dimension ofAis bounded by d . Then for X∈D≥0and Y ∈D<−d−2we have Hom(X,Y)=0.
Proof. We apply Lemma 4. ThusXfits into an exact triangle given by the truncationsτ≤pX, and it suffices to show that Hom(τ≤pX,Y) and Hom(Στ≤pX,Y) vanish for allp. On the other hand, Y fits into an exact triangle given by the truncationsτ≥qY, and therefore it suffices to show that Hom(τ≤pX,τ≥qY), Hom(Στ≤pX,τ≥qY), and Hom(Στ≤pX,Σ−1τ≥qY) vanish for allpandq. This
holds by Lemma 3 since both arguments belong toDb(A).
3. Tilting for D(A)
LetA be a Grothendieck category andD(A) its unbounded derived category. Recall that the categoryD(A) has arbitrary (set-indexed) coproducts given by coproducts in the category of complexes. Notice that the right derived product functor yields arbitrary products inD(A). In particular, the product of a family of left bounded complexes with injective components is also their product inD(A).
Lemma 7. If C is a compact object ofD(A), then the cohomology HpC vanishes for all but finitely many integers p.
Proof. For eachp∈Z, choose a monomorphismip:HpC →Ip into an injective object. Using the identification
HomD(A)(C,Σ−pI)=HomA(HpC,I)
valid for each injectiveIofA, theip yield a morphismi fromCto the product (in the category of complexes and in the derived category) of theΣ−pIp. Clearly, in the category of complexes (and hence in the derived category), this product is canonically isomorphic to the corresponding coproduct. So we obtain a morphism fromCto the coproduct of theΣ−pIpwhich in cohomology induces theip. By the compactness ofC, this morphism factors through a finite subcoproduct of theΣ−pIpso that all but finitely many of theip have to vanish. Since they are monomorphisms,
the same holds for theHpC.
Now letTbe atilting objectofD(A). ThusTis compact, the group Hom(T,ΣpT) vanishes for allp6=0, andD(A) equals its localizing subcategory generated byT.
LetΛbe the endomorphism ring ofT. ThenΛis quasi-isomorphic to the derived endomor- phism algebraRHom(T,T) and so the functorRHom(T,−) yields a triangle equivalence
D(A)−−→∼ D(ModΛ),
cf. [13]. We use it to identifyD(A) withD(ModΛ). The canonical t-structure onD(A) is denoted by (D≤0,D>0), while the canonical t-structure onD(ModΛ) is denoted by (D(Λ)≤0,D(Λ)>0).
Lemma 8. Suppose that A and ModΛ have finite global dimension. Then the functor RHom(T,−)restricts to an equivalenceDb(A)−→∼ Db(ModΛ).
Proof. Given objectsX,Y ∈Db(A) we have Hom(X,ΣiY)=0 for almost alli sinceA has finite global dimension. This is easily shown by induction on the number of integersn such that Hn(X⊕Y)6=0. It follows thatRHom(T,−) restricts to a functorF:Db(A)→Db(ModΛ), since
HiRHom(T,X)∼=Hom(T,ΣiX)
568 Bernhard Keller and Henning Krause
andT ∈Db(A) by Lemma 7. On the other hand, Db(ModΛ) equals the thick subcategory of D(ModΛ) that is generated by the category ProjΛof projectiveΛ-modules, viewed as complexes concentrated in degree zero, sinceΛhas finite global dimension. It follows thatF is essentially surjective sinceFidentifies the closure ofT under arbitrary coproducts and direct summands
with ProjΛ.
From now on suppose that the global dimension ofA is bounded byd, and fixt≥0 such that HpT=0 for allp6∈[−t, 0], cf. Lemma 7.
Lemma 9. We haveD(Λ)≤0⊆D≤0.
Proof. ForX∈D>0andi≤0 we have Hom(T,ΣiX)=0 sinceT∈D≤0. It follows thatX∈D(Λ)>0, since D(A) −→∼ D(ModΛ) identifies T with Λ and HiX ∼= Hom(Λ,ΣiX) in D(ModΛ). Thus
D(Λ)≤0⊆D≤0.
Lemma 10. We haveD(Λ)≥0⊆D≥−d−t−2.
Proof. LetX∈D≤0. ThenHiT=0 for alli6∈[−t, 0] implies Hom(T,ΣiX)=0 for alli>d+t+2 by Lemma 6. It follows thatD≤0⊆D(Λ)≤d+t+2, and thereforeD(Λ)≥0⊆D≥ −d−t−2. Proof of Theorem 2. LetX,Y ∈ModΛandi>2d+t+4. Then
X∈D(Λ)≥0⊆D≥ −d−t−2 and ΣiY ∈D(Λ)< −2d−t−4⊆D< −2d−t−4
by Lemmas 9 and 10. It follows from Lemma 6 that Exti(X,Y)=Hom³
X,ΣiY´
=0.
