Tilting categories with applications to stratifying systems

Journal of Algebra 302 (2006) 419 449 www.elsevier.com/locate/jalgebra Tilting categories with applications to stratifying systems Octavio Mendoza a,, Corina Sáenz b a Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, CP 04510, México D.F., Mexico b Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, Ciudad Universitaria, C.P. 04510, México D.F., Mexico Received 8 June 2005 Available online 24 January 2006 Communicated by Kent R. Fuller Dedicated to María Ines Platzeck on her 60th birthday Abstract In the study of standardly stratified algebras and stratifying systems, we find an object which is either a tilting module or one whose properties strongly remind us of a tilting module. This tilting module appeared already in Dlab and Ringel s work on quasi-hereditary algebras (see [V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: Repr. Theory and Related Topics, in: London Math. Soc. Lecture Note Ser. 168 (1992) 200 224] and [C.M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991) 209 223]). Also, this tilting module appears on the work on standardly stratified algebras of I. Ágoston, D. Happel, E. Lukács, and L. Unger, and in the paper of M.I. Platzeck and I. Reiten (see [I. Ágoston, D. Happel, E. Lukács, L. Unger, Standardly stratified algebras and tilting, J. Algebra 226 (2000) 144 160] and [M.I. Platzeck, I. Reiten, Modules of finite projective dimension for standardly stratified algebras, Comm. Algebra 29 (3) (2001) 973 986]). Inspired by them, we introduce the notion of tilting category in order to give a unified approach of these situations for stratifying systems. To do so, we use the ideas of M. Auslander, O. Buchweitz and I. Reiten related to approximation theory (see [M. Auslander, R.O. Buchweitz, The homological theory of maximal Both authors thank the financial support received from Project PAPIIT-UNAM IN115905. * Corresponding author. E-mail addresses: omendoza@matem.unam.mx, omendoza@math.unam.mx (O. Mendoza), ecsv@lya.fciencias.unam.mx (C. Sáenz). 0021-8693/$ see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2006.01.006

420 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 Cohen Macaulay approximations. Mem. Soc. Math. Fr. (N.S.) 38 (1989) 5 37; M. Auslander, I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991) 111 152]). 2006 Elsevier Inc. All rights reserved. 0. Introduction Let R be a finite-dimensional and basic k algebra over an algebraically closed field k. Let (X, ) be a total ordered set which is in bijective correspondence with the iso-classes of simple R-modules. For each i X, we denote by S(i) the simple R-module corresponding to i, and P(i) the projective cover of S(i). The standard module R Δ(i), which depends on the poset (X, ), is the maximal quotient of P(i) with composition factors amongst S(j) with j i. Let R Δ ={ R Δ(i)} i X, and F( R Δ) be the full subcategory of R-modules whose objects are all the R-modules having a R Δ-filtration. The algebra R is called standardly stratified if R R F( R Δ), and a standardly stratified algebra R is called quasi-hereditary if dim k End( R Δ(i)) = 1 for all i X. Quasi-hereditary algebras were introduced by E. Cline, B.J. Parshall and L.L. Scott in [6] and standardly stratified algebras by I. Ágoston, V. Dlab and E. Lukács in [2]. Observe that in [2], instead of considering a partial order on the iso-classes of simple modules (as was done in [6]), it is considered a total order. In 1991, C.M. Ringel introduced in [17] the characteristic module T associated to a quasi-hereditary algebra. Moreover, the characteristic module associated to a standardly stratified algebra was studied; on one hand, by I. Ágoston, D. Happel, E. Lukács and L. Unger in [3], and, on the other hand, by M.I. Platzeck and I. Reiten in [16]. This module T is a generalized tilting module and has the property that the endomorphism ring of T is again a quasi-hereditary algebra (respectively a standardly stratified algebra). Furthermore, the characteristic module T is very closely connected with homological properties of F( R Δ), and with the computation of the finitistic or the global dimension of R. Later on, given an algebra R, K. Erdmann and C. Sáenz in [8] introduced the concept of stratifying system (θ, Y, ) of size t, where is a total order on the set [1,t]= {1, 2,...,t}. The set θ ={θ(i)} t i=1 consists of non-zero R-modules and Y ={Y(i)}t i=1 of Ext-injective indecomposable R-modules in the category F(θ) whose objects are the R- modules having a θ-filtration. Moreover, they showed in [8] that the algebra A = End R (Y ) is standardly stratified, where Y := t i=1 Y(i). Afterwards, in [11] E.N. Marcos, O. Mendoza and C. Sáenz gave a characterization of the notion of stratifying system of size t, depending only on the system (θ, ). In fact, for any algebra, we can always consider a stratifying system. For instance, let s be the number of iso-classes of simple R-modules and X ={1, 2,...,s}. Then the pair ( R Δ, ) is always a stratifying system of size s, and for this reason, we say that ( R Δ, ) is the canonical stratifying system. In this way, we have that the notion of stratifying system (θ, ) generalizes the notion of the standard modules RΔ which depend on the poset (X, ). Furthermore, the module Y, which is related to the triple (θ, Y, ), is the corresponding generalization for stratifying systems of the characteristic module T. In general, the module Y is not a generalized tilting even not a partial tilting R-module. However, under certain conditions we will get that Y is a generalized tilting module (see Theorem 6.7).

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 421 Recently, in order to calculate the global dimension of the Schur algebra for GL 2 and GL 3, A.E. Parker introduced in [15] the notion of R Δ-good filtration dimension for quasihereditary algebras. Afterwards, B. Zhu and S. Caenepeel did the same in [5] for standardly stratified algebras. Since, as we have seen above, the notion of stratifying system generalizes the concept of standardly stratified algebra, it would be very interesting to extend such filtration dimension and to get one that makes sense in any algebra. Moreover, we would like to know if the generalized characteristic module Y is closely connected with homological properties of F(θ) and mod R. In order to give a unified approach of those situations for stratifying systems, we introduce in this paper the notion of tilting category. To obtain this new approach, we use the ideas due to M. Auslander, O. Buchweitz and I. Reiten related to approximation theory. The idea that we had in mind when we started working in this paper was to get a definition of tilting category in such a way that our typical examples (a) addt for any generalized tilting R-module, and (b) F( R Δ) for any standardly stratified algebra fitted into this definition. The theory that we develop here can be applied to stratifying system, on one hand, and to the theory of classical tilting modules on the other hand. For example, if (θ, ) is a stratifying system of size t, and I(θ) := I F(θ) is a coresolving subcategory of mod R, then we prove that pd F(θ) t and that F(θ) is a partial tilting category. So, in many cases, the stratifying systems are an important source of examples of tilting and partial tilting categories. 1. Preliminaries We start this section by collecting all the background material that will be necessary for the development of the paper. First, we introduce some general notation; afterwards, we recall the definition of contravariantly finite and of resolving subcategories in mod R and also the definition of relative projective dimension and resolution dimension of a given module. In this paper, algebra means finite-dimensional basic algebra over an algebraically closed field k.letr be an algebra, R-module means finitely generated left R-module. We denote by mod R the category of all finitely generated left R-modules and by D :modr mod R op the usual duality Hom k (,k). Given morphisms f : M N and g : N L in mod R, we denote the composition of f and g by gf which is a morphism from M to L. Throughout the paper, all the subcategories of mod R to be considered will be full and closed under isomorphisms. We denote by P R (respectively I R ) the category of all projective (respectively injective) R-modules. Let X be a subcategory of mod R. Associated to X, we have the following subcategories of mod R: I X = { M mod R: Ext 1 R (,M) X = 0 }, P X = { M mod R: Ext 1 R (M, ) X = 0 }, ω X := I X X and Z X := P X X.

