art%3a10.1007%2fs10485-011-9266-z(1)

Appl Categor Struct (2013) 21:311–348DOI 10.1007/s10485-011-9266-z

Tilting Theory and Functor Categories II.Generalized Tilting

Roberto Martínez-Villa · Martin Ortiz-Morales

Received: 16 December 2010 / Accepted: 26 August 2011 / Published online: 22 September 2011© Springer Science+Business Media B.V. 2011

Abstract In this paper we continue the project of generalizing tilting theory to thecategory of contravariant functors Mod(C), from a skeletally small preadditive cate-gory C to the category of abelian groups, initiated in [15]. We introduce the notionof a generalized tilting category T , and we concentrate here on extending Happel’stheorem to Mod(C); more specifically, we prove that there is an equivalence oftriangulated categories Db (Mod(C)) ∼= Db (Mod(T )). We then add some restrictionson our category C, in order to obtain a version of Happel’s theorem for the categoriesof finitely presented functors. We end the paper proving that some of the theoremsfor artin algebras, relating tilting with contravariantly finite categories proved inAuslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151,1991), can be extended to the category of finitely presented functors mod(C), withC a dualizing variety.

Keywords Classical tilting · Functor categories

Mathematics Subject Classifications (2010) Primary 05C38 · 15A15;Secondary 05A15 · 15A18

This paper is in final form, and no version of it will be submitted for publication elsewhere.

R. Martínez-Villa · M. Ortiz-Morales (B)Instituto de Matemáticas UNAM, Universidad Nacional Autonoma de Mexico,Unidad Morelia, Mexicoe-mail: [email protected]: http://www.matmor.unam.mx

R. Martínez-Villae-mail: [email protected]

312 R. Martínez-Villa, M. Ortiz-Morales

1 Introduction and Basic Results

In this article we continue the project initiated in [15] of generalizing tilting theoryto functor categories. We deal with the notion of a generalized tilting subcategory Tof the category Mod(C) of contravariant functors from a skeletally small preadditivecategory C, to the category of abelian groups.

The paper consists of three parts:In the first section we fix the notation and recall some basic results from [1, 15].

We then concentrate on proving Happel’s theorem: given a tilting subcategoryT of Mod(C), the derived categories of bounded complexes, Db (Mod(C)) andDb (Mod(T )) are equivalent, and we discuss a partial converse. The proof is basedon a paper by Cline et al. [9].

In the next section we prove, that for a dualizing variety C and a tilting categoryT with peseudokernels, the categories of finitely presented functors mod(C) andmod(T ) have equivalent derived categories, Db ( mod(C)) ∼= Db (mod(T )).

We end the paper showing that for a dualizing variety C, there are relationsbetween contravariantly finite categories and generalized tilting subcategories ofmod(C), in the spirit of the relations given in [2, 6, 7] for artin algebras.

1.1 Functor Categories

In this section C will be an arbitrary skeletally small preadditive category, andMod(C) will denote the category of contravariant functors from C to the categoryof abelian groups. The aim of the paper is to show that the notions of generalizedtilting can be extended to Mod(C), obtaining generalizations of the main theoremson tilting for rings.

The subcategory of Mod(C) consisting of all finitely generated projective objects,p(C), is a skeletally small additive category in which idempotents split. The func-tor P : C → p(C), P(C) = C( , C), is fully faithful and induces by restriction, res :Mod(p(C)) → Mod(C), an equivalence of categories. For this reason we may assumethat our categories are skeletally small, additive categories, such that idempotentssplit. Such categories were called annuli varieties in [1].

We prove a version of Happel’s theorem in full generality, but in order to obtaincategorical versions of some tilting theorems for finite dimensional algebras, we needto add restrictions on our category C such as the following: existence of pseudokernels, Krull-Schmidt, Hom-finite, dualizing, etc. To fix the notation, we recallknown results on functors and categories that we use throughout the paper, referringthe proofs of the papers by Auslander and Reiten [3, 6, 7, 15].

Definition 1 Given a preadditive, skeletally small category C, we say C has pseudok-ernels; if given a map f : C1 → C0, there exists a map g : C2 → C1 such that the

sequence of representable functors C( , C2)( ,g)→ C( C1)

( , f )→ C( , C0) is exact.A functor M is finitely presented, if there exists an exact sequence C( C1) →

C( , C0) → M → 0.

We denote by mod(C) the full subcategory of Mod(C) consisting of finitelypresented functors. It was proved in [3] that mod(C) is abelian if and only if C haspseudokernels.

Generalized Tilting 313

We will say indistinctly that M is an object of Mod(C) or that M is a C-module.A representable functor C( , C) will be sometimes denoted by ( , C).

1.2 Krull-Schmidt Categories

We start by giving some definitions from [7].

Definition 2 Let R be a commutative artin ring. A R-category C is a preadditivecategory such that C(C1, C2) is an R-module, and the composition is R-bilinear.Under these conditions Mod(C) is a R-category, which we identify with the categoryof functors (Cop, Mod(R)).

A R-category C is Hom-finite, if for each pair of objects C1, C2 in C, the R-moduleC(C1, C2) is finitely generated. We denote by (Cop, mod(R)), the full subcategory of(C, Mod(R)) consisting of the C-modules, such that for every C in C, the R-moduleM(C) is finitely generated. (Cop, mod(R)) is an abelian category, and the inclusion(Cop, mod(R)) → (Cop, Mod(R)) is exact.

The category mod(C) is a full subcategory of (Cop, mod(R)). The functors D :(Cop, mod(R)) → (C, mod(R)) and D : (C, mod(R)) → (Cop, mod(R)), given for anyC in C, by D(M)(C) = HomR(M(C), I(R/r)), with r the Jacobson radical of R, andI(R/r) is the injective envelope of R/r. The functor D defines a duality between(C, mod(R)) and (Cop, mod(R)). If C is a Hom-finite R-category and M is in mod(C),then M(C) is a finitely generated R-module and in mod(R).

Definition 3 A Hom-finite R-category C is called dualizing, if the functor D :(Cop, mod(R)) → (C, mod(R)) induces a duality between the categories mod(C) andmod(Cop).

It is clear from the definition that for a dualizing category C, mod(C) has enoughinjectives.

To finish, we recall the following definitions:

Definition 4 An additive category C is Krull-Schmidt, if every object in C decom-poses in a finite sum of objects whose endomorphism rings are local.

Definition 5 A ring A is semiperfect if every finitely generated A-module has aprojective cover.

In the ring situation, the notions of semiperfect and Krull-Schmidt are related, aswe can see from the following:

Proposition 1 [4, 18] For a ring A, the following conditions are equivalent:

(a) The category of f initely generated projective left A-modules is Krull-Schmidt.(b) Every simple left A-module has a projective cover.(c) Every f initely generated left A-module has a projective cover.

The proof of the next theorem can be found in [1, Theorem 4.12].


Theorem 1 Let C be an annuli variety. Then the following conditions are equivalent:

(a) Every object in mod(C) has a minimal projective presentation.(b) For each object C in C, every f initely presented End(C)op-module has a minimal

projective presentation.

Corollary 1 If C is a category, such that for every object C in C End(C)op is semiper-fect, then every object in mod(C) has a minimal projective presentation.

We will see next that for a Krull-Schmidt variety C the finitely presented functorshave projective covers.

Theorem 2 Let C be an annuli R-variety. Then the following statements hold:

(a) If C is Krull-Schmidt, then for every object C in C the endomorphism ring,EndC(C), is semiperfect.

(b) If C is a dualizing Krull-Schmidt R-variety, then mod(C) is a dualizing Krull-Schmidt R-variety.

Proof

(a) Let C be an object in C and C(C) the annuli subvariety of C generated by C.According to [1, Proposition 2.7], there is an equivalence of categories:

C(C)P−→ p(C(C))

eC−→ p(Endop(C)).

Let Q be an object in p(End(C)op). By the above equivalence, there exists anobject C′ in C(C) such that eC(P(C′)) = eC(C( , C′)) = C(C, C′) ∼= Q. Since Cis Krull-Schmidt, C′ decomposes in a finite sum of objects C′ = ∐n

i=1 C j, suchthat each EndC(C j) is a local ring. Thus Q ∼= ∐n

i=1(C, C j), and End((C, C j)) ∼=EndC(C j) is local, that is: p(End(C)op) is Krull-Schmidt. By Proposition 1, thering Endop

C (C) is semiperfect.(b) It was proved in [7, Proposition 2.6], that mod(C) is a dualizing R-variety; we

must only see that every object in mod(C) decomposes in a finite sum of objectswith local endomorphism rings. Let M be an object in mod(C). By (a) andCorollary 1, M has a projective cover ( , C) → M → 0. By the uniqueness ofthe projective cover, the fact that C is Krull-Schmidt and Yoneda’s Lemma, itfollows that M decomposes in a finite number of indecomposable objects Mi inmod(C), that is: M = ∐n

i=1 Mi.We only need to see that for each i, EndC(Mi) is local.Let (M) be the annuli subvariety of mod(C) generated by M and � =Endop

C (M). By [1, Proposition 2.7], there is an equivalence of categories:

(M)P−→ p((M))

eM−→ p(�).

Thus, it is clear that eM(P(M)) = eM(P(∐n

i=1 Mi)) = HomC(M,∐n

i=1 Mi) =∐i∈I(M, Mi) is a decomposition of � in indecomposable projective �-modules,

(M, Mi).Since C is Hom-finite and M finitely presented, it is easy to see that � is an artinalgebra.


It follows that (M, Mi) has local endomorphism ring, and the claim is aconsequence of the isomorphism End�((M, Mi)) ∼= EndC(Mi). ��

We will need the following result from [15]:

Theorem 3 Let C be an annuli variety and T a skeletally small full subcategory ofMod(C). Let’s def ine the following functor:

φ : Mod(C) → Mod(T ), φ(M) = Hom( , M)T .

Then,

(i) the functor φ has a left adjoint − ⊗ T : Mod(T ) → Mod(C),(ii) the action of − ⊗ T on representable functors is as follows: the map ( , T1) ⊗

T ( , f )⊗T→ ( , T0) ⊗ T is isomorphic to the map: T1f→ T0.

Proposition 2 If M is f initely presented functor in Mod(C), then the functorExt1

C(M, ) commute with arbitrary sums.

Proof There is an exact sequence

0 → Ker(α) → C( , C)α−→ M → 0 (1.1)

with Ker(α) finitely generated and C an object in C. Let F = {Fi}i∈I be a family ofobjects in Mod(C). After applying HomC( ,

∐i∈I Fi) to Eq. 1.1, it follows by the long

homology sequence that the sequence

0 → HomC

(

M,∐

i∈I

Fi

)

→ HomC

(

C( , C),∐

i∈I

Fi

)

→

→ HomC

(

Ker(α) ,∐

i∈I

Fi

)

→ Ext1C

(

M,∐

i∈I

Fi

)

→ 0

is exact. Since both M and Ker(α) are finitely generated and for finitely generatedobjects X the functor HomC(X,−) commutes with arbitrary sums, it follows from theabove sequence that Ext1

C(M,∐

i∈I Fi) is naturally isomorphic to∐

i∈I Ext1C(M, Fi).

��

Corollary 2 If M is an object in Mod(C) which has a projective resolution con-sisting of f initely generated projectives, then for an integer n > 0, the functorsExtn

C(M , −) commutes with arbitrary sums.

Proof By induction. ��

In order to fix the notation, we will recall some basic results on derived categoriesthat we will use throughout the paper, referring to [12–14, 19, 20] for more detailsand proofs.

For a given abelian category A we associate the category K(A) of complexesmodule the homotopy relation, and its full subcategories K∗(A), where the stardenotes +, −, b , ∅; and + means complexes that are zero for all degrees small


enough, − means that the complexes are zero in degrees large enough, and bmeans the complexes that are non zero for only a finite number of indexes, theseare the bounded complexes, ∅ means nothing. In addition, we denote by K+,b (A)

(respectively K−,b (A)) the full subcategory of K+(A) (respectively K−(A)) of allcomplexes with bounded cohomology. These categories are triangulated categories.