Thus the global dimension ofΛis bounded by 2d+t+4. In order to improve this bound, observe thatRHom(T,−) restricts to an equivalenceDb(A)−→∼ Db(ModΛ) by Lemma 8. Then we compare t-structures onDb(A) and use Lemma 3 instead of Lemma 6. It follows that the global dimension
ofΛis bounded by 2d+t.
4. Tilting for Db(A)
LetAbe an abelian category andT∈Db(A) a tilting object; recall this means Hom(T,ΣiT)=0 for alli6=0 andDb(A) equals the thick subcategory generated byT. SetΛ=End(T) and denote by projΛthe category of finitely generated projectiveΛ-modules. By [17, Theorem 3.2], the inclusion addT ,→Db(A) extends to a triangle functorKb(addT)→Db(A). Then it is straightforward to show that the composite projΛ−→∼ addT,→Db(A) extends to a triangle equivalence
Db(projΛ)−−→∼ Kb(addT)−−→∼ Db(A).
We deduce Theorem 1 from Theorem 2 whenA isnoetherian, that is, each object in A is noetherian. To this end, we fix an essentially small abelian categoryA and let ¯A:=Lex(Aop, Ab) denote the category of left exact functorsAop →Ab. ThenA¯is a Grothendieck category and the Yoneda embeddingA →A¯which sends X ∈A to Hom(−,X) is fully faithful and exact, cf. [6, Chapter II].
Lemma 11. Suppose thatAis noetherian and of finite global dimension. ThenD( ¯A)is compactly generated (so equals the localizing subcategory generated by all compact objects) and the inclusion A → A¯ induces a fully faithful functor Db(A) →D( ¯A) that identifies Db(A) with the full subcategory of compact objects.
C. R. Mathématique,2020, 358, n5, 563-570
Proof. The inclusionA→A¯identifiesAwith the full subcategory of noetherian objects in ¯A. It is well-known that an objectIof ¯A is injective if and only if Ext1(−,I) vanishes on all noetherian objects. This implies that the global dimension of ¯Aequals that ofA.
Let Inj ¯A denote the full subcategory of injective objects and K(Inj ¯A) the category of complexes up to homotopy. Then the canonical functorK(Inj ¯A)→D( ¯A) is an equivalence, cf. [18, Proposition 3.6]. It follows thatD( ¯A) is compactly generated and thatDb(A) identifies with the full subcategory of compact objects, cf. [18, Proposition 2.3].
Proof of Theorem 1. We apply Lemma 11. The functorDb(A)→D( ¯A) identifies a tilting object T ofDb(A) with a tilting object ofD( ¯A). LetΛ=End(T). Then Theorem 2 provides the bound for the global dimension ofΛ. WhenΛis right coherent, then the triangle equivalenceD( ¯A)
−→∼ D(ModΛ) restricts to an equivalence
Db(A)−→∼ Db(projΛ)−→∼ Db(modΛ)
on the full subcategory of compact objects.
5. Concluding remarks
We end this paper with some remarks. Let us fix an essentially small abelian categoryA with a tilting objectT∈Db(A), and setΛ=End(T).
Recall that aΛ-moduleXispseudo-coherentif it admits a projective resolution
· · · −→P1−→P0−→X−→0
such that eachPi is finitely generated. We denote by pcohΛthe full subcategory of pseudo- coherentΛ-modules; it is a thick subcategory of the category of allΛ-modules, so closed under direct summands, extensions, kernels of epis, and cokernels of monos.
Remark 12. Suppose thatA is noetherian and of finite global dimension. ThenRHom(T,−) induces a triangle equivalenceDb(A)−→∼ Db(pcohΛ).
For each pair of objects X,X0 ∈A we have Exti¡ X,X0¢
=0 fori À0. This provides some restriction on the global dimension ofA.
Remark 13. LetA be alength category; thus each object has finite composition length. Then gl.dimA= inf
S,S0 simple
n i∈N¯
¯
¯Exti+1¡ S,S0¢
=0o
< ∞
since the number of isoclasses of simple objects is bounded by the length ofH∗T.
Remark 14. The global dimension ofA need not be finite whenDb(A) admits a tilting object.
LetΛbe a right noetherian ring and set A =modΛ. ThenΛ∈Db(A) is tilting if and only if each object inA has finite projective dimension. In this case the global dimension ofA equals the (small) finitistic dimension ofΛ, which may be infinite (even whenΛ is commutative), cf. [20, Appendix, Example 1].
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