422 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 It is clear that P R P X and I R I X. Moreover, P X and I X are closed under extensions and direct summands. Let R be an algebra. A morphism f : X C is said to be right minimal if any morphism g : X X with f = fg is an automorphism. We say that f : X C is a right X -approximation of C if X X and Hom R (X,f):Hom R (X,X) Hom R (X,C) is surjective. A right X -approximation h : X C of C is said to be a minimal right X - approximation of C if h is a right minimal morphism. The subcategory X is said to be contravariantly finite in mod R if each R-module C has a right X -approximation. We say that X is a resolving subcategory of mod R if it satisfies the following three conditions: (a) closed under extensions, (b) closed under kernels of surjections, and (c) contains the projective R-modules. We will use freely the notions of left and minimal left X -approximations, X being covariantly finite and coresolving in mod R, which are the duals of the notions given above, see [4]. If X is a class of R-modules, we denote by X the subcategory of mod R whose objects are those R-modules X for which there exists a finite X -resolution, that is, there is a long exact sequence 0 X u X 1 X 0 X 0 with X i X for all 0 i u. Dually, X is the subcategory of mod R whose objects have a finite X -coresolution. We denote by pd X the projective dimension of X. Similarly we have id X, which is the injective dimension of X. Given a subcategory C of mod R, we denote by pd C the projective dimension of C, that is, pd C = sup{pd X: X C}. Dually, id C = sup{id X: X C} is the injective dimension of C. We also consider the subcategories P < (C) ={X C: pdx< } and I < (C) = {X C: idx< }.Theprojective finitistic dimension of the category C, denoted by pfd C, is equal to pd P < (C). Dually, ifd C = id I < (C) is the injective finitistic dimension of C. We abuse notation and use ifd R and pfd R for the ifd(mod R) and the pfd(mod R), respectively. Also we set P < (R) = P < (mod R) and I < (R) = I < (mod R). Following Auslander and Buchweitz in [1], we recall the definition of relative projective dimension and the resolution dimension of a given module. Definition 1.1. Let X be a class of objects in mod R and M be an R-module. (a) We shall denote by pd X M the relative projective dimension of M with respect to X. That is, pd X M := if M = 0, and pd X M := min{n: Ext j R (M, ) X = 0 for any j>n 0} for M 0. Dually, we denote by id X M the relative injective dimension of M with respect to X. (b) We shall denote by resdim X M the X -resolution dimension of M. That is, resdim X M := if M = 0, resdim X M := + if M/ X, and resdim X M := min{r: there is an exact sequence 0 X r X 0 M 0 with X i X } if M X. Dually, we have coresdim X M the X -coresolution dimension of M. (c) For any class C of R-modules we set pd X C := sup{pd X M: M C} and resdim X C := sup{resdim X M: M C}. Dually we define id X C and coresdim X C. Remark 1.2. Let X and Y be two classes of objects in mod R, and M be an R-module. It can be seen that

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 423 (a) pd X (Y) = id Y (X ) and pd X M = sup{pd {X} M: X X }, (b) if Y X then coresdim X (M) coresdim Y (M). Let X be an R-module. Associated to X, we consider the following subcategories of mod R: the right (respectively left) perpendicular category X (respectively X) with objects X satisfying Ext i R (X, X ) = 0 (respectively Ext i R (X,X)= 0) for all i>0. On the other hand, we recall that T is a generalized tilting R-module if pd T is finite, T T and RR (add T). We recall that a classical tilting R-module M satisfies by definition the following conditions: (a) pd M 1, (b) Ext 1 R (M, M) = 0, and (c) coresdim add M( R R) 1. 2. General results In this section, we continue with the study started by M. Auslander, R.O. Buchweitz and I. Reiten in [1,4] of the relationship between the relative injective dimension and the coresolution dimension of a given module. The aim of this section is to establish some general results that can be applied both to tilting modules and to stratifying systems. One motivation for doing so is that for a given stratifying system (θ, Y, ) of size t we have that the coresdim add Y (F(θ)) is bounded by t 1 (see [8]). Finally, we point out that there are clear dual analogues of the statements proved in this section; for this reason, we shall use them freely. Theorem 2.1. Let X and Y be subcategories of mod R such that id X (Y) is finite. For any L Y we have that: (a) id X (L) id X (Y) + coresdim Y (L), (b) let Y be equal to I X or let Y X be closed under direct summands. If id X (Y) = 0 then id X (L) = coresdim Y (L). Proof. Let id X (Y) = α. We may assume that L 0 otherwise we have nothing to prove. (a) We proceed by induction on d = coresdim Y (L). If d = 0 then L Y and so id X (L) α. If d = 1 then we have the exact sequence 0 L I 0 I 1 0 with I 0 and I 1 in Y. (1) Applying the functor Hom R (M, ) to (1) with M X, we get the exact sequence Ext i 1 R (M, I 1) Ext i R (M, L) Exti R (M, I 0). Since Ext i 1 R (M, I 1) = 0fori>α+ 1 and Ext i R (M, I 0) = 0fori>αwe have that Ext i R (,L) X = 0fori>α+ 1 and so id X (L) α + 1.