For each K∗(A), there is a corresponding derived category D∗(A), which is alsotriangulated, and it is obtained from K∗(A) by inverting the quasi-isomorphisms. Inprinciple we must worry about a logical problem, because the quasi-isomorphismsmay not be a set; however in the case that we have enough projectives or enoughinjectives, the problem is avoided since the homotopy category of complexes ofprojectives ( respectively, of injectives) K−(P) ( respectively K+(I)) and the derivedcategory D−(A) (respectively D+(A)) are equivalent, and in the bounded caseK−,b (P) and K+,b (I) are both equivalent to the bounded category Db (A).

Given an additive functor F : A → B between abelian categories such that A hasenough injectives, there is a right derived functor R+ F : D+(A) → D(B) which canbe considered a lifting of F.

For each integer n, Rn F is the composition A i−→ D+(A)R+ F→ D(B)

Hn−→ B, where iis the inclusion as a complex concentrated in degree zero, and Hn is the functor givingthe nth cohomology group. An object X in A is called F-acyclic, if Rn F(X) = 0, forn > 0. The F-acyclic objects form a so-called adapted class [see 13, 14], and R+ F canbe computed using resolutions with objects in the adapted class.

The functor R+ F has finite cohomological dimension ≤ n, if Rk F = 0, when k > n.If A has enough injectives, it is well-known [see 14] that any additive functor F :

A → B, with R+ F of finite cohomological dimension, has a right derived functorRb F : Db (A) → Db (B).

Case B is an abelian category with enough projectives, left derived functors L−Gof an additive functor G : B → A are defined in an analogous way. If G : B → Ais an additive functor such that the left derived functor L−G : D−(B) → D(A)

is of finite cohomological dimension, then there is a left derived functor Lb G :Db (B) → Db (A).

For an adjoint pair (G, F), the derived functors (Lb G, Rb F) form an adjoint pair[11, 12].

1.3 Generalized Tilting and Happel’s Theorem

Happel’s theorem [10] can be seen as a generalization of Morita’s theorem, since itanswers in terms of generalized tilting the following question:

Given a ring R, a finitely generated left R-module T and the endomorphismring S = EndR(T)op, there is a pair of adjoints functors between the correspond-ing categories of left modules: F = HomR(T, −): ModR → ModS and G = T ⊗ − :ModS → ModR. When there exist derived functors Rb F : Db (ModR) → Db (ModS)

and Lb G : Db (ModS) → Db (ModR) which are quasi inverse of each other, this is anequivalence of triangulated categories?.

Our aim is to extend this theorem to annuli varieties C.We will introduce first the notion of a generalized tilting subcategory T of Mod(C),

then consider the pair of adjoint functors

φ : Mod(C) → Mod(T ), − ⊗ T : Mod(T ) → Mod(C)


given above, and look for the corresponding question. Our proof will follow veryclosely the paper by Cline et al. [9].

Definition 6 Let C be an annuli variety. A full subcategory T of Mod(C) is general-ized tilting if the following is true:

(i) There exists a fixed integer f such that every object T in T has a projectiveresolution

0 → P f → · · · → P1 → P0 → T → 0,

with each Pi finitely generated.(ii) For each pair of objects T and T ′ in T and any positive integer n, we have

ExtnC(T, T ′) = 0.

(iii) Each functor ( , C), C ∈ C, has a resolution

0 → ( , C) → T0C → · · · → TgC

C → 0,

with TiC in T .

In this paper, a generalized tilting category will be called, for short, a tiltingcategory.

Condition (i) can be interpreted as saying that the category T has finite projectivedimension, and condition (ii) as T has no self extensions. But in order to proveHappel’s theorem, we need to assume the stronger condition:

(iii’) There is a fixed integer g such that each functor ( , C) with C ∈ C has aresolution

0 → ( , C) → T0C → · · · → Tg

C → 0,

with TiC in T .

We will discuss later when condition (iii) implies condition (iii’). We will also seethat to assume conditions (iii’) it is necessary for the proof we are giving.

Following [9], the strategy of the proof of Happel’s theorem is as follows: we willprove first that conditions (i), (ii) and (iii’) imply that the derived functors R+φ andL− − ⊗T have finite cohomological dimension, then we will make use of the lemmabelow to prove that Rb φ and Lb − ⊗T are quasi-inverse of each other.

Lemma 1 [9, 1.2] Let D and E be triangulated categories, and F : D → E , G : E → Dbe exact functors. Assume that

(i) F is right adjoint to G,(ii) the adjunction functor θ : idE → FG is an isomorphism,

(iii) F is an embedding (this means F X ∼= 0 implies X ∼= 0).

Then, the adjunction functor η : GF → idD is an isomorphism, and hence F and Gare equivalences.

Given categories C and T as above, the functor φ : Mod(C) → Mod(T ) has anth- right derived functor, denoted by Extn

C( , −)T . The functor − ⊗ T : Mod(T ) →Mod(C) has a nth-left derived functor, denoted by TorTn (−,T ).


The following proposition is proved in [15]:

Proposition 3 For each C in C and M in Mod(T ), there exists an isomorphism

TorTn ((C( , C), )T , M) ∼= TorTn (M,T )(C), (1.2)

which for any non-negative integer n, it is natural in C and M.

We can prove now the following:

Proposition 4 Let C be an annuli variety and T a generalized tilting subcategory ofMod(C) satisfying condition (iii’). Then,

(a) the functor R+,b φ : Db (Mod(C)) → D(Mod(T )) has f inite cohomological di-mension, in fact dimR+,b φ ≤ f ,

(b) the functor L−,b − ⊗T : Db (Mod(T )) → D(Mod(C)) has f inite cohomologicaldimension, in fact dimL−,b − ⊗T ≤ g.

Proof

(a) Let M be an object in Mod(C). The object M can be seen as complex inD+,b (Mod(C)) concentrated in degree zero, which is quasi-isomorphic to acomplex of injective objects I. ∈ K+(Mod(C)), and there are isomorphisms:

Rn(φ)(M) = Hn(R+φ(M)) = Hn(φ(I.)) = Hn(( , I.)T ) = ExtnC( , M)T .

Since pdim(T ) = f , then for n > f we have Rn(φ)(M) = ExtnC( , M)T = 0.

(b) Let C be an object in C. By condition (iii’), there is an exact sequence

0 → ( , C) → T0 d1−→ T1 → · · · → Tg−1 dg−→ Tg → 0,

which gives rise to the short exact sequence:

0 → Ker(dg) → Tg−1 → Tg → 0.

By the long homology sequence and property (ii) for each positive integer n,

we have ExtnC(Ker(dg), )T = 0. Using backward induction we obtain for n > 0

and i = 1, . . . , g that ExtnC(Ker(di), )T = 0. We thus obtain an exact sequence

in Mod(T op)

0 → (Tg, ) → (Tg−1, ) → · · · → (T0, ) → (( , C), )T → 0, (1.3)

which implies pdim(( , C), )T ≤ g.Let N be an object in Mod(T ).As usual, we consider the object N as complex in D−,b (Mod(T )) concentratedin degree zero quasi-isomorphic to the complex P. ∈ K−(Mod(T )) of projectiveobjects. Then there are isomorphisms:

Ln(N ⊗ T ) = Hn(L−(N ⊗ T )) = Hn(P. ⊗ T ) = TorTn (N,T ).

From the equation

TorTn (N,T )(C) = TorTn ((( , C), )T , N),


given in Proposition 3, and the fact that pdim(( , C), )T ≤ g, it followsTorTn (N,T )(C) = 0 for n > g. ��

We can prove now the main theorem of the section:

Theorem 4 (Happel) Let C be an annuli variety and T a generalized tilting subcate-gory of Mod(C) satisfying condition (iii’). Then,

(a) the functors

R+,b φ : Db (Mod(C)) → Db (Mod(T )) and

L−,b − ⊗T : Db (Mod(T )) → Db (Mod(C)),

induce an equivalence of derived categories,(b) the full subcategory {(( , C), )T }C∈C of Mod(T op) is a tilting category equivalent

to the category Cop, satisfying the stronger condition (iii’).

Proof

(a) To prove our first claim, we proceed to prove each of the conditions inLemma 1.

(a1) By Proposition 4, the functor R+,b φ : Db (Mod(C)) → Db (Mod(T )) isright adjoint to the functor L−,b − ⊗T : Db (Mod(T )) → Db (Mod(C)).

(a2) idDb (B) → R+,b φL−,b ⊗ T is an isomorphism:(a21) Observe that an arbitrary sum

∐i∈I Ti of objects in T is φ-acyclic.

Indeed, from the description of the derived functors given aboveRnφ(T) = Extn

C( , T)T . The fact that objects in T have no selfextensions, implies Rnφ(T) = 0, for n > 0. Since the objects in Thave projective resolutions consisting of finitely generated projec-tives, the functor Extn

C(T, ) commutes with arbitrary sums, thereforeExtn

C( ,∐

i∈I Ti)T = Rnφ(∐

i∈I Ti) = 0, for n > 0.(a22) Let Y . be an object in Db (Mod(T )) quasi-isomorphic to a complex

P. : · · · →∐

i∈I

( , Ti) →∐

j∈J

( , T j) → · · ·

of projective objects in Mod(T )

Then we claim that the complexes P. and R+,b φ(P. ⊗ T ) areisomorphic.By Theorem 3 the complex P. ⊗ T is:

P. ⊗ T : · · · →∐

i∈I

Ti →∐

j∈J

T j → · · · .

Since the objects in T are finitely generated, there is a naturalisomorphism

(

,∐

i∈I

Ti

)

T

∼=∐

i∈I

( , Ti)T , therefore: φ(P. ⊗ T ) ∼= P.


However we proved in part (a21), P. ⊗ T is a complex of φ-acyclicobjects, which according to [14] form an adapted class, hence thederived functor R+,b φ evaluated in (P. ⊗ T ) is just R+,b φ(P. ⊗ T ) =φ(P. ⊗ T ) ∼= P.

Now (a2) follows from the isomorphisms: R+,b φL−,b ⊗ T (Y) ∼=R+,b φL−,b ⊗ T (P.) ∼= R+,b φ(P. ⊗ T ) = φ(P. ⊗ T ) ∼= P. ∼= Y.

(a3) The derived functor R+,b φ is an embedding.(a31) For any object C in C the evaluation functor eC : Mod(C) → Ab is exact,

hence inducing a functor of triangulated categories

eC : Db (Mod(C)) → Db (Ab).

We claim that for any object C in C there is an isomorphism

RHom.(R+,b φ(C( , C)), )R+,b φ ∼= eC.

Indeed, from condition (iii’), C( , C) is isomorphic in Db (Mod(C)) to a complexof φ-acyclic objects:

T . : 0 → T0 → T1 → · · · → Tg → 0,

which implies that R+,b φ(C( , C)) = φ(T .) is a complex of projective objects inMod(T ).If X . in Db (Mod(C)) is quasi-isomorphic to a complex

Q. : 0 → Q0 → Q1 → · · · → Qi → · · ·in K+(Mod(C)), of φ-acyclic objects, then R+,b φ(X .) ∼= φ(Q.). Furthermore, byYoneda’s Lemma, and by using the fact that φ(T .) consists of projective objectsin Mod(T ), there are natural isomorphisms:

RHom.(R+,b φ(C( , C)), )R+,b φ(X .) ∼= RHom.(R+,b φ(C( , C)), )φ(Q.)

∼= RHom.(φ(T .), )φ(Q.)

∼= Hom.(φ(T .), φ(Q.))

∼= Hom.(T ., Q.).