424 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 f 0 f 1 Let d 2 and consider the exact sequence 0 L I 0 I 1 fd 1 I d 0 with I i Y. LetK 0 = Im f 0. Then coresdim Y (K 0 ) = d 1 and so by induction we get id X (K 0 ) α +d 1. Let M X, applying the functor Hom R (M, ) to the exact sequence 0 L I 0 K 0 0 we obtain the exact sequence Ext i 1 R (M, K 0) Ext i R (M, L) Exti R (M, I 0). Since Ext i 1 R (M, K 0) = 0fori α +d +1 and Ext i R (M, I 0) = 0fori α +1 we conclude that id X (L) α + d. (b) We will proceed by induction on d = coresdim Y (L). Ifd = 0 then L Y and hence id X (L) = 0. If d = 1 then we have the exact sequence (1). Hence L/ Y and by (a) we conclude that id X (L) 1. We assert that Ext 1 R (,L) X 0. Indeed, suppose that Ext 1 R (,L) X = 0. Then L I X.IfY = I X then we have a contradiction, since L/ Y. Assume that Y X and that Y is closed under direct summands. Then the fact that Ext 1 R (,L) X = 0 give us that (1) splits and so L Y giving a contradiction, proving that id X (L) = 1. f 0 f 1 Let d 2 and consider the exact sequence 0 L I 0 I 1 fd 1 I d 0 with I i Y for any i.letk 0 = Im f 0. Hence coresdim Y (K 0 ) = d 1, and so by induction we have that id X (K 0 ) = d 1. On the other hand, by (a) we obtain that id X (L) d. To prove that id X (L) = d it is enough to check that Ext d R (,L) X 0. Suppose that Ext d R (,L) X = 0 and let M X. Applying the functor Hom R (M, ) to the exact sequence 0 L I 0 K 0 0 we get the exact sequence Ext d 1 R (M, I 0) Ext d 1 (M, K 0) Ext d R (M, L). R Since Ext d 1 R (,I 0) X = 0 and Ext d R (,L) X = 0 then Ext d 1 R (,K 0) X = 0. Therefore id X (K 0 ) d 2, which is a contradiction. Thus id X (L) = d = coresdim Y (L). Corollary 2.2. Let X be a subcategory of mod R. If I R X then id X (M) = id(m) for any M I < (R). Proof. Using that id X (I R ) = 0, we get the result from 2.1 since I R is closed under direct summands and coresdim IR (M) = id(m). Corollary 2.3. Let X and Y be subcategories of mod R such that id X (Y) is finite. Then: (a) id X (X ) id X (Y) + coresdim Y (X ), (b) if id X (I X ) = 0 and X is closed under direct summands then id X (M) = coresdim IX (M) = coresdim ωx (M) = id ωx (M) for all M ω X.

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 425 Proof. (a) We can assume that coresdim Y (X )<+ (otherwise we have nothing to prove). Therefore X Y, and so we get the result from 2.1. (b) Since id X (I X ) = 0, we get that id ωx (ω X ) = 0 = id X (ω X ). Hence, the result follows from 2.1. The following corollary will be very useful in the proof of the main result in Section 3. Corollary 2.4. Let X be a subcategory of mod R. Then (a) if X is non-zero then id IX (X ) id IX (I X ) + coresdim IX (X ), (b) id IX (P X ) id IX (I X ) + coresdim IX (P X ). Proof. (a) We may assume that coresdim IX (X ) and id IX (I X ) are finite (otherwise there is nothing to prove). Then X (I X ), and so replacing X by I X and Y by I X in 2.1 we get the result. The proof of (b) is very similar to the one given in (a). The equality id X (I X ) = 0 is used very frequently in the forthcoming results. That is the reason why it would be useful to have some equivalent conditions of this fact. Lemma 2.5. Let X be a subcategory of mod R. The following conditions are equivalent: (a) I X is a coresolving subcategory of mod R, (b) id X (I X ) = 0, (c) Ext 2 R (X, I X ) = 0. Proof. (a) (b): Assume that I X is closed under cokernels of injections. Let M X and N I X, we prove that Ext i R (M, N) = 0 for any i>0. Consider the exact sequence 0 N I 0 (N) I 1 (N) I i 2 (N) Ω i+1 (N) 0 with I m (N) an injective R-module for all m = 0, 1,...,i 2 and Ω i+1 (N) I X. Therefore Ext i R (M, N) Ext1 R (M, Ω i+1 (N)) = 0. (b) (c): It is trivial. (c) (a): Assume that Ext 2 R (X, I X ) = 0. Consider the exact sequence 0 M E N 0 with M,E I X. (2) Applying the functor Hom R (X, ) to (2) with X X, we get the exact sequence Ext 1 R (X, E) Ext1 R (X, N) Ext2 R (X, M). Since Ext 1 R (,E) X = 0 and Ext 2 R (,M) X = 0 we have that Ext 1 R (,N) X = 0, proving that N I X and therefore I X is closed under cokernels of injections.

426 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 We recall the following results due to M. Auslander and I. Reiten in [4]. Those statements help us to establish some connections between projective and relative projective dimensions. The main property that we will use is the existence of the exact sequences given by the following proposition and its dual. Proposition 2.6 (Auslander Reiten). Let X be a subcategory of mod R, which is closed under extensions and direct summands. The following conditions are equivalent for any C mod R: (a) the functor Ext 1 R (C, ) X is finitely generated, (b) there is an exact sequence 0 X C Q C C 0 with Q C P X and X C X. Remark 2.7 (Auslander Reiten). Let X be a covariantly finite subcategory of mod R.IfX is closed under extensions and direct summands then Ext 1 R (C, ) X is finitely generated for any C mod R and P X is contravariantly finite. Lemma 2.8. Let X and Y be subcategories of mod R, Ext 1 R (,C) X be finitely generated for all C Y and M be an R-module. If X is closed under extensions and direct summands then (a) id Y (M) max{id X (M), id IX (M)}, (b) if X I X Y then id Y (M) = max{id X (M), id IX (M)}. Proof. (a) We may assume that id X (M) and id IX (M) are finite. So we prove that Ext i R (,M) Y = 0fori>max{id X (M), id IX (M)}. Using the dual of 2.6 we have that for any N Y there exists an exact sequence 0 N Y N N 0 with Y N I X and N X. (3) Applying the functor Hom R (,M)to (3) we obtain the exact sequence Ext i R (N,M) Ext i R (Y N,M) Ext i R (N, M) Exti+1 (N,M). So we have that Ext i R (Y N,M) Ext i R (N, M) for i>id X (M). Since Ext i R (Y N,M)= 0 for i>id IX (M), we get that Ext i R (,M) Y = 0fori>max{id X (M), id IX M}. (b) Using that X I X Y we obtain id Y (M) max{id X (M), id IX (M)}. Then the result follows from (a). Corollary 2.9. Let X be a subcategory of mod R, which is closed under extensions and direct summands, and let M be an R-module: (a) if Ext 1 R (,C) X is finitely generated for all C mod R then id(m) = max { id X (M), id IX (M) },

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 427 (b) if Ext 1 R (C, ) X is finitely generated for all C mod R then pd(m) = max { pd X (M), pd PX (M) }. Proof. We prove only (a), since (b) follows from (a) by duality. Since Ext 1 R (,C) X is finitely generated for all C mod R, we can replace Y by mod R in 2.8. Hence the result follows because of the fact id mod R (M) = id(m). Proposition 2.10. Let X be a contravariantly finite subcategory of mod R closed under extensions and direct summands. If P R X then (a) [Auslander Reiten, 1991] X = P IX, (b) I X is coresolving if and only if X is resolving, (c) id(m) = max{id X (M), id IX (M)} for any M mod R, (d) pd(m) = max{pd X (M), pd IX (M)} for any M mod R. Proof. (a) This is Proposition 1.10 in [4]. (b) Suppose that I X is coresolving. Then, by (a) and Lemma 3.1 in [4] we conclude that X is resolving. Assume now that X is resolving. Hence by the dual of Lemma 3.1 in [4] we have that I X is coresolving. (c) and (d): Since X is contravariantly finite we get from the dual result of 2.7 that I X is covariantly finite. Hence by 2.7 and its dual, we have that the functors Ext 1 R (,C) X and Ext 1 R (C, ) I X are finitely generated for any C mod R. Thus, the result follows from 2.9 and the fact that X = P IX. Corollary 2.11. Let X be a covariantly finite subcategory of mod R. IfX is closed under extensions and direct summands then (a) Y := I PX is covariantly finite and P X = P Y, (b) pd M = max{pd X (M), pd PX (M)}=max{pd Y (M), pd PX (M)}, (c) id M = max{id Y (M), id PX (M)}. Proof. By 2.7 we have that Ext 1 R (C, ) X is finitely generated for any C mod R and P X is contravariantly finite. So we can apply 2.10 to the category P X and 2.9 to X, proving the result. As an application of 2.3 and 2.6, we get our first theorem that relates the relative injective dimension of Y with respect to X and the I X -coresolution dimension of X. Moreover, a bound is given for the injective finitistic dimension of an algebra R by using some suitable subcategories of R-modules. Theorem 2.12. Let X and Y be subcategories of mod R such that the functor Ext 1 R (,C) X is finitely generated for all C Y. IfX is closed under extensions and direct summands then

428 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 (a) id X (Y) id X (I X ) + coresdim IX (X ) + 1, (b) if Y = mod R, id X (I X )<+ and coresdim IX (X )<+ then id IX (X) =+ for all X mod R \ I X, (c) if I < (R) Y then ifd R max(ifd I X, id X + 1). Proof. (a) We can assume that id X (I X ) = α<+ and coresdim IX (X ) = β<+. Let X Y. Using that Ext 1 R (,X) X is finitely generated we obtain, by the dual result of 2.6, an exact sequence 0 X Y X X 0 with Y X I X and X X.Let M X, applying the functor Hom R (M, ) to that sequence we get the exact sequence (M, Y X) Ext i 1 (M, X ) Ext i R (M, X) Exti R (M, Y X). Then we have that Ext i 1 R R Ext i 1 R (,X ) X Ext i R (,X) X for any i α + 2. (4) On the other hand, since X X, we have by 2.3(a), that b := id X (X ) α + β. Therefore from (4) we get that So we obtain Ext i R (,X) X = 0 for any i max(α + 2,b+ 2). id X (X) max(α + 2,b+ 2) 1 max(α + 2,α+ β + 2) 1 α + β + 1, proving that id X (Y) α + β + 1. (b) Assume that Y = mod R, id X (I X )<+ and coresdim IX (X )<+. LetX mod R \ IX. Since I R I X we obtain that id(x) =+. This means, by 2.9, that id X (X) =+ or id IX (X) =+. On the other hand, by (a) we have that id X (X) is finite, and so we obtain that id IX (X) =+. (c) Assume that id X is finite and I < (R) Y.LetX I < (R). Since Ext 1 R (,X) X is finitely generated we obtain, by the dual result of 2.6, an exact sequence 0 X Y X X 0 with Y X I X and X X.Soid(Y X ) is finite, since id(x) and id(x ) are finite. Hence id(x) max ( id(y X ), id X + 1 ) max(ifd I X, id X + 1). Lemma 2.13. Let X, Y and Z be subcategories of mod R. Then (a) pd Y (X ) = pd Y (X ), (b) if X Z X then pd Y (Z) = pd Y (X ), (c) if X ω X then pd X = pd ω X. Proof. (a) Assume that pd Y (X ) = α<+. So, by induction on d = coresdim X (M) for M X, it can be seen that pd Y (X ) α. Then (a) follows, since pd Y (X ) pd Y (X ). Finally, we have that (b) follows from (a), and (c) follows from (a) and (b).

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 429 Proposition 2.14. Let X and Y be subcategories of mod R such that X is closed under extensions and direct summands, X ω X and Ext1 R (,C) X is finitely generated for all C Y. Then (a) if pd ω X is finite and Y = P < (R) then pfd R = pfd I X, (b) if Y = mod R then gl.dim R = pd I X. Proof. (a) Assume that pd ω X is finite and Y = P < (R). Since pd ω X is finite, we get by 2.13 that pd X is so. Let M Y, using that Ext 1 R (,M) X is finitely generated, we have from the dual result of 2.6 an exact sequence 0 M Y X M X 0 with Y X I X,M X X. (5) Hence pd Y X is finite, since pd M<+ and pd M X pd X < +. Therefore pd Y X pfd I X. So by (5) we get pd M max(pd Y X, pd M X 1) pfd I X, since pd M X pd X = pd ω X pfd I X. Thus pfd R pfd I X, proving that pfd R = pfd I X. (b) The proof of the equality gl.dim R = pd I X is very similar to the one given in (a). The following theorem is the main result of this section. This theorem relates different kinds of homological dimensions by using suitable subcategories of mod R. Furthermore, in Section 3 we will see that this result can be strengthened if we assume in addition that X is a partial tilting category. Theorem 2.15. Let X be a contravariantly finite subcategory of mod R, which is closed under extensions and direct summands. Then (a) pd X id X (I X ) + coresdim IX (X ) + 1, (b) ifd R max(ifd I X, id X + 1), (c) if I X is coresolving then id IX (I X ) = id I X. Moreover, pd X 1 + coresdim IX (X ) and id X id I X + coresdim IX (X ) + 1, (d) if X ω X then (i) gl.dim R = pd I X, (ii) if pd ω X is finite then pfd R = pfd I X, (iii) if P < (I X ) ω X and pd ω X is finite then pfd R = pd ω X. Proof. Since X is contravariantly finite, closed under extensions and direct summands, we get from the dual result of 2.7 that Ext 1 R (,C) X is finitely generated for any R-module C. Hence (a) and (b) follows from 2.12, since id X (mod R) = pd X. (c) We already know that the functor Ext 1 R (,M) X is finitely generated for any M mod R. On the other hand, I X coresolving implies by 2.5 that id X (I X ) = 0. So by 2.9(a) we get that id I X = max(id X (I X ), id IX (I X )). Thus id IX (I X ) is equal to id I X, since id X (I X ) = 0.