Let us consider the bicomplex Hom.(T ., Q.). Since each Qi is φ-acyclic, thecomplex

Hom(T ., Qq) : 0→ Hom(Tg, Qq)→ · · ·→ Hom(T1, Qq)→ Hom(T0, Qq)→ 0

has cohomology:

Hi(Hom(T ., Qq)) ={

Hom(C( , C), Qq) = Qq(C) if i = 0;0 if i �= 0.

There is a collapsing spectral sequence {E∗pq} with E2

pq = Hvp Hh

q Hom(T ·, Q·),which converges to the homology of the total complex, Hom.(T ·, Q.),which implies that the homology of the total complex is the homologyof Hom(C( , C), Q·) = Q·(C) [17, 11.20]. We have proved the complexes


Hom.(φ(T ·), φ(Q.)) and eC(Q.) = Q.(C) are quasi-isomorphic, and in Db (Ab)

there are isomorphisms:

RHom.(R+,b φ(C( , C)), )R+,b φ(X)∼= Hom.(φ(T ·), φ(Q.))∼= eC(Q.)∼= eC(X .).

We can now prove (a3) below:If X . is a complex in Db (Mod(C)) such that R+,b φ(X) ∼= 0 in Db (Mod(T )),then for every C in C we have eC(X .) = X .(C) ∼= 0 in Db (Ab). Then X . is anacyclic complex.

(b) (i) It was proved in Eq. 1.3 that pdim{(( , C), )T } ≤ g. (ii) Let C and C′be objects in C. We compute Extn

T op(( , C), )T , (( , C′), )T ) using the pro-jective resolution of (( , C), )T given in Eq. 1.3, that is: Extn

T op(( , C), )T ,

(( , C′), )T ) is the n-th homology group of the complex

0 → ((T0, ), (( , C′), )T ) → · · · → ((Tg, ), (( , C′), )T ) → 0.

By Yoneda’s lemma, ExtnT op(( , C), )T , (( , C′), )T ) is the n-th homology

group of the complex:

0 → T0(C′) → T1(C′) → ... → Tg(C′) → 0.

However, this complex has non zero homology only in degree zero. It follows

ExtnT op(( , C), )T , (( , C′), )T ) = 0, for n > 0.

(iii’) Condition (i) implies the sequence

0 → (T, ) → (P0, ) → · · · → (P f , ) → 0

is exact with (Pi, ) ∈ add{(( , C), )T }, for i = 0, . . . , f .For the rest of the proof consider the exact sequence:

0 → ( , C) → T0 → T1 → · · · → Tg−1 → Tg → 0. (1.4)

In the proof of (b) of the Theorem 4, we obtained the exact sequence Eq. 1.3.By applying the functor (−, (( , C′), )T ), to the exact sequence Eq. 1.3 andthe functor (( , C′), ) to the exact sequence Eq. 1.4, and by comparing bothsequences, there is an induced isomorphism

C(C′, C) ∼= ((( , C), )T , (( , C′), )T ),

proving that Cop and {(( , C), )T }C∈C , are equivalent. ��

We will see next that for the proof we gave, it is necessary to assume condition(iii’); in fact we will see that if L− − ⊗T has finite cohomological dimension, thencondition (iii’) holds with g = dimL−,b − ⊗T .

We will need some preliminary results:

Lemma 2 Let M be in Mod(C) and N in Mod(Cop), D : Mod(C) → Mod(Cop), thefunctor given by (DM)(C) = Hom(M(C), Q/Z). There is then a natural isomorphism

ψ : HomC(M, DN) → HomCop(N, DM),

given by (ψ(η))C(x)(y) = (ηC)(y)(x), with η ∈ HomC(M, DN), x ∈ N(C) and y ∈M(C).


Proof We leave the proof to the reader. ��

We will need the following:

Lemma 3 Let D : Mod(C) → Mod(Cop) be the functor given by DM(C) =HomZ(M(C), Q/Z) and E a f initely presented object in Mod( C). Then, there is anatural isomorphism of functors

D( ) ⊗C E ∼= HomZ((E, ), Q/Z).

Proof Let ( , C1) → ( , C0) → E → 0 be a projective presentation of E. By apply-ing DB ⊗C − we obtain the following commutative diagram:

By applying the functor ( , B) to the projective presentation of E, we obtain byYoneda’s Lemma the following isomorphism of exact sequences:

By applying HomZ( , Q/Z) to the bottom sequence and by comparing it with theexact sequence above we obtain the desired isomorphism.

To prove the naturality of the isomorphism we make use of Lemma 2. We leavethe details to the reader. ��

We use this lemma to prove the following generalization of a well-known resulton modules.

Lemma 4 Any f initely presented f lat object in Mod(C) is projective.

Proof Let E be a finitely presented flat object and A → A′ → 0 an exact sequencein Mod(C). We want to prove that the sequence (E, A) → (E, A′) → 0 is exact. SinceE is flat, the sequence 0 → DA′ ⊗C E → DA ⊗C E is exact. It follows by Lemma 3that the sequence

0 → HomZ((E, A′), Q/Z) → HomZ((E, A), Q/Z)

is exact.Finally, from the fact Q/Z is an injective cogenerator in the category of abelian

groups, it follows that (E, A) → (E, A′) → 0 is exact. ��

We have the results needed to prove our proposition:


Proposition 5 Let T be a generalized tilting subcategory of Mod(C). If the functorL−(− ⊗ T ) has f inite cohomological dimension g. Then for each object C in C there isan exact sequence

0 → ( , C) → T0 → T1 → · · · → Tg−1 → Tg → 0,

with Ti ∈ T .

Proof Let

0 → ( , C) → T0 d1−→ T1 → · · · → Tn−1 dn−→ Tn → 0

be a resolution of ( , C), with Ti in T and n > g.Proceeding as in the proof of Proposition 4, we get a projective resolution of

(( , C), )T in Mod(T op)

0 → (Tn, ) → (Tg, ) → · · · → (T1, )(d1, )−−−→ (T0, ) → (( , C), )T → 0.

If L−,b ( − ⊗ T ) has finite cohomological dimension ≤ g, then there areisomorphisms:

Lm(− ⊗ T )(G)(C) = Hm(L−,b (G ⊗ T ))(C)

= TorTm (G,T )(C)

= TorTm ((( , C), ), G) = 0,

for m ≥ g + 1 and G ∈ Mod(T ).However, then for all G in Mod(T ) there are isomorphisms:

TorTg+1((( , C), )T , G) ∼= TorT1 (Ker(dg−1, ), G),

which means Ker((dg−1, )) is flat. There is an exact sequence

0 → (Tn, ) → (Tn−1, ) → · · · → Ker(dg−1, ) → 0,

which implies that Ker(dg−1, ) is finitely presented. By Lemma 4, it follows thatKer(dg−1, ) is projective, and the sequence

0 → Ker(dg, ) → (Tg, ) → Ker(dg−1, ) → 0

splits. Thus this implies the sequence 0 → Im(dg+1) → Tg → Im(dg) → 0 splits andIm(dg) ∈ T .

Finally the resolution

0 → ( , C) → T0 → · · · → Tg−1 → Im(dg) → 0

has the desired properties. ��

1.4 A Converse of Happel’s Theorem

In this subsection we will give a proof of a converse of the main theorem of theprevious subsection. To motivate the way in which we stated the theorem observethe following:


Remark 1 Let C be an annuli variety and T a tilting subcategory of Mod(C), denoteby p(C) and p(T ) the full subcategories of finitely generated projectives of Mod(C)

and Mod(T ), respectively. The homotopy categories Kb (p(C)) and Kb (p(T )) arefully embedded in Db (Mod(C)) and Db (Mod(T )), respectively [11]. It follows fromthe proof given above that the derived functor R+,b φ :Db (Mod(C)) → Db (Mod(T ))

is an equivalence, which induces an equivalence of the triangulated categories ofperfect complexes Kb (p(C)) and Kb (p(T )).

In the spirit of the paper by Cline, Parshall and Scott we state next the converse ofHappel’s Theorem, and follow in the proof the same line of arguments as they did.

Theorem 5 Let C andB be annuli varieties; p(C) and p(B) denote the full subcategoriesof f initely generated projectives of Mod(C) and Mod(B), respectively, and let

F : Mod(C) → Mod(B)

be a left exact functor. Assume the right derived functor R+,b F : Db (Mod(C)) →D(Mod(B)) restricts to a functor Db (Mod(C)) → Db (Mod(B)), inducing an equiv-alence of triangulated categories and an equivalence between the homotopy categoriesof perfect complexes Kb (p(C)) and Kb (p(B)). Then, B is equivalent to a generalizedtilting subcategory T of Mod(C), via a functor

F : B → T .

In particular, there exist equivalences of triangulated categories

R+,b F ′(Mod(C))R+,b F−−−→ Db (Mod(B))

R+,b F−−−→ Db (Mod(T )),

with F ′ : Mod(C) → Mod(T ), def ined by F ′(X) = ( , X)T .

Proof Let G be the quasi inverse of R+,b F. By [9, 3.1], the functor F has a left adjointG given by the composition

Mod(B)i−→ Db (B)

G−→ Db (Mod(C))H0−→ Mod(C).

By the uniqueness of the adjunction, the functors L−,b G and G are isomorphic.For each object B in B, let us consider the functor ( , B) ∈ Mod(B), and let TB be

the object TB = G( , B) and T the full subcategory of Mod(C) with objects {TB}B∈B.Since F is left exact, R0 F = H0(R+,b F) ∼= F. Moreover, since G and L−,b G are

isomorphic, for each object B there is an isomorphism of functors

R+,b FL−,b G( , B) ∼= ( , B). (1.5)

Since ( , B) is G-acyclic L−,b G( , B) ∼= G( , B). Computing the zero homologygroup in both sides of the equality 1.5 we get isomorphisms:

H0(R+,b FL−,b G( , B)) ∼= H0(( , B)) ∼= ( , B).

However, H0(R+,b FL−,b G( , B)) = (H0R+,b F)(L−,b G( , B)) = F(G( , B)), thatis:

FG( , B) ∼= ( , B). (1.6)


We saw above that there is another pair of adjoint functors:

φ : Mod(C) → Mod(T ),

− ⊗ T : Mod(T ) → Mod(C).

We want to prove that the compositions of functors

Mod(B)G−→ Mod(C)

φ−→ Mod(T ) and

Mod(T )−⊗T−−−→ Mod(C)

F−→ Mod(B)

induce an equivalence of categories. It will be sufficient to prove they induce anequivalence between the categories of finitely generated projectives p(B) and p(T )

[see 1, Proposition 2.6].Let ( , B) be in p(B). Then there exists a chain of isomorphisms:

(F ◦ − ⊗ T ) ◦ (φ ◦ G)( , B) = (F ◦ − ⊗ T )( , TB)T

= F(( , TB)T ⊗ T ))

= F(TB) = FG( , B) = ( , B), by Eq. 1.6.

For a functor ( , TB) in p(T ), there are isomorphisms:

(φ ◦ G) ◦ (F ◦ − ⊗ T )( , TB) = (φ ◦ G)(F(( , TB) ⊗ T ))

= (φ ◦ G)F(TB)

= (φ ◦ G)FG( , B)

= (φ ◦ G)( , B)

= φ(G( , B)) = φ(TB) = ( , TB).

Observe that the equivalence between p(B) and p(T ) induce an equivalencebetween B and T , given by the functors

G : B → T , G(B) = TB,

F : T → B, F(TB) = B,

such that the following diagram

commutes. Thus, we have functors

F : Mod(B) → Mod(T ),

G : Mod(T ) → Mod(B),

defined in objects as F(M)(TB) = M(F(TB)) and G(N)(B) = N(G(B)), with M inMod(B) and N in Mod(T ). It is easy to see that they are inverse equivalences.