430 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 The first inequality pd X 1 + coresdim IX (X ) follows from (a). Now, we prove the second one: By 2.9 we have that id X = max(id X (X ), id IX (X )). So the inequality id X id(i X ) + coresdim IX (X ) + 1 follows from 2.4, since id X (X ) = pd X (X ) pd X 1 + coresdim IX (X ). (d) The proof of (i) and (ii) follows from 2.14(a), since X is contravariantly finite, closed under extensions and direct summands. To prove (iii) we assume that P < (I X ) ωx. We assert that pfd I X pd ω X. Indeed, by 2.13 we get pfd I X = pd P < (I X ) pd ωx = pd ω X. On the other hand, the fact that pd ω X is finite implies that pd ω X pfd I X. Therefore pfd I X = pd ω X, proving the result. As an easy consequence of 2.15 we have the following result. Corollary 2.16. Let T be an R-module such that Ext 1 R (T, T ) = 0 and pd T is finite. Then (a) pfd R = pfd I add T and gl.dim = pd I add T, (b) if P < (I add T ) (add T) then pfd R = pd T. Proof. It follows from 2.15(d) by taking X := add T. The following well-known result can also be obtained from the previous theorem. Corollary 2.17. If the global dimension of R is finite then gl.dim R = id R R. Proof. Assume that gl.dim R is finite. We take X = mod R. Then ω X = I X = I R and X IR. So by 2.15(d) we get gl.dim R = pd I R = pd D(R R ) = id R R. As another application of 2.1 and 2.15 we get the following result. We point out, that the item (c) below is a generalization of Corollary 2.11 in [5]. Proposition 2.18. Let T be a generalized tilting R-module. Then (a) id R R id T + coresdim add T ( R R), (b) coresdim add T ( R R) = id add T ( R R) pd T, (c) if gl.dim R is finite then gl.dim R pd T + id add T ( R R) pd T + id T. Proof. (a) It follows from 2.1 by taking Y = add T, X = mod R and L = R. (b) We set X = add T. Then by 2.1 (b) we have that coresdim add T ( R R) = id add T ( R R) id add T (mod R) = pd T. (c) follows from (a) and 2.17. 3. Tilting subcategories in mod R In this section we introduce the notion of tilting category. This concept inglobes the main homological properties of two classical categories. That is, add T with T a generalized tilting R-module and F( R Δ), where R is a standardly stratified algebra. If we assume

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 431 that the category under consideration is tilting, then we will see, in this section, that the results obtained in the previous section can be strengthened, and as a consequence we will get interesting results in the following sections. Definition 3.1. Let X be a subcategory of mod R, which is closed under extensions and direct summands. We say that X is a partial tilting category if pd X (X ) and pd PX (X ) are finite. A partial tilting category X is said to be tilting if R R ωx. Dually, we say that X is a partial cotilting category if id X (X ) and id IX (X ) are finite. A partial cotilting category X is called cotilting if D(R R ) ZX. Lemma 3.2. Let X be a subcategory of mod R, which is closed under extensions and direct summands. If X is covariantly finite then (a) X is partial tilting if and only if pd X is finite, (b) if R is self-injective and P R X then X is partial tilting if and only if X = P R. Proof. (a)by2.11wehavethatpdx = max{pd X (X ), pd PX (X )}. So the result follows. (b) Assume that R is self-injective and P R X. Since R is self-injective we apply the dual result of 2.2 to X to get that pd X is finite if and only if X = P R. Hence the result follows from (a). Lemma 3.3. Let M be an R-module and X be a subcategory of mod R. Then coresdim IX (M) max { id X (M), 1 } and coresdim IX (mod R) max{pd X, 1}. Proof. Assume that d = id X (M) is finite. Consider the exact sequence 0 M I 0 I 1 I s 1 Ω s (M) 0, where s = 1ifd = 0 and s = d otherwise. Then Ext 1 ( R,Ω s (M) ) X ExtR s+1 (,M) X = 0. Therefore Ω s (M) I X and coresdim IX (M) s, proving the first equality. The second one follows from the first and the fact that id X (mod R) = pd X. Proposition 3.4. Let X be a partial tilting category. Then (a) coresdim IX (X ) max{pd X (X ), 1} < +, (b) resdim PX (X ) max{pd X (X ), 1} < +, (c) coresdim IX (P X ) max{pd PX (X ), 1} < +, (d) if I X is coresolving then (i) pd PX (X ) = coresdim IX (P X ), (ii) if X ω X then pd X (X ) = coresdim I X (X ) = coresdim ωx (X ) = id ωx (X ).

432 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 Proof. (a), (b) and (c) follow from 3.3, since id X (X ) = pd X (X ) and id X (P X ) = pd PX (X ). To prove (d) we assume that I X is coresolving. By (c) we have that P X IX. Then (i) follows from 2.1, since pd PX (X ) = id X (P X ). Finally, (ii) follows easily from 2.3. Proposition 3.5. Let X be a contravariantly finite subcategory of mod R, which is closed under extensions and direct summands. If I X is coresolving then (a) the following conditions are equivalent: (i) coresdim IX (X ) is finite, (ii) pd X is finite, (iii) coresdim IX (mod R) is finite. (b) In case that pd X be finite we have that: (i) coresdim IX (X ) = pd X (X ) pd X = coresdim IX (mod R), (ii) if P R X then coresdim IX (X ) = pd X (X ) = pd X = coresdim IX (mod R). Proof. (a) The assumption that coresdim IX (X ) is finite implies, by 2.15, that pd X is finite and therefore IX = mod R (see 3.3). Hence using 2.1 we get that pd X = coresdim IX (mod R), proving (a). (b) Let pd X be finite and P R X. We get by 2.1 and the dual of 2.2 that coresdim IX (X ) = pd X (X ) = pd X. So (b) follows from the equality pd X = coresdim IX (mod R), which was obtained in the proof of (a). Proposition 3.6. Let X be a contravariantly finite subcategory of mod R which is closed under extensions and direct summands. If id X (ω X ) = 0 then (a) if pd X (X ) is finite then X ω X and coresdim ω X (X ) pd X (X ), (b) id X (M) = coresdim ωx (M) for any M ω X. Proof. (a) Since X is a contravariantly finite subcategory and it is closed under extensions we get from the dual result of 2.6, that for any X X, there exists an exact sequence 0 X W X 0 with W ω X and X X. (6) Let X X,X 1 := X and d := pd X (X ). Using the exact sequence (6) we can construct exact sequences ε i :0 X i 1 W i X i 0 with W i ω X and X i X. Applying the functor Hom R (X d, ) to the exact sequence ε i we get that Ext j R (X d,x i ) Ext j+1 R (X d,x i 1 ), since id X (ω X ) = 0. It follows that Ext 1 R (X d,x d 1 )

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 433 Ext 2 R (X d,x d 2 ) Ext d+1 R (X d,x 1 ) = 0, because d := pd X (X ). Hence the exact sequence ε d splits and so the module W d := X d 1 is a direct summand of W d ω X. Hence we get an exact sequence 0 X W 0 W 1 W d 1 W d 0 with W i ω X for all i = 0, 1, 2,...,d 1 and W d ω X, proving the result. (b) It follows from 2.1, since id X (ω X ) = 0. Theorem 3.7. Let X be a contravariantly finite partial tilting subcategory of mod R, such that id X (I X ) is finite. Then (a) pd X id X (I X ) + coresdim IX (X ) + 1 < +, (b) coresdim IX (mod R) max{pd X, 1} < +, (c) if id X (I X ) = 0 then (i) id X (M) = coresdim ωx (M) = id ωx (M) for any M ω X, (ii) id X (M) = coresdim IX (M) for all M mod R, (iii) coresdim IX (mod R) = pd X = pd PX (X ) = coresdim IX (P X ), (iv) id ωx (X ) = coresdim ωx (X ) = pd X (X ) = coresdim IX (X ). Proof. (a) From 2.15 we have that pd X id X (I X )+coresdim IX (X )+1. So (a) follows, since by 3.4 we know that coresdim IX (X ) is finite. (b) By (a) we have that pd X is finite. So the result follows from 3.3. (c) (i) It follows from 2.3(b). (ii) It can be obtained from (b) and 2.1. (iii) From (ii) we have that coresdim IX (mod R) = pd X and pd PX (X ) = id X (P X ) = coresdim IX (P X ). On the other hand, since pd X is finite we conclude from the dual result of 2.2 that pd X = pd PX (X ). (iv) Since id X (I X ) = 0 we get from 3.6 that X ωx. Thus the result follows from 3.4(d). Theorem 3.8. Let X be a covariantly finite partial tilting subcategory of mod R, such that pd X (P X )<+. Then (a) id X pd X (P X ) + resdim PX (X ) + 1 < +, (b) resdim PX (mod R) max{id X, 1} < +, (c) if pd X (P X ) = 0 then (i) pd X (M) = resdim ZX (M) = pd ZX (M) for all M Z X, (ii) pd X (M) = resdim PX (M) for all M mod R, (iii) resdim PX (mod R) = id X = pd X (I X ) = resdim PX (I X ), (iv) pd ZX (X ) = resdim ZX (X ) = pd X (X ) = resdim PX (X ). Proof. It can be proven in a very similar way as the one given in 3.7.

434 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 The following theorem is the main result of this section. The goal is that given a contravariantly finite and partial tilting category X, we want to construct a functorially finite, resolving and tilting category Y which contains X and is very closely connected to X in the sense of homological dimensions. As we will see below, if I X is coresolving then the category Y = P IX is the indicated one. In order to prove the main result, we will make use of the following lemma. Lemma 3.9. Let X be a contravariantly finite subcategory of mod R which is closed under extension and direct summands, and let Y be equal to P IX.IfI X is coresolving then (a) Y is a resolving and functorially finite subcategory of mod R, (b) I Y = I X and pd IX (Y) = 0. Proof. Assume that I X is coresolving. We know, by the dual result of 2.7, that I X is covariantly finite. Then by Lemma 3.1 in [4] (using that I X is also coresolving) and 2.7, we get that P IX is resolving and contravariantly finite. So by the dual result of 2.5 we conclude that pd IX (Y) = 0. On the other hand, the dual result of 2.11 gives us the equality I Y = I X. Finally, since Y is a resolving and contravariantly finite subcategory of mod R, we get from Corollary 2.6 in [10] that Y is also covariantly finite, proving the result. Theorem 3.10. Let X be a contravariantly finite partial tilting subcategory of mod R and Y := P IX.IfI X is coresolving then (a) coresdim IX (Y) = pd Y (Y) = pd Y = coresdim IX (mod R) = pd X = pd ω X < +, (b) Y is a resolving functorially finite and tilting subcategory of mod R, (c) pd IX (M) = resdim Y (M) for any M Y, (d) Y ={M mod R: pd IX (M) < } = P < (R), (e) pd M pd ω X + resdim Y (M) for any M P < (R), (f) pfd R = pfd I X pd ω X + resdim Y (P < (I X )) pd ω X + resdim Y (Y ), (g) gl.dim R = pd I X = max(id I X, pd Y (I X )). Moreover, if X is covariantly finite then gl.dim R pd ω X + id I X, (h) the following conditions are equivalent (1) id I X is finite, (2) pd I X is finite, (3) I X is a partial tilting category, (4) Y = mod R. (i) If pd I X is finite then gl.dim R pd ω X + id I X < +. Moreover, we have that (1) pd IX (M) = resdim ZIX (M) = pd ZIX (M) for any M ZI X, (2) pd IX (M) = resdim Y (M) for any M mod R, (3) resdim Y (mod R) = id I X = pd IX (I IX ) = resdim Y (I IX ) = resdim Y (I X ) = resdim ZIX (I X ) = pd ZIX (I X ). Proof. Assume that I X is coresolving. Since X is partial tilting we get the inclusion X ωx from 3.6(a). That inclusion will be used in the proof of the main theorem.