Moreover, F|p(B) = φ ◦ G and G|p(T ) = F ◦ − ⊗ T . Hence, the compositions

Mod(C)F−→ Mod(B)

F−→ Mod(T ),

Mod(T )G−→ Mod(B)

G−→ Mod(T ),

give an adjoint pair, which induces an equivalence of triangulated categories

Db (Mod(C))R+,b F−−−→ Db (Mod(B))

R+,b F−−−→ Db (Mod(T )).

Then, for each X in Mod(C) and ( , TB) in Mod(T ), we have an isomorphisminduced by the adjunction

(( , TB), F F(X)) ∼= (GG( , TB), X).

However, we have the following chain of isomorphisms:

GG( , TB) = G(F ◦ − ⊗ T )( , TB)

= (G ◦ F)(( , TB) ⊗ T ) ∼= G(F(TB))

= G(FG( , B)) = G( , B) = TB,

and (( , TB), F F(X)) ∼= F F(X)(TB), by Yoneda’s lemma. Setting F ′ = F F one partof the theorem is proved.

We only must check that the subcategory T of Mod(C) is a tilting category.

(i) The representable functors ( , B) considered as complexes concentrated indegree zero are objects of Kb (p(B)). By the equivalence between Kb (p(B))

and Kb (p(C)) given by the functors R+,b F and L−,b G, there exists an objectP. in Kb (p(C)) such that P. ∼= L−,b G( , B) = G( , B) = TB, that is: there is aquasi-isomorphism P. → TB.The differentials in non-negative degrees of the complex

P. : 0 → Pm′dm′−→ Pm′+1 → · · · → P0

d0−→ P1 → · · · → Pm → 0,

are splittable epimorphisms; hence, each Im(di) is projective for m ≥ i ≥ 0.

It follows from the exact sequence 0 → Ker(d0)j−→ P0 → Im(d0) → 0 that

Ker(d0) is projective, and the composition τ≤0 P. → P. → TB is a quasi-isomorphism. In this way we can change P. for a complex in Kb (p(C)) con-centrated in degrees ≤ 0. This is a projective resolution for TB,

0 → Pm′ → · · · → Kerd0 → TB → 0.

By the first part of the proof, we can identify the categories T and B and as-sume that F = φ : Mod(C) → Mod(T ) induces an equivalence of triangulatedcategories

R+,b φ : Db (Mod(C)) → Db (Mod(T )),

also between the corresponding subcategories Kb (p(C)) and Kb (p(T )). Thus,− ⊗ T : Mod(T ) → Mod(C) is the right adjoint of φ. We had observed thefollowing isomorphism:

(φ ◦ − ⊗ T )( , T) ∼= ( , T).


Hence, φ(( , T) ⊗ T ) ∼= φ(T) = ( , T).(ii) In the proof of Happel’s Theorem 4 we make use of the isomorphism

Rnφ(T) = Hn(R+,b φ(T)) ∼= ExtnC( , T)T .

Using the equivalence, we obtain the isomorphism

R+,b φL−,b (− ⊗ T )( , T) ∼= ( , T),

and computing the n-th homology group in both sides of the equation, for n > 0we get:

Hn(R+,b FL−,b (− ⊗ T )( , T)) = Hn(( , T)) = 0.

However, there is a chain of isomorphisms:

Hn(R+,b φL−,b (− ⊗ T )( , T)) = (HnR+,b φ)(L−,b (− ⊗ T )( , T))

= (HnR+,b φ)(( , T) ⊗ T )) = (HnR+,b φ)(T)

= ExtnC( , T)T ,

which means that for any T ∈ T and n > 0, ExtnC( , T)T = 0 or rather T has

no self extensions.(iii) In a similar way, for each functor ( , C) ∈ Mod(C) considered as a complex in

Kb (p(C)) concentrated in degree zero, there exists a complex X in Kb (p(T ))

such that

( , C) ∼= L−,b (− ⊗ T )(X) = G(X)

in Db (Mod(C)). Since G send Kb (p(T )) to Kb (T ), there is a quasiisomorphism

with each Ti ∈ T .Let us define N = T0/Im(d−1). Since ( , C) ∼= Ker(d0)/Im(d−1), then N/

( , C) ∼= T0/Ker(d0) ∼= Im(d0). Therefore, there is a short exact sequence:

0 → ( , C) → N → Im(d0) → 0. (1.7)

(iii.a) We claim for any positive integer n, ExtnC(Im(d0), )T = 0. Indeed, let

us consider the exact sequence

0 → Im(d0) → T1 d1−→ T2 → · · · → Tg−1 dg−1−−→ Tg → 0. (1.8)

By the long homology sequence obtained after applying the repre-

sentable functor to the short exact sequence 0 → Im(dg−1) → Tg−1 dg−→Tg → 0, and part (ii), it follows that Extn

C(Im(dg−1), )T = 0. By back-ward induction, clearly Extn

C(Im(d0), )T = 0.(iii.b) Using part (iii.a), Extn

C(( , C), )T = 0 and the sequence in Eq. 1.7, itfollows that Extn

C(N, )T = 0.


(iii.c) Let us consider the exact sequence

0 → T−hd−h−→ T−h+1 → · · · → T−2 d−2−→ T−1 → N → 0.

It follows from the exact sequence 0 → Im(d−2) → T−1 → N → 0, the fact thatExtn

C(N, )T = 0 proved in (iii.b), and the long homology sequence, thatExtn

C(Im(d−1), )T = 0. Using backward induction again, it follows thatExtn

C(Im(d−h+1), )T = 0, and in particular Ext1C(Im(d−h+2, T−h)T = 0. Hence

the exact sequence

0 → T−h → T−h+1 → Im(d−h+2) → 0

splits, and Im(d−h+1) ∈ T . Continuing by induction we obtain Im(d−1) ∈ T , andfinally that the short exact sequence

0 → Im(d−1) → T−1 → N → 0

splits, proving that N ∈ T .We have gotten from Eqs. 1.7 and 1.8 an exact sequence:

0 → ( , C) → N → T1 → T2 → · · · → Tg → 0,

finishing the proof of the theorem. ��

2 Happel’s Theorem for Finitely Presented Functors

In this section we analyze the possibility of extending Happel’s theorem to thecategories of finitely presented functors.

We observe that given an annuli variety C and a tilting subcategory T of Mod(C),in order to have both categories of finitely presented functors mod(C) and mod(T )

abelian, we need to assume C and T have pseudokernels. We prove first that thiscondition is enough to restrict the functors φ and − ⊗ T to the categories of finitelypresented functors. Moreover, we will prove that it is equivalent to assume T haspseudokernels and to assume T is contravariantly finite in mod(C).

In order to have an analogous result to the main theorem of the previous section,we need to have enough injective objects in mod(C), to ensure this property, we willassume C is dualizing and T has pseudokernels. We will see that these conditionsare sufficient to obtain an equivalence of triangulated categories Db (mod(C)) ∼=Db (mod(T )). We end the section studying the case in which T is also a dualizingcategory, and we prove that in this case, there is an equivalence of triangulatedcategories Db (mod(Cop)) ∼= Db (mod(T op)).

Lemma 5 Let C be an annuli variety with pseudokernels and T a tilting subcategoryof mod(C) with pdimT = f . If T has pseudokernels, then for i > 0 the T -modules( , ( , C))T and Exti

C( , ( , C))T are f initely presented.


Proof We will make use of the fact that mod(T ) is abelian. For each C in C there is

a resolution 0 → ( , C)d0−→ T0 d1−→ T1 → · · · → Tg−1 dg−→ Tg → 0, which splits in short

exact sequences:

0 → Imdg−1 → Tg−1 dg−→ Tg → 0,

0 → Imdg−2 → Tg−2 → Imdg−1 → 0,

...

0 → ( , C) → T0 → Imd1 → 0.

By the long exact sequence, there are short exact sequences:

0 → ( , Imdg−1)T → ( , Tg−1)T( ,dg)−−−→ ( , Tg)T → Ext1

C( , Imdg−1)T → 0

0 → ( , Imdg−2)T → ( , Tg−2)T → ( , Imdg−1)T → Ext1C( , Imdg−2)T → 0

...

0 → ( , ( , C))T → ( , T0)T → ( , Imd1)T → Ext1C( , ( , C))T → 0. (2.1)

From the exact sequence (Eq. 2.1), it follows ( , Imdg−1)T and Ext1C( , Imdg−1)T

are finitely presented, and for i ≥ 2, ExtiC( , Imdg−1)T = 0. By applying induction it

follows that for g ≥ i > 0 the functors Ext1C( , Imdi−1)T and ( , ( , C))T are finitely

presented. Moreover,

for i ≥ 3, ExtiC( , Imdg−2)T = 0;

for i ≥ 4, ExtiC( , Imdg−3)T = 0;

...

for i ≥ g + 1, ExtiC( , ( , C))T = 0.

The above equalities together with the following isomorphisms:

Ext jC( , Imdg−1)T ∼= Ext j+1

C ( , Imdg−2)T , . . . , Ext jC( , Imd1)T ∼= Ext j+1

C ( , ( , C))T ,

for j ≥ 1, implies

ExtiC( , ( , C))T ∼=

{Ext1

C( , Imdi−1)T if g ≥ i > 0,

0 if i > g.

Hence, for i > 0, ExtiC( , ( , C))T is finitely presented. ��

Next we can prove the following result:

Proposition 6 Assume C is an annuli variety having pseudokernels and T a tiltingsubcategory of mod(C) with pdimT = f . If T has pseudokernels, then φ|mod(C) :


mod(C) → Mod(T ) has image in mod(T ), and in consequence, T is contravariantlyf inite in mod(C).

Proof Let M be a finitely presented C-module and

· · · ( , f3)−−→ ( , C2)( , f2)−−→ ( , C1)

( , f1)−−→ ( , C0) → M → 0

a projective resolution of M. Set Ki = Im( , fi) and split the resolution in short exactsequences: 0 → Ki+1

qi−→ ( , ( , Ci))pi−→ Ki → 0. Since pdimT = f , it follows from the

long homology sequence that for each i ≥ 1, there is an exact sequence

Ext f−1( , Ki+1)TExt f−1( ,qi)−−−−−−→ Ext f−1( , ( , Ci))T

Ext f−1( ,pi)−−−−−−→ Ext f−1( , Ki)T →

→ Ext f ( , Ki+1)TExt f ( ,qi)−−−−−→ Ext f ( , ( , Ci))T

Ext f ( ,pi)−−−−−→ Ext f ( , Ki)T → 0.

We claim that for each i, j ≥ 1, Ext j( , Ki)T is finitely presented. Since for j > f ,Ext j( , Ki)T = 0, we only need to prove the claim for 1 ≤ j ≤ f . Indeed, by Lemma 5we know that each Ext j( , ( , Ci))T is finitely presented; hence we have the following:

(1) Since for i, j ≥ 1, Ext fT ( , ( , Ki))T and Im(Ext j( , pi)) are epimorphic images of

finitely presented functors, they are finitely generated.(2) By (1), Ext f

C( , Ki+1)T is finitely generated; hence Im(Ext f ( , qi)) is finitelygenerated. Since for i ≥ 1, Ext f

C( , ( , Ki))T is the cokernel of a map betweena finitely generated object and a finitely presented object, it follows that fori > 0 each Ext f

C( , ( , Ki))T is finitely presented.(3) There is an exact sequence

0 → Im(Ext f ( , qi)) → Ext fC( , ( , Ci))T → Ext f

C( , Ki)T → 0.

By (2), Ext fC( , ( , Ki))T is finitely presented. Therefore, from the above exact

sequence and the fact that mod(T ) is abelian, it follows that Im(Ext f ( , qi)) isfinitely presented.

(4) Each Ext f−1C ( , Ki)T is an extension of the finitely presented functors

Im(Ext f−1( , pi)) and Im(Ext fC( , qi)); hence for i ≥ 1 the functor Ext f−1

C ( , Ki)Tis finitely presented. Continuing by backward induction we obtain the claim.

From the exact sequence 0 → K1 → ( , C0) → M → 0 and the long homologysequence we obtain an exact sequence

( , K1)T → ( , ( , C0))T → ( , M)T → Ext1C( , K1)T → Ext1

C( , ( , C0))T → · · · .

Using the fact ( , ( , C0))T , Ext1C( , K1)T and Ext1

C( , ( , C0))T are finitely presentedit follows ( , M)T is finitely presented, in particular finitely generated, which provesT is contravariantly finite in mod(C). ��

The converse of the previous proposition is contained in the following:

Proposition 7 Let C be an annuli variety having pseudokernels and T a full subcate-gory of mod(C). Then the following statements hold:

(a) If T is contravariantly f inite in mod(C), then T has pseudokernels.


(b) If T is covariantly f inite in mod(C), then T has pseudocokernels.(c) Assume T is a tilting subcategory of mod(C). The category T then has pseudok-

ernels if and only if it is contravariantly f inite in mod(C).

Proof

(a) Let T1g−→ T0 be a morphism in T . There is then an exact sequence in mod(C),

0 → Ker(g)j−→ T1

g−→ T0. Let T2h−→ Ker(g) be a T -approximation. Hence,

the sequence ( , T2)T( ,hj)−−→ ( , T1)T

( ,g)−−→ ( , T0)T is an exact sequence of T -modules, and we have proved T has pseudokernels.

(b) Is the dual of (a).(c) Follows from (a) and the previous proposition. ��

Let us assume C is a dualizing annuli variety and T a tilting subcategory of mod(C)

with pseudokernels. Then mod(C) has enough projective, proj(C), and enoughinjective objects, inj(C), while mod(T ) has enough projectives, proj(T ). Denote byK+,b (inj(C)) and K−,b (proj(T )) the corresponding homotopy categories of com-plexes with bounded homology. By standard arguments [see 12], there exist equiva-lences of triangulated categories: Db (mod(C)) ∼= K+,b (inj(C)) and Db (mod(T )) ∼=K−,b (proj(T )). The pair of adjoint functors φ|mod(C) and − ⊗ T |mod(T ) have de-rived functors R+,b φ : Db (mod(C)) → Db (mod(T )), L−,b − ⊗T : Db (mod(T )) →Db (mod(C)), forming an adjoint pair. All the line of arguments used to prove thegeneral version of Happel’s theorem can then be used to prove the following:

Theorem 6 (Happel) Let C be a dualizing variety and T a generalized tilting subcate-gory of mod(T ) satisfying condition (iii’ ) and having pseudokernels. Thus,

(a) the pair of functors R+,b φ : Db (mod(C)) → Db (mod(T )), L−,b − ⊗T :Db (mod(T )) → Db (mod(C)) induce an equivalence of derived categories,

(b) the full subcategory {(( , C), )T }C∈C of mod(T op) is a tilting category satisfyingthe stronger condition (iii ’), which is equivalent to the category Cop.

Observe that the theorem given above lacks symmetry; in order to recover it weneed to assume T is a dualizing variety. In the next propositions we give enoughconditions on T to be a dualizing variety. We will need Proposition 2.6 from [7],which proves that if C is a dualizing variety, then mod(C) is also dualizing.

Proposition 8 Let C be a dualizing variety and T a functorially f inite subcategory ofmod(C). The category T is then a dualizing variety.

Proof To simplify the notation, set B = mod(C). Since B is dualizing by [7, Theorem2.4] it has pseudokernels; hence, mod(B) is abelian. Let D : mod(B) → mod(Bop)

be a duality. Therefore, for each object T in T , the functor B( , T) is an object inmod(B), and DB( , T) is an object in mod(Bop).

Let

B(B1, ) → B(B0, ) → DB( , T) → 0

be the presentation of DB( , T) in mod(Bop).


(1) We claim DB( , T) is in mod(T op). Indeed, since T is covariantly finite, thereexists T0 in T and an exact sequence

B0g−→ T0 → Coker(g) → 0,

which induces a short exact sequence of T op-modules

0 → B(Coker(g), )T → B(T0, )T → B(B0, )T → 0.

In a similar way, using the fact T is contravariantly finite, there isa morphism Coker(g) → T ′

0, which induces an epimorphism B(T ′0, )T →

B(Coker(g), )T → 0. That is: there is a short exact sequence of T op-modules

B(T ′

0,)T → B(T0, )T → B(B0, )T → 0.

Interchanging B0 and B1, we obtain a short exact sequence

B(T ′

1,)T → B(T1, )T → B(B1, )T → 0.

Since finitely presented functors are closed under cokernels, it follows thatDB( , T) is finitely presented.

(2) If N is a finitely presented T -module, there is an exact sequence

B( , T1)T → B( , T0)T → N → 0.

Dualizing it, we get the exact sequence of T op-modules

0 → DN → DB( , T0)T → DB( , T1)T .

Since mod(T ) is abelian, and DB( , T0)T and DB( , T1)T are by (1) finitelypresented functors, it follows that DN is finitely presented.The proof that if N is a finitely presented T op-module, then DN is a finitelypresented T -module is dual. ��

We do not know the answer to the following:

Question Let C be a dualizing variety and T a tilting subcategory of mod(C),assuming that T is a dualizing variety, is T then functorially finite in mod(C)?

In the last theorem of the section we see that by assuming T is dualizing,for example if it is functorially finite, we recover the symmetry: there is a tiltingsubcategory of mod(T op) equivalent to Cop. More precisely:

Theorem 7 Let C be a dualizing variety and T a tilting subcategory of mod(C)

satisfying condition (iii’). Assume T is a dualizing variety, and let θ be the fullsubcategory of mod(T op) with objects {(( , C), )T }C∈C . The category θ is then atilting subcategory of mod(T op) equivalent to Cop and the functor φθ : mod(T op) →mod(θ) ∼= mod(Cop) induces an equivalence of triangulated categories R+,b φθ :Db (mod(T op)) → Db (mod(Cop)).

Proof Since θ and Cop are equivalent, θ is a dualizing variety; in particular it haspseudokernels, and we have the same conditions as in Theorem 6. ��


In the next section we will see that for dualizing varieties C, any tilting subcategoryof mod(C) satisfies condition (iii’).

3 Relations Between Tilting and Contravariantly Finite Categories

In the previous section we saw that for a dualizing variety C, there is a definitionof tilting category which allows us to prove a version of Happel’s theorem in thecategories of finitely presented functors. It is then natural to look for characteri-zations of such tilting categories. This question was considered in the case of artinalgebras by Auslander and Reiten in [6], where they used results from Auslander andBuchweitz [2].

Our aim in this section is to extend these results to dualizing varieties. Moreprecisely: let C be a Krull-Schmidt dualizing variety and T a tilting subcategory ofmod(C). We will see that if T is contravariantly finite in mod(C), then condition (iii’)is satisfied.

We will next give a bijection between tilting subcategoríes T of mod(C) withpdimT ≤ n, such that T is a generator in T ⊥, and covariantly finite coresolvingsubcategories Y such that pdim(⊥Y ) ≤ n.

The proof of these results will be along the lines of the papers [2] and [6]. For thebenefit of the reader, we sketch in the first two subsections the definitions and resultsfrom these papers that we will use, and show the modifications we need to do so thatthe theorems mentioned above hold.

3.1 Approximations and Projective Generators.

Through this subsection C is an annuli variety with pseudokernels. The definitionsgiven below are taken from the paper by Auslander and Buchweitz [2].

By a subcategory A of mod(C), we will understand A is full, closed under finitesums and isomorphic objects of mod(C).

We will say that a chain of morphisms · · · → Ai+1 → Ai → Ai−1 → · · · in A isexact , if it is an exact sequence in mod(C). For each object M in mod(C), we defineA − left.resol.dimM as the least n such that there is an exact sequence

0 → An → An−1 → · · · → A0 → M → 0,

with Ai in A , if such an integer exists. We say that A − left.resol.dimM < ∞ if A −left.resol.dimM = n for some non-negative integer n. The subcategory of mod(C)

consisting of the objects M in mod(C) such that A − left.resol.dimM < ∞ will bedenoted by Â .

Analogously we define A − right.resol.dimM as the least non-negative integer nsuch that there is an exact sequence 0 → M → A0 → · · · → An−1 → An → 0, withAi in A , if such an integer exists, and with ˇA we will denote the subcategory ofmod(C) with objects M such that A − right.resol.dimM < ∞.

For each non-negative integer n, we denote by An the full subcategory ofmod(C) whose objects M satisfy A − left.resol.dimM ≤ n, and put Â−1 = 0. Ob-serve that A0 = A and Âi−1 ⊂ Ai for i ≥ 0. Dually, for each non-negative inte-ger n we denote by An the full subcategory of mod(C), whose objects M satisfy


A − right.resol.dimM ≤ n, and we put ˇA−1 = 0. Similarly, A0 = A and ˇAi−1 ⊂ Ai

for i ≥ 0. Finally, a subcategory B of A is a generator for A , if for each object Ain A , there exists a short exact sequence 0 → A′ → B → A → 0 in A with B in B.We have dually the concept of a subcategory which is a cogenerator of A .

By Y we will denote in this section an additively closed subcategory closed underextensions, and by ω ⊂ Y , an additively closed generator of Y . We will only useright Y -resolutions; hence for any object M in mod(C), Y − resol.dimM will denotethe right resolution dimension of M.

Here we give some results from [2] with sometimes small variations.

Theorem 8 [2, Theorem 1.1] For each M in Y there are exact sequences

0 → M → Y M → X M → 0, (3.1)

0 → YM → XM → M → 0,

with YM and Y M in Y and X M and XM in ω.

Here there is a small variation in the way we state Theorem 8 :

Theorem 9 For each M in Yn, n ≥ 0 there are exact sequences

0 → M → Y M → X M → 0, (3.2)

0 → YM → XM → M → 0,

with YM and Y M in Y and X M in ωn−1, and XM in ωn.

For an object M in Y , the sequence 3.1 is called a left Y -approximation of M,while if M is an object in Yn the sequence 3.2 will be called a left Yn-approximation ofM. The connection between these sequences and contravariantly finite subcategoriesof mod(C) will be clarified below.

Let M and N be objects in mod(C). If there exists an integer n such that fori > n, Exti

C(M, N) = 0, then the least of such integers will be called the M-injectivedimension of N, denoted by M − inj.dimN, or the N- projective dimension of Mand denoted by N − proj.dimM, otherwise we define M − inj.dimN = ∞ = N −proj.dimM. If B is a subcategory of mod(C), then for each M in mod(C) we defineM − inj.dimB as the maximum of M − inj.dimB with B in B. Dually, for each N inmod(C), we define N − proj.dimB as the maximum of N − proj.dimB with B in B.Clearly M − inj.dimB = B − proj.dimM.

Assume A and B are subcategories of mod(C). We then define A − proj.dimBas the maximum of A − proj.dimB with A in A and B in B. Dually we define A −inj.dimB as the maximum of A − inj.dimB with A in A and B in B. It is clear againthat A − inj.dimB = B − proj.dimA .

If for such categories we have A − inj.dimB = 0 = B − proj.dimA , then we sayA is left orthogonal to B and B is right orthogonal to B. If A consists of all objectsA for which A − inj.dimB = 0, we say that A is the left orthogonal complement of Bin mod(C), which is denoted b y A =⊥ B. Dually, the right orthogonal complement


of A in mod(C) is denoted by A ⊥ and is the subcategory consisting of all objects Bin mod(C) with A − inj.dimB = 0.

It is easy to see that orthogonal complements are additive closed and closedunder extensions; the left orthogonal complement ⊥B is closed under kernels ofepimorphisms, while the right orthogonal complement A ⊥ is closed under cokernelsof monomorphisms. Moreover, ⊥B contains all projective objects of mod(C). Dually,A ⊥ contains all injective objects of mod(C).