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 435 (a) We have by 3.9 that Y is contravariantly finite, closed under extensions and direct summands, I Y = I X and P R Y. We will apply 3.5 to Y. We start by proving that coresdim IY (Y) is finite. That is, coresdim IY (Y) = coresdim IX (Y) coresdim IX (mod R) = pd X < +, see 3.7. On the other hand, the inclusion X ωx implies that pd X = pd ω X (see 2.13(c)). Hence by 3.5(a) we get that pd Y is finite and so the result follows from 3.5(b). (b) By 3.9 we know that Y is a resolving and functorially finite subcategory of mod R. On the other hand, from (a) we conclude that Y is partial tilting. Finally, since I Y is coresolving we get that Y ωy (see 3.6(a)). Hence Y is a tilting category because P R Y ωy. (c) Since pd IX (Y) = 0 (see 3.9) we get (c) from the dual result of 2.1(b). (d) Let P I < X (R) := {M mod R: pd IX (M) < }. By the dual of 3.3 we conclude that resdim Y (M) max(pd IX (M), 1) for any M mod R. Thus P I < X (R) Y. Hence by (c) we get P I < X (R) = Y. Since P R Y we have that resdim Y (M) resdim PR (M) = pd M. Therefore P < (R) Y. On the other hand, since pd Y is finite (see (a)) then Y P < (R). (e) By the dual of 2.1(a), we obtain pd Y (M) pd Y (Y)+resdim Y (M) for any M Y. On the other hand, Y = P < (R) and pd Y = pd ω X is finite (see (a) and (d)). So the result follows from the dual of 2.2, since P R Y. (f) We have that X ωx and pd ω X is finite. So we have by 2.15(d) that pfd R = pfd I X. Then the result follows from (e), since P < (I X ) Y (see (d)). (g) Since X ωx we obtain from 2.15(d) that gl.dim R = pd I X. We know that id I X = id IX (I X ) (see 2.15(c)) since I X is coresolving. So by the dual result of 2.7 and 2.9 we get pd I X = max(id I X, pd Y (I X )), since I X is covariantly finite and pd IX (I X ) = id IX (I X ) = id I X. Assume that X is covariantly finite. Then by the dual result of 2.7 and 2.9 we have pd I X = max ( pd X (I X ), pd PX (I X ) ). (7) On the other hand, from 2.4 we have id IX (X ) id IX (I X ) + coresdim IX (X ) and id IX (P X ) id IX (I X ) + coresdim IX (P X ). Then by (7) and the equalities pd X (X ) = coresdim IX (X ) and pd PX (X ) = coresdim IX (P X ) (see 3.7) we get pd I X id I X + max ( pd X (X ), pd PX (X ) ) = id I X + pd X. (8) Hence by (8) we conclude that gl.dim R pd ω X + id I X, since pd X = pd ω X (see (a)). (h) (1) (2): Assume that id I X is finite. Then pd IX (I X ) = id IX (I X ) = id I X < + (see 2.15(c)). Therefore by the dual of 3.3 we get I X Y. So by (c), (d) and (e) we conclude that pd I X pd ω X + id I X < +. (2) (1): It follows from id I X = id IX (I X ) = pd IX (I X ) pd I X. (2) (3): It follows from 3.2(a), since I X is covariantly finite. (4) (2): Assume that Y = mod R. Then by (d) we have that P < (R) = mod R. Hence by Corollary 3.10 in [4] we obtain that gl.dim R is finite. (2) (4): Suppose that pd I X is finite. So by (g) we conclude that gl.dim R is finite. Therefore Y = mod R, since P R Y.

436 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 (i) Assume that pd I X is finite. Then We have that pd I X pd ω X + id I X < + (see the proof of (1) (2) in (h)). On the other hand, we have that I X is covariantly finite, partial tilting and pd IX (Y) = 0. So we can apply 3.8(c) to I X. Hence the result follows, since pd IX (I X ) = id I X. Some reduction can be made in Theorem 3.10 if we assume the extra condition: P R X. Corollary 3.11. Let X be a contravariantly finite partial tilting subcategory of mod R. If I X is coresolving and P R X then (a) X is a resolving, functorially finite and tilting subcategory of mod R, (b) id ωx (X ) = coresdim ωx (X ) = pd X (X ) = coresdim IX (X ) = pd X = pd ω X = pd PX (X ) = coresdim IX (P X ) = coresdim IX (mod R) < +, (c) pd IX (M) = resdim X (M) for any M X, (d) X ={M mod R: pd IX (M) < } = P < (R), (e) pd M pd ω X + resdim X (M) for any M P < (R), (f) pfd R = pfd I X pd ω X + resdim X (P < (I X )) pd ω X + resdim X (X ), (g) gl.dim R = pd I X pd ω X + id I X, (h) if pd I X is finite then (1) pd IX (M) = resdim ωx (M) = pd ωx (M) for any M ω X, (2) pd IX (M) = resdim X (M) for any M mod R, (3) resdim X (mod R) = id I X = pd IX (I IX ) = resdim X (I IX ) = resdim X (I X ) = resdim ωx (I X ) = pd ωx (I X ). Proof. Assume that I X is coresolving and P R X. Then by 3.10(a) we have that X = P IX = Y and so Z IX = I X Y = ω X. Hence, we can replace Y by X and Z IX by ω X in the main theorem to obtain the corollary. Observe that not all the equalities of (b) are obtained in this way; but, if we use 3.7(c) we get (b). 4. Tilting categories and tilting modules Let R be a quasi-hereditary algebra. In 1991, C.M. Ringel proved in [17] that there exists a generalized tilting R-module T (called by Ringel the characteristic module ) associated to the category F( R Δ) (see [17, Theorem 5]). Since F( R Δ) is a typical example of a tilting category, it is natural to expect that a generalization of Theorem 5 can be done for tilting categories. In this section, we establish this generalization for a tilting category X. Moreover, we get interesting homological relationships between this tilting module T associated to X and the algebra R. Lemma 4.1. Let X be a subcategory of mod R closed under extensions and direct summands. If X ω X and there is a generalized tilting R-module T such that I X = T, then I X is coresolving and pd X = pd ω X pd T.