Concerning the subcategories ω and Y of mod(C) we say ω is a projectivegenerator of Y if Y − proj.dimω = 0, that is: ω ⊂⊥ Y .

Our aim in this subsection is to prove ωn =⊥ Y ∩ Yn.

Proposition 9 [2, Proposition 2.1] Given an object M in an exact additive subcategoryexact Y of mod(C) and a projective generator ω of Y , for any integer n ≥ 0, thefollowing conditions are equivalent:

(a) Y − resol.dimM = n.

(b) M − proj.dimω = n.

(c) M − proj.dimω = n.

(d) For every X in ω, Extn+1C (X, M) = 0.

Corollary 3 [2, Corollary 2.2] Y − proj.dimω = 0.

Proof For every M in Y we have Y − resol.dimM = 0, and by Proposition 9 itfollows M − proj.dimω = 0. Hence, Y − proj.dimω = 0. ��

Observe that Y − proj.dimω = 0 means for i > 0 , ExtiC(ω,Y ) = 0. Since for all

n ≥ 0, ωn ⊂ ω, it follows ExtiC(ωn,Y ) = 0, for i > 0.

We have the following variation of Corollary 3.

Corollary 4 For any n ≥ 0, Y − proj.dimωn = 0.

In [2, Theorem 2.5] the following is proved:

Theorem 10 Let 0 → MjM−→ Y M → X M → 0 be a Y -approximation of M in Y .

Then for each Y in Y , the following statements hold:

(a) The sequence 0 → HomC(X M, Y) → HomC(Y M, Y) → HomC(M, Y) → 0 isexact.

(b) For any i > 0 the map jM induces an isomorphism

ExtiC(Y M, Y) ∼= Exti

C(M, Y).

Proof By Corollary 3 we know Y − proj.dimω = 0. Since X M is in ω, it implies thatfor all i > 0 Exti

C(X M, Y) = 0. ��

Using the fact that Y − proj.dimωn = 0, for n ≥ 0, we have the following result:


Theorem 11 Let 0 → MjM−→ Y M → X M → 0 be a Yn-approximation of M in Yn.

Then for any Y in Y the following is true:

(a) The sequence 0 → HomC(X M, Y) → HomC(Y M, Y) → HomC(M, Y) → 0 isexact.

(b) For any i > 0 the map jM induces an isomorphism

ExtiC(Y M, Y) ∼= Exti

C(M, Y).

In Proposition 9 it was proved that for any n ≥ 0 every M in Yn has a Yn-approximation 0 → M → Y M → X M → 0, with X M in ωn−1.

The following variation [2, Proposition 3.6] is very important in the situation weare considering, since it proves ωn =⊥ Y ∩ Yn in particular.

Proposition 10 For an object M in Yn the following statements are equivalent:

(a) M is an object in ωn.

(b) M is in ⊥Y ∩ Yn.

(c) If 0 → M → Y M → X M → 0 , is a Yn-approximation; that is: Y M is in Y anX M is in ωn−1, then Y M is in ω.

Proof Observe that if Y is in Y and Y − proj.dimY = 0 , then Y is in ω. Indeed,since ω is a projective generator of Y , there exists an exact sequence (∗)0 → Y ′ →W → Y → 0 in Y , with W in ω. If Y − proj.dimY = 0, then for i > 0 Exti

C(Y,Y ) =0, and the exact sequence (*) splits; hence, Y is a summand of W.

We have two different cases: n = 0 and n > 0.Assume n = 0.(a) implies (b).If M is in ω0 = ω, then by the definition of a projective generator, ω ⊂⊥ Y , that

is: Y − proj.dimM = 0.(b) implies (c).Since M is in Y , then Y − proj.dimM = 0 implies, by the above observation, M

is in ω. Consequently, if

0 → M → Y M → 0 → 0

is a Y0-approximation, then Y M = M is in ω.(c) implies (a) is clear.Assume n > 0.(a) implies (b).If M is in ωn, then by Corollary 4, Y − proj.dimM = 0.(b) implies (c).Suppose for i > 0, Exti

C(M,Y ) = 0. If 0 → M → Y M → X M → 0 is an Yn-approximation, that is: Y M is in Y and X M is in ωn−1, then by Corollary 4, we haveExti

C(X M,Y ) = 0. By the long homology sequence, it follows that ExtiC(Y,Y ) = 0

for i > 0. By the above observation, Y M is in ω. Hence, in the Yn-approximation0 → M → Y M → X M → 0, M is in ωn.

(c) implies (a) is clear. ��


By dual arguments, we have analogous results using left approximations, injectivecogenerators, etc.

3.2 Covariantly and Contravariantly Finite Subcategories of mod(C)

In the remaining of section C will be a dualizing Krull-Schmidt R-variety. Accordingto Theorem 2 mod(C) is a dualizing Krull-Schmidt R-variety. From the fact thatmod(C) is Krull-Schmidt, it follows any additively closed subcategory C of mod(C)

is Krull-Schmidt, and in consequence, the objects in mod(C ) have projective covers[16, Lemma 3.5].

We will see that under the above conditions, there is a bijection between thecovariantly finite coresolving and the contravariantly finite resolving subcategoriesof mod(C), generalizing results from [6].

We begin the subsection recalling some definitions from [6, 8].In this subsection X and Y denote subcategories of mod(C), closed under

summands and isomorphisms. A morphism f : X → M in mod(C), with X in X , is

a right X -approximation of M, if ( , X)X( ,h)X−−−→ ( , M)X → 0 is an exact sequence.

Dually, let Y be a subcategory of mod(C), a morphism g : M → Y with Y in Y is a

left Y -approximation of M, if (Y, )Y(g, )Y−−−→ (M, )Y → 0 is exact.

Recall from [5] that a morphism f : X → Y is said to be right minimal if anendomorphism g : X → X is an automorphism whenever f = fg. We have the dualconcept of a left minimal approximation and a left minimal morphism.

It follows from the definition that two minimal right X -approximations (left Y -approximations) are isomorphic.

We will use fM : XM → M (gM : M → Y M ) to denote a right X -approximationof M (left Y -approximation of M). A subcategory X of mod(C) is contravari-antly (covariantly) finite in mod(C) if every object M in mod(C) has a right X -approximation (left Y -approximation).

The results of this subsection will be proved closely following the paper [6]. Weneed to prove first the existence of minimal morphisms in mod(C).

Proposition 11 Let f : X → M be a map in mod(C). Then there exists a decomposi-tion of X such that X = B0

∐B1, and f |B0 is right minimal and f |B1 = 0.

Proof Let be F = Coker(( , X)( , f )−−→ ( , M)). Then there is an exact sequence in

mod(mod(C))

( , X)( , f )−−→ ( , M)

π−→ F → 0.

Since mod(C) is a Krull-Schmidt variety, there exist projective covers in the categorymod(mod(C)); hence, there is a decomposition of M = M1

∐M0, such that π0 =

π |( , M0) : ( , M0) → F → 0 is a projective cover of F. If K0 = Kerπ0, then thereis an epimorphism ( , X) → Kerπ0; thus a decomposition of X = X0

∐X1 such that


( , X0) → Kerπ0 is a projective cover. Therefore, there is the following commutativeexact diagram with splitting columns:

The map f decomposes as

f =(

f1 00 f0

)

: X1

∐X0 → M1

∐M0,

and ( , X0)( , f0)−−→ ( , M0)

π0−→ F → 0 is a minimal projective presentation of F and

( , X1)( , f1)−−→ ( , M1) → 0 is a splitting epimorphism. It follows the existence of a

decomposition X1 = X ′1

∐X ′′

1 such that f1 = (h 0) : X ′1

∐X ′′

1 → M1, and h is anisomorphism. Putting B0 = X0

∐X ′

1 and B1 = X ′′1 , we have

f |B0 =(

f0 00 h

)

,

and f |B1 = f1|B1 = 0.We claim f |B0 is right minimal. Let g be a map

g =(

a bc d

)

: X0

∐X ′

1 → X0

∐X ′

1

such that ( f |B0)g = f |B0. Therefore,(

f0a f0bhc hd

)

=(

f0 00 h

)

and f0a = f0. We have the following commutative diagram

By the properties of the minimal projective presentations, it follows ( , a) is anisomorphism, and since h is also isomorphism, it follows c = 0 and d = 1X ′

1. Therefore

g is an isomorphism. ��

Corollary 5 Let X be a closed under summands contravariantly f inite subcategory ofmod(C). Every object in mod(C) then has a minimal right X -approximation.

3.3 Covariantly Finite and Resolving Categories.

Let X and Y be subcategories of mod(C). It is easy to verify X ⊂⊥ (X ⊥) and Y ⊂(⊥Y )⊥, but we do not necessary have equality. When dealing with artin algebras,


conditions for these equalities are given in [6]. Using the same arguments, one caneasily verify that analogous results hold in mod(C).

Let us recall some definitions from [6].A subcategory X of mod(C) is resolving , if it satisfies the following three condi-

tions: (a) it is closed under extensions, (b) it is closed under kernels of epimorphisms,and (c) it contains the projective objects. Dually, a subcategory Y of mod(C), iscoresolving if it satisfies the dual conditions: (a) it is closed under extensions, (b)it is closed under cokernels of monomorphisms, (c) it contains the injective objects.For any subcategory Z of mod(C) the left (right) orthogonal complement ⊥Z (Z ⊥)is a resolving (coresolving) subcategory of mod(C). Following the proofs in [6] wehave the following:

Proposition 12 [6, Proposition 3.3] Let Y be a coresolving covariantly f inite subcat-egory of mod(C). Then the following statements are true:

(a) X =⊥ Y is a contravariantly f inite resolving subcategory of mod(C).(b) Y = X ⊥ = (⊥Y )⊥.(c) For each M in mod(C), there is a unique up to isomorphism exact sequence 0 →

Mg−→ Y M → X M → 0 satisfying the following:

(i) g is a minimal left Y -approximation of M.(ii) X M is in X .

(iii) g induces an isomorphism ExtiC(Y M, Y) ∼= Exti

C(M, Y), for all i > 0 andY in Y .

(d) For each M in mod(C), there is a unique up to isomorphism exact sequence 0 →YM → XM

f−→ M → 0 satisfying the following:

(i) f is a minimal right X -approximation of M.(ii) YM is in Y .

(iii) f induces an isomorphism ExtiC(X, XM) ∼= Exti

C(X, M), for all i > 0 andX in X .

Given a covariantly finite coresolving subcategory Y , we denote ⊥Y by X , andwe denote Y ∩ X by ω. With this notation we have the following:

Proposition 13 [6, Proposition 3.4] Let Y be a covariantly f inite coresolving subcat-egory of mod(C). Then ω = Y ∩ X has the following properties:

(a) ω is self-orthogonal.(b) For each X in X there is an exact sequence 0 → X → W → X ′ → 0 with W in

ω and X ′ in X .(c) For each Y in Y there is an exact sequence 0 → Y ′ → W → Y → 0 with W in ω

and Y ′ in Y .

Remark 2 We could have considered in Proposition 12 instead of a covariantly finitecoresolving subcategory Y of mod(C) a contravariantly finite resolving subcategoryX and obtain the dual statements. This proves that there exists a bijection betweenthe class of covariantly finite coresolving subcategories and the class of contravari-


antly finite resolving subcategories of mod(C) , given by Y �→⊥ Y and with inverseX �→ X ⊥.

3.4 Tilting and Covariantly Finite Categories.

In this subsection the generalization of some theorems given in [6] will continue; wewill concentrate on the relations between tilting subcategories and covariantly finitecoresolving subcategories of mod(C).

Let ω be a closed under summands self-orthogonal subcategory of mod(C). Definethe subcategory Yω of ω⊥ whose objects are the C-modules M for which there existsan exact sequence

· · · → Ti+1di−→ Ti → · · · → T1

d0−→ T0 → M → 0,

with Ti in ω, and Im(di) in ω⊥. Observe that by definition ω is a generator of Yω andsince Yω ⊂ ω⊥, then ω ⊂⊥ Yω; this means, ω is a projective generator of Yω.

We will assume ω is closed under summand. The following result will be veryuseful:

Proposition 14 Let ω be a self-orthogonal subcategory of mod(C). If ω is a generatorof ω⊥, then Yω = ω⊥.

Proof We only need to prove ω⊥ ⊂ Yω. Since ω is a generator of ω⊥, then for eachM in ω⊥, there is a short exact sequence 0 → M′

0 → W0 → M → 0 in ω⊥, with W0

in ω. By recursiveness there is an exact sequence 0 → M′i → Wi → M′

i−1 → 0 in ω⊥,with Wi in ω , for all i > 1. Thus, from the existence of the exact sequence, · · · →W1 → W0 → M → 0 it follows M is in Yω. ��

We will use the dual of [6, Proposition 5.1] which for the benefit of the reader westate next.

Proposition 15 Given a self-orthogonal category ω, the category Yω is closed underextensions, cokernels of monomorphisms and direct summands.

It is of interest to see when for some integer n, (Yω)n = mod(C). Since thiscondition will be used to relate covariantly finite coresolving subcategories and tiltingsubcategories of mod(C), we will use the following:

Proposition 16 Let Y be an additively closed and closed under extensions subcat-egory of mod(C) with a projective generator ω. If Yn = mod(C), then the followingstatements hold:

(a) Y is covariantly f inite in mod(C).(b) X =⊥ Y = ωn.


Proof

(a) If mod(C) = Yn, then each M in mod(C) has a left Yn-approximation 0 → M →Y M → X M → 0 with Y M in Y and X M in ωn−1. By Theorem 11, there is anexact sequence

0 → (X M, )Y → (Y M, )Y → (M, )Y → Ext1C(X M, )Y = 0,

which implies Y is covariantly finite.(b) By Proposition 10 we have ωn =⊥ Y ∩ Yn. However, if Yn = mod(C), then

ωn =⊥ Y ∩ mod(C) =⊥ Y . ��

For n ≥ 0, Pn(C) denotes the full subcategory of mod(C), whose objects haveprojective dimension ≤ n. Analogously In(C) denotes the full subcategory of mod(C)

whose objects have injective dimension ≤ n.

Proposition 17 Let Y be a covariantly f inite coresolving subcategory of mod(C), andX =⊥ Y the corresponding contravariantly f inite resolving subcategory of mod(C).Then Yn = mod(C) if and only if X ⊂ Pn(C).

Proof Assume first that Yn = mod(C). Let X be an object in X . We will prove thatfor i > n, Exti(X, Yn) = 0. Let M be in Yn. There is then an exact sequence

0 → M → Y0 d1−→ Y1 → · · · dn−1−−→ Yn−1 → Yn → 0.

From, the exact sequence 0 → Imdn−1 → Yn−1 → Yn → 0, and the long homologysequence, we obtain for i > 0 the exact sequence

0 = ExtiC(X, Yn) → Exti+1

C (X, Imdn−1) → Exti+1C (X, Yn−1) = 0,

and ExtiC(X, Imdn−1) = 0, for i > 1. Going backwards, we obtain in a similar way that

for i > n − 1, ExtiC(X, Imd1) = 0. Using this fact and the long homology sequence,

we have for i > n, ExtiC(X, M) = 0.

Assume now that X ⊂ Pn(C). Let M be in mod(C). We will prove M is in Yn. ByProposition 12 X is a contravariantly finite resolving subcategory of mod(C), andX ⊥ = (⊥Y )⊥ = Y . For i > 0 then, Exti+n

C (X , M) = 0.Consider the resolution of M

0 → M → I0 → · · · → In−1 → �−n M → 0,

with each Ii injective.Then for i > 0, Exti

C(X , �−n M) ∼= Extn+iC (X , M) = 0, which implies �−n M is in

X ⊥ = (⊥Y )⊥ = Y , and M is in Yn, since Y , being resolving, contains the injectiveobjects. ��


We have next the Proposition:

Proposition 18 Let Y be a covariantly f inite coresolving subcategory of mod(C). SetX =⊥ Y , and ω = Y ∩ X . The following statements are true:

(a) Y = ω⊥ if and only if X ∩ ω⊥ = ω.(b) If ⊥Y ⊂ Pn(C), then Y = ω⊥.

Proof

(a) If Y = ω⊥, then it is clear that X ∩ ω⊥ = ω. Since ω ⊂⊥ Y , then Y =(⊥Y )⊥ ⊂ ω⊥. We only need to prove that ω⊥ ⊂ Y whenever X ∩ ω⊥ = ω.Let M be in ω⊥. Since Y is covariantly finite and coresolving there is an exactsequence (*) 0 → M → Y → X → 0, with Y in Y and X in X . Since M andY are in ω⊥, it follows that X is in ω⊥; thus, X is in ω⊥ ∩ X = ω, which impliesthat the sequence (*) splits, and M is in Y .

(b) If ⊥Y is contained inPn(C), then it follows by Proposition 17 that mod(C) = Yn.By Proposition 16, we have the equality ωn =⊥ Y ∩ Yn =⊥ Y ; thus X ∩ ω⊥ =ωn ∩ ω⊥. By part (a) it is enough to verify ω = ωn ∩ ω⊥. Clearly, ω ⊂ ωn ∩ ω⊥.Let Y be in ωn ∩ ω⊥. We claim Y is in ω. We use induction on n. If n = 0, thenω0 = ω, and the claim is true. Suppose the claim is true for n − 1 > 0. Let Y bein ωn ∩ ω⊥. There is then an exact sequence

0 → Y → W0d1−→ W1 → · · · → Wn−1

dn−→ Wn → 0,

with Wi in ω. Since Y and W0 are in ω⊥, and ω⊥ is closed under cokernelsof monomorphisms, then Im(d1) is in ω⊥. It follows W ′ = Im(d1) is in ωn−1 ∩ω⊥ and, by induction hypothesis, W ′ = Im(d1) is in ω; hence there is an exactsequence

0 → Y → W0 → W ′ → 0 ,

with Y in ω⊥ and W ′ in ω. The short exact sequence above splits and Y isa summand of ω. By assumption ω is closed under summands, and we haveproved Y is in ω, as required. ��

Remark 3 With the notation of this section, we see that it is equivalent to say ω isa tilting subcategory of mod(C) to saying that ω is a self-orthogonal subcategory ofmod(C) with pdimω ≤ n and such that ω contains the projective objects. By Corollary2 the functor Exti

C(ω, ) commutes with arbitrary sums, in particular ω is closed undersummands.

We have the following version of the results in [6].

Proposition 19 Let ω be a self-orthogonal subcategory of mod(C). Then the followingstatements hold:

(a) If ω is a tilting subcategory of mod(C) with pdimω ≤ n , and ω is a generator ofω⊥, then (Yω)n = mod(C).


(b) If there is an equality (Yω)n = mod(C), then the following is true:

(i) Yω is covariantly f inite and coresolving.(ii) The subcategory ω of mod(C) is tilting, with pdimω ≤ n, and ωn contains the

projective objects.

Proof

(a) Let us suppose ω is a tilting subcategory and, ω is a generator of ω⊥. Therefore,by Proposition 14, Yω = ω⊥. Let M be an object in mod(C). We will prove M isin (ω⊥)n. Since pdimω ≤ n, for i > n, Exti

C(ω, M) = 0.Let 0 → M → I0 → I1 → · · · → �−n M → 0 be a resolution of M with each Ii

injective. Then for i > 0, ExtiC(ω, �−n M) = Exti+n

C (ω, M) = 0. Hence �−n M isin ω⊥ and M is in (ω⊥)n.

(b)

(i) Let us suppose (Yω)n = mod(C). It follows by Proposition 16, that Yω iscovariantly finite and ⊥Yω = ωn. By Proposition 15, then Yω is closedunder extensions, cokernels of monomorphisms and direct summands. Weneed to see that Yω contains the injective objects. Since (Yω)n = mod(C),then for any injective object I there is a splitting exact sequence 0 → I →Y → Y ′ → 0 with Y in Yω and Y ′ in (Yω)n−1. Since Yω is closed undersummands, I is in Yω.

(ii) By (i), the category Yω is covariantly finite, coresolving and ⊥Yω = ωn. ByProposition 17, pdim(⊥Yω) = n. Thus, pdim(ωn) = pdim(⊥Yω) = n, andsince ω ⊂ ωn, then pdimω ≤ n. It is clear that every project object is in⊥Yω = (ω)n.

��

Let Λ be an artin algebra and T a self-orthogonal module in mod(Λ). It wasproved in [6, Teo.5.5] that the assignments: T �→ T⊥, Y �→ Y ∩⊥ Y induce abijection between the class of tilting modules and the class of all covariantly finitecoresolving subcategories mod(Λ). The main theorem of the section is the followinganalogous result:

Theorem 12 The assignments ω �→ ω⊥ and Y �→ Y ∩⊥ Y induce a bijection betweenequivalence classes of tilting subcategories ω of mod(C), with pdimω ≤ n, such that ω

is a generator of ω⊥ and classes of subcategories Y of mod(C) which are covariantlyf inite, coresolving, and whose orthogonal complement ⊥Y is contained in Pn(C).

Proof

(1) Let ω be a tilting subcategory of mod(C) with pdimω ≤ n and ω a projectivegenerator of ω⊥. By Proposition 14, Yω = ω⊥, and by Proposition 19, (Yω)n =mod(C), which in turn implies Yω = ω⊥ is covariantly finite and coresolving. ByProposition 17, it follows that ⊥Yω ⊂ Pn(C).

(2) Let Y be a covariantly finite coresolving subcategory of mod(C), such that⊥Y ⊂ Pn(C) and ω =⊥ Y ∩ Y . Then by Proposition 17, (Y )n = mod(C).Moreover, we have the inclusion Y ⊂ Yω by Proposition 13, it follows that


(Yω)n = (Y )n = mod(C). By Proposition 19, ω is a tilting subcategory ofmod(C), with pdimω ≤ n, and Yω is a covariantly finite coresolving subcategoryof mod(C). By Proposition 16, we have ⊥Yω =⊥ Y = (ω)n, and in consequence,Y = Yω.

By Proposition 18, condition ⊥Y ⊂ Pn(C) implies Y = ω⊥. Therefore, ω is agenerator of Yω = Y = ω⊥.

To prove that the assignment is a bijection we verify that the compositions

ω �→ ω⊥ �→ (ω⊥) ∩⊥ (ω⊥), Y �→ (Y ∩⊥ Y ) �→ (Y ∩⊥ Y )⊥

are the identity.Let ω be a tilting subcategory of mod(C) with pdimω ≤ n. We proved in (1)

Yω = ω⊥ is a covariantly finite coresolving subcategory such that (Yω)n = mod(C).We claim Yω ∩⊥ Yω = ω. By Proposition 16, we have ⊥Yω = ωn. Hence, by the proofof Proposition 18 we have equalities: Yω ∩⊥ Yω = ωn ∩ ω⊥ = ω.

Let Y be a covariantly finite subcategory of mod(C) such that ⊥Y ⊂ Pn(C). Bypart (b) of Proposition 18, the equality ω =⊥ Y ∩ Y implies that ω⊥ = Y . ��

We end the subsection stating the dual of Theorem 12.

Theorem 13 The assignments ω �→ ω⊥ and X �→ X ∩ X ⊥ induce a bijection be-tween the equivalence class of cotilting subcategories ω of mod(C), with idimω ≤ n,such that ω is a cogenerator of ⊥ω and classes of subcategories X of mod(C) which arecontravariantly f inite, resolving, and whose orthogonal complement X ⊥ is containedin In(C).

Corollary 6 Let ω be a tilting subcategory of mod(C), such that ω is a generator of ω⊥.If gdimC < ∞, then ω is a cotilting subcategory of mod(C).