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 437 Proof. Assume X ω X and there is a generalized tilting R-module T such that I X = T. In particular we obtain that I X is coresolving, since T is so. On the other hand, by 2.13(c) we know that the inclusion X ω X implies that pd X = pd ω X. We assert that pd ω X pd T. Indeed, let M ω X I X = T. Then we have an exact sequence (see [13, Lemma 3.1]) 0 K T 0 M 0 with T 0 add T and K T = I X. (9) Then the exact sequence (9) splits and so M is a direct summand of T, proving that pd M pd T. Theorem 4.2. Let X be a contravariantly finite partial tilting subcategory of mod R.IfI X is coresolving then there is a generalized tilting R-module T such that (a) I X = T and pd X pd T, (b) pfd R = pfd T pd T + resdim add T (add T), (c) pfd R is finite if and only if resdim add T (add T) is finite, (d) the following conditions are equivalent: (i) id T is finite, (ii) gl.dim R is finite, (iii) T = (add T) and pfd R is finite, (iv) pd T is finite, (e) if gl.dim R is finite then id I X = id T. Proof. Assume that I X is coresolving. (a)wehavethatix = mod R (see 3.10(b)) and I X is covariantly finite (see the dual result of 2.7). Hence the dual result of Theorem 5.5 in [4] implies that there is a generalized tilting R-module T such that I X = T. Finally, the inequality pd X pd T follows from 4.1. (b) Since (add T) = P < (T ) (see [13, Lemma 3.1]), pd ω X pd T (see 4.1) and add T P T the result follows from 3.10(f). (c) By Lemma 3.1 in [13] we have that resdim add T (add T) pfd T. Thus the result follows from (b). (d) (i) (iv): It follows from 3.10(h); (iv) (ii): It follows from 3.10(i); (ii) (iii): Follows from the equality (add T) = P < (T ) (see [13, Lemma 3.1]); (iii) (iv): We have pd T = pd(add T) pd T + resdim add T (add T), so the result follows from (c). (e) Assume that gl.dim R is finite. Then by (d) we have (add T) = T. Therefore id T = id T (see the dual result of 2.13). Definition 4.3. Let X be a subcategory of mod R, which is closed under extensions and direct summands. A basic R-module Y X is said to be the characteristic R-module associated to X if ω X = add Y X.Dually,theco-characteristic R-module associated to X is a basic R-module Q X such that Z X = add Q X.

438 O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 Given a subcategory X of mod R the characteristic R-module may not exist. However, it does exist for the category F(θ), where (θ, ) is a stratifying system, see [11, Theorem 2.4]. The following results give sufficient conditions for the existence of a characteristic R-module. Moreover, they give sufficient conditions to get that it is a generalized tilting R-module. Lemma 4.4. Let X be a subcategory of mod R which is closed under extensions and direct summands, and let 0 R R X 0 X m 0 be an exact sequence with X i ω X for any i. Ifid X (ω X ) = 0 and pd X is finite then T := m i=0 X i is a generalized tilting R-module such that ω X = add T. Proof. Since pd X is finite we have that pd T is so. On the other hand, id X (ω X ) = 0 implies that Ext i R (T, T ) = 0 for any i>0. Then T is a generalized tilting R-module. Moreover, if M ω X then T M is a generalized tilting R-module. By tilting theory we must have M add T. Therefore ω X = add T. Theorem 4.5. Let X be a contravariantly finite tilting subcategory of mod R and let 0 RR X 0 X m 0 be an exact sequence with X i ω X for any i. IfI X is coresolving then T := m i=0 X i is a generalized tilting R-module such that ω X = add T, I X = T and pd X = pd T. Proof. Assume that I X is coresolving. By 3.10(b) we have that pd X is finite. Therefore from 4.4 we conclude that T is a generalized tilting R-module and ω X = add T.Onthe other hand, by 4.2(a) there is a generalized tilting R-module T such that I X = T. Since T ω X I X = T we obtain an exact sequence 0 K T 0 T 0 with T 0 add T and K T = I X. (10) Then the exact sequence (10) splits and so T is a direct summand of T, proving that T = T. On the other hand, since X ωx (see 3.6(a)) we get by 2.13 that pd X = pd ω X = pd T,since by 4.4 we know that ω X = add T. Theorem 4.6. Let X be a resolving contravariantly finite and partial tilting subcategory of mod R. Then, there exists a generalized tilting R-module T such that (a) I X = T,ω X = add T, X = (add T) and pd X = pd T, (b) X = T and T is also a generalized cotilting R-module if and only if gl.dim R is finite. Proof. Since X is a resolving and contravariantly finite subcategory of mod R, we get from 2.10(b) that I X is coresolving. (a) We start by proving the equality X = ωx. The inclusion X ω X follows from 3.6 since I X is coresolving. Using that X is closed under kernels of surjections, it is not difficult to see that ωx X. Therefore (a) follows from 4.5. (b) The proof is very similar to the proof of Theorem 2.4 in [3].

O. Mendoza, C. Sáenz / Journal of Algebra 302 (2006) 419 449 439 ( ) Suppose that X = T and T is a generalized cotilting R-module. Hence by the Theorem 5.5 in [4] we infer that X = mod R. Therefore P < (R) = mod R since by (a) we know that pd X is finite. So we can use Corollary 3.10 in [4] to conclude that gl.dim R is finite. ( ) Assume now that gl.dim R is finite. We prove that X = T and T is a cotilting R-module. Indeed, it is well know that over an algebra of finite global dimension any generalized tilting module is also cotilting (see [9]). On the other hand, using that I X is coresolving we have by 2.5 that id X (I X ) = 0. Hence id X (ω X ) = 0, and so the inclusion X T follows since ω X = add T. To prove the inclusion T X, we start by taking an element X in T.Consider a right minimal X -approximation of X, which exists and is surjective since X is contravariantly finite and resolving. So by Wakamatsu s Lemma (see [4, Lemma 1.3]) we get an exact sequence ε: 0 K X F X X 0 with K X I X = T and F X X. We assert that K X X. To prove that it is enough to see that K X P IX, since by 2.10(a) we know that P IX = X. Let N I X = T and r := pd N. Then there is a long exact sequence (see the proof of (a) and (b) of Lemma 3.1 in [13]) 0 T r T r 1 f r 1 T 1 f 1 T 0 f 0 N 0 with T i add T and K i := Ker f i T for any i. Therefore, the fact that K X T implies that Ext 1 R (K X,N) Ext r+1 R (K X,T r ) = 0, and so K X P IX = X. Then K X X I X = ω X = add T, which implies that the exact sequence ε splits, proving that X belongs to X. Corollary 4.7. Let X = add M with pd M finite. If I X is coresolving then (a) I X = T, where T is a generalized basic tilting R-module, (b) if R R ωx then (b1) ω X = add T, (b2) if Ext 1 R (M, M) = 0 then T is a direct summand of M. Moreover, we have that M is a generalized tilting R-module. Proof. add M is a functorially finite subcategory of mod R. On the other hand, pd M finite implies that X is a partial tilting subcategory of mod R. So by 4.2(a) there is a generalized basic tilting R-module T such that I X = T. Assume that R R ωx. Then X is a functorially finite tilting subcategory of mod R. Hence by 4.5 we obtain that ω X = add T. Suppose that Ext 1 R (M, M) = 0. Hence X I X and so add M = add T, proving the result. As we have seen in the previous corollary, we needed to know when the category I X is coresolving for X = add M. The following proposition gives an answer to this question. Then, in the following corollary, we apply this result to the particular case when M is a generalized tilting R-module.