Proof We know Yω is a covariantly finite coresolving subcategory of mod(C). SincegdimC < ∞, we establish that X =⊥ Yω ⊂ Im(C) is a contravariantly finite resolvingsubcategory of mod(C), with m = gdimC. By Theorem 12, we know ω = Yω ∩⊥ Yω.However Yω ∩⊥ Yω = X ∩ X ⊥ implies, by Theorem 13, that ω is a cotilting subcat-egory of mod(C). ��

3.5 Tilting Subcategories in mod(C).

In this subsection we will use the results from the previous one to prove that acontravariantly finite tilting subcategory T of mod(C) satisfies condition (iii’). Wewill need the following:

Proposition 20 Let T be a contravariantly f inite tilting subcategory of mod(C). Thenfor each M in T ⊥ there is a short exact sequence

0 → M′ → Th−→ M → 0,

such that, T is a T -approximation of M, and M′ ∈ T ⊥. In other words, T is a generatorof T ⊥, and in consequence, YT = T ⊥.


Proof Let M be an object in T ⊥, and ( , C) → M → 0 its projective cover. Morever,( , C) is in T , and hence, we have the commutative exact diagram

with Ti in T , and L = Im(g). Since L is in T, then for i > 0, Exti(L,T ⊥) = 0. Thus,

the exact sequence 0 → M → W → L → 0 splits, and there is an epimorphism T0f−→

M → 0.Since T is covariantly finite in mod(C) we can choose a minimal right T -approxi-

mation Th−→ M. Then there exists a morphism T0

k−→ T such that the followingdiagram commutes:

It follows that the minimal right T -approximation h is an epimorphism.We have an exact sequence

0 → Ker(h) → Th−→ M → 0,

with h a minimal right T -approximation; however by Wakamatsu’s Lemma, we haveExt1

C(T , Ker(h)) = 0.We have a short exact sequence

0 → ( , Ker(h))T → ( , T)T( ,h)−−→ ( , M)T → 0.

By the long homology sequence, for i > 0, Exti(T , Ker(h)) = 0 or Ker(h) is in T ⊥.The rest of the proposition was proved in Proposition 14. ��

We have now the main theorem of the subsection:

Theorem 14 Let C be a dualizing Krull-Schmidt R-variety and T a tilting subcategoryof mod(C) with pseudokernels. Then T satisf ies condition (iii’).

Proof It follows from Propositions 19 and 20. ��

We can get rid of condition (iii’) to have the following:

Corollary 7 (Happel) Let C a dualizing Krull-Schmidt R-variety and T a generalizedtilting subcategory of mod(C) with pseudokernels. Then

(a) the pair of functors R+,b φ : Db (mod(C)) → Db (mod(T )), L−,b − ⊗T :Db (mod(T )) → Db (mod(C)) induce an equivalence of derived categories,


(b) the full subcategory θ = {(( , C), )T }C∈C of mod(T op) is a tilting category satis-fying the stronger condition (iii ’), which is equivalent to the category Cop.

To prove the remaining results of the subsection we need the following result from[6, Proposition 5.9]:

Proposition 21 Let (F, G) be a pair of adjoint functors F : mod(C) → mod(T ),G : mod(T ) → mod(C). Let Y be a subcategory of mod(C), such that the naturalmorphism GF → I is an isomorphism on objects from Y , and Z is a subcategory ofmod(T ) such that the natural morphism I → FG is an isomorphism in objects fromZ , and assume that F(Y ) ⊂ Z and G(Z ) ⊂ Y . Then the following is true:

(a) If Z is contravariantly f inite in mod(T ), then Y is contravariantly f inite inmod(C).

(b) If Y is covariantly f inite in mod(C), then Z is covariantly f inite in mod(C).

Let us suppose T is a tilting contravariantly finite subcategory of mod(C), withpdimT = f . By Proposition 20, T is a generator of T ⊥. Then, by Proposition 19, (T ) f

contains the projective objects of mod(C), and YT = T ⊥ is covariantly finite core-solving. It follows by Theorem 4 that the category with objects θ = {(( , C), )T }C∈C isa tilting subcategory of mod(T op). Therefore, if T is functorialy finite, T is dualizingby Proposition 8; and in consequence Dθ is a cotilting subcategory of mod(T ).

In fact, we have the following:

Proposition 22 Let T be a tilting subcategory of mod(C), with pdimT = f . If T isfunctorialy f inite, then the following statements hold:

(a) The pair of adjoint functors (φ, − ⊗ T ) induce an equivalence of categoriesbetween YT ⊂ mod(C) and ⊥ Dθ ⊂ mod(T ).

(b) Dθ is a generator of ⊥ Dθ , and in consequence, ⊥ Dθ is contravariantly f inite.(c) YT is functorially f inite and coresolving.

Proof

(a) Let us suppose M is in YT . There is then an exact sequence

T. → M → 0 : · · · → T2d1−→ T1

d0−→ T0 → M → 0, (3.3)

such that Im(di) is in T ⊥. After applying φ to the exact sequence Eq. 3.3, we getan exact sequence

( , T.) → φ(M) → 0 : · · · ( ,d1)−−→ ( , T1)( ,d0)−−→ ( , T0) → ( , M)T → 0,

giving a projective resolution of φ(M). Let D(( , C), )T be an object in Dθ . Wethen have isomorphisms:

ExtiT (φ(M), D(( , C), )T ) = Hi(HomT (( , T.), D(( , C), )T ) = Hi(DT.(C)).

Since Hi(T.) = 0 if |i| > 0, it follows Hi(DT.) = 0 if |i| > 0, and for each C in Cand i > 0, Exti

T (φ(M), D(( , C), )T ) = 0, which proves φ(M) is in ⊥ Dθ . Since⊗T is right exact, M ∼= φ(M) ⊗ T .


Let us suppose N is a T -module in ⊥ Dθ . Then for all i > 0 and C in C,Exti

T (N, D(( , C), )T ) = 0. Let us consider a projective resolution of N

( , T.) → N → 0 : · · · → ( , T1)( ,d0)−−→ ( , T0) → N → 0. (3.4)

Then there are isomorphisms

ExtiT (N, D(( , C), )T ) = Hi(HomT (( , T.), D(( , C), )T ) = Hi(DT.(C)),

and it follows that for the complex

T. : · · · → T3d2−→ T2

d1−→ T1d0−→ T0 → 0 → · · ·

we have Hi(T.)=0, if |i| > 0. Putting Ki = Imdi, we split the sequence in shortexact sequences 0 → Ki+1 → Ti+1 → Ki → 0, for i ≥ 0. We claim Ki is in T ⊥.Indeed, since there are no self extensions in T and pdimT = f , there is a chainof isomorphisms:

Ext1C(T , K0) ∼= Ext2

C(T , K1) ∼= Ext3C(T , K2) ∼= · · · ∼= Ext f+1

C (T , K f ) = 0.

Analogously, for j > 1 there are isomorphisms:

Ext jC(T , K0) ∼= Ext j+1

C (T , K1) ∼= Ext j+2C (T , K2) ∼= · · · ∼= Ext f+1

C (T , K j+ f+1) = 0.

Therefore K0 is in T ⊥. It follows by induction that for i > 1, Ki is in T ⊥.By applying ⊗T to the sequence 3.4 we get the exact sequence

· · · → T1d0−→ T0 → N ⊗ T → 0.

By applying HomC(T , ) to the short exact sequences

0 → K0 → T0 → N ⊗ T → 0,

0 → K1 → T1 → K0 → 0,

it follows, by the long homology sequence and the fact that K0 and K1 are inT ⊥, that

0 → ( , K1)T → ( , T1)T → ( , T0)T → ( , N ⊗ T )T → 0 (3.5)

is exact, and ExtiC(T , N ⊗ T ) = 0, for i > 0, that is: N ⊗ T is in T ⊥. It follows

by Eqs. 3.5 and 3.4 that φ(N ⊗ T ) ∼= N.(b) Observe first, for any C in C, φ(D(C, )) ∼= D(( , C), )T . Let N be in ⊥ Dθ . Then

for the equivalence given in part (a), there is an object M in T ⊥, with N ∼=φ(M). Let (C, ) → DM → 0 be the projective cover of DM. There is then ashort exact sequence

0 → Mf−→ D(C, ) → Coker( f ) → 0.

Observe that D(C, ) is injective. Since both M and D(C, ) are in T ⊥, Coker( f )is in T ⊥. Set N′ = φ(Coker( f )). It follows by the long homology sequence thatthe sequence

0 → ( , M)T → ( , D( , C))T → ( , Coker( f ))T → Ext1( , M)T = 0,


is exact. Hence,

0 → N → D(( , C), )T → N′ → 0

is exact. In other words, Dθ is a cotilting subcategory of mod(T ) such thatDθ is a cogenerator of ⊥ Dθ . It follows by Theorem 13 that ⊥ Dθ = XDθ iscontravariantly finite.

(c) By Proposition 21, (c) follows from (a) and (b). ��

Question In view of this theorem it is natural to ask the following question: Let Tbe a contravariantly finite subcategory of mod(C). If YT is coresolving functoriallyfinite, then is T functorially finite?

Acknowledgement Martin Ortiz-Morales thanks CONACYT for the financial support given tohim during his graduate studies.

References

1. Auslander, M.: Representation theory of Artin algebras I. Commun. Algebra 1(3), 177–268(1971)

2. Auslander, M., Buchweitz, R.: The homological theory of maximal Cohen–Macaulay approxi-mations. Colloque en l’honneur de Pierre Samuel (Orsay 1987). M’em. Soc. Math. France (N.S.)38, 5–37 (1989)

3. Auslander, M.: Representation dimension of Artin algebras I. Queen Mary College MathematicsNotes. Selected Works of Maurice Auslander Part 1, pp. 505–574. AMS (1971)

4. Anderson, F.W., Fuller, K.R.: Rings and categories of modules. Graduate Texts in Mathematics,p. 13. Springer (1974)

5. Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras, no. 36. Cam-bridge University Press (1995)

6. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1),111–151 (1991)

7. Auslander, M. Reiten. I.: Stable equivalence of dualizing R-varieties. Adv. Math. 12(3), 306–366(1974)

8. Auslander, M., Smalø, S.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454(1981)

9. Cline, E., Parshall, B., Scott, L.: Derived categories and Morita theory. J. Algebra 104, 397–409(1986)

10. Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras.London Mathematical Society Lecture Note Series, no. 119. Cambridge University Press (1988)

11. Hartshorne, R.: Residues and duality. Lecture Notes in Math, vol. 20, pp. 19–84. Springer, Berlin(1966)

12. Miyachi, J.: Derived categories with applications to representations of algebras. www.u-gakugei.ac.jp/∼miyachi/papers/ChibaSemi.pdf

13. Gelfand, S.I., Manin, Yu I.: Methods of homological algebra. EMS, vol. 38, pp. 86–139. Springer(1996)

14. Milicic, D.: Lectures on derived categories. www.math.utah.edu/∼milicic/Eprints/dercat.pdf15. Martínez-Villa, R., Ortiz-Morales, M.: Tilting theory and functor categories III. The Maps Cate-

gory. Int. J. Algebra, 5(11), 529–561 (2011)16. Martínez-Villa, R., Solberg, Ø.: Graded and Koszul categories Applied Categorical Structures.

doi:10.1007/s10485-009-9191-617. Rotman, J.J.: An introduction to homological algebra. Pure and Applied Mathematics, vol. 85,

pp. 608–688. Academic Press (1979)18. Stenström, B.: Rings of Quotients, no 217. Springer, New York (1975)19. Verdier, J.L.: Des catégories dériv ées des catégories abéliennes. Edited and with a note by

Georges Maltsiniotis, no. 239. Asterisque (1996)20. Charles, Ch.A.: An Introduction to Homological Algebra, vol. 38. Cambridge Studies in Ad-

vanced Mathematics (1994)

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