On finitistic dimension of stratified algebras

On finitistic dimension of stratified algebras

In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.

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Zitieren:On finitistic dimension of stratified algebras / V. Mazorchuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 77–88. — Бібліогр.: 13 назв. — англ.

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spelling irk-123456789-1564582019-06-19T01:28:32Z On finitistic dimension of stratified algebras Mazorchuk, V. In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems. 2004 Article On finitistic dimension of stratified algebras / V. Mazorchuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 77–88. — Бібліогр.: 13 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16E10, 16G10. http://dspace.nbuv.gov.ua/handle/123456789/156458 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.
format Article
author Mazorchuk, V.
spellingShingle Mazorchuk, V.
On finitistic dimension of stratified algebras
Algebra and Discrete Mathematics
author_facet Mazorchuk, V.
author_sort Mazorchuk, V.
title On finitistic dimension of stratified algebras
title_short On finitistic dimension of stratified algebras
title_full On finitistic dimension of stratified algebras
title_fullStr On finitistic dimension of stratified algebras
title_full_unstemmed On finitistic dimension of stratified algebras
title_sort on finitistic dimension of stratified algebras
publisher Інститут прикладної математики і механіки НАН України
publishDate 2004
url http://dspace.nbuv.gov.ua/handle/123456789/156458
citation_txt On finitistic dimension of stratified algebras / V. Mazorchuk // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 77–88. — Бібліогр.: 13 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT mazorchukv onfinitisticdimensionofstratifiedalgebras
first_indexed 2025-07-14T08:49:48Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2004). pp. 77 – 88 c© Journal “Algebra and Discrete Mathematics” On finitistic dimension of stratified algebras Volodymyr Mazorchuk Communicated by Yu. Drozd Abstract. In this survey we discuss the results on the fini- tistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems. 1. Introduction and preliminaries Let A be a finite-dimensional, associative, and unital algebra over an alge- braically closed field k, and A−mod be the category of finite-dimensional left A-modules. Assume that the isomorphism classes of simple A- modules are indexed by Λ = {1, 2, . . . , n} and denote by L(λ), P (λ), I(λ), λ ∈ Λ, the corresponding simple module, its projective cover, and its injective envelope respectively. Remark that the elements of Λ are ordered in the natural way. For λ ∈ Λ set P>λ = ⊕µ>λP (µ) and de- fine the standard module ∆(λ) = P (λ)/TraceP >λ(P (λ)). Denote by F(∆) the full subcategory of A−mod, which consists of all modules, having a filtration with subquotients isomorphic to standard modules. Call the algebra A strongly standardly stratified (or an SSS-algebra) if AA ∈ F(∆). The class of SSS-algebras contains the very important sub- class of quasi-hereditary algebras, and forms a subclass of the class of standardly stratified algebras, introduced in [CPS]. SSS-algebras (some- times also called just standardly stratified in the literature, which makes everything somewhat confusing) were intensively studied during the last decade, see [AHLU1, AHLU2, Ma] and references therein. Such algebras arise naturally in Lie theory, see [Ma]. In [AHLU1] it has been shown 2000 Mathematics Subject Classification: 16E10, 16G10. Key words and phrases: stratified algebra, finitistic dimension, tilting module. Jo u rn al A lg eb ra D is cr et e M at h .78 On finitistic dimension that both the projectively and the injectively defined finitistic dimensions of such algebras do not exceed 2n − 2. Though this bound is exact for certain algebras, in most cases this estimate is very rough. For example any hereditary algebra is stratified (even quasi-hereditary) with respect to any order on Λ, see [DR, Theorem 1], and has global dimension 1. In the present paper we try to approach rather non-symmetric sit- uations, i.e. the one for which projective and injective dimensions can be different. Let P<∞(A) and I<∞(A) denote the full subcategories of A−mod, which consists of all modules M having finite projective or in- jective dimension respectively. We denote by fdim(A) the projectively defined finitistic dimension of A, that is the supremum of pd(M), taken over all M ∈ P<∞(A); and by ifdim(A) the injectively defined finitis- tic dimension of A, that is the supremum of i. d.(M), taken over all M ∈ I<∞(A). For λ ∈ Λ define the proper standard module ∆(λ) = ∆(λ)/TraceP (λ)(rad∆(λ)). Dually one defines the costandard modules ∇(λ) and the proper costan- dard modules ∇(λ), λ ∈ Λ. The categories F(∇), F(∆) and F(∇) are defined analogously to F(∆). For all modules indexed by λ ∈ Λ the no- tation without index will mean the direct sum over all λ ∈ Λ, for example L = ⊕n λ=1L(λ) etc. According to [Dl2, La], an alternative description of SSS-algebras can be given requiring I ∈ F(∇). Varying the requirements one gets many other classes of stratified al- gebras. The ones, which are important for the present paper, are properly stratified algebras, defined in [Dl1] via AA ∈ F(∆) ∩ F(∆), or, alterna- tively, via I ∈ F(∇) ∩ F(∇); and quasi-hereditary algebras, defined as those properly stratified algebras, for which ∆(λ) = ∆(λ) for all λ, which is equivalent to requiring ∇(λ) = ∇(λ) for all λ (see for example [DR]). 2. General approach via tilting modules 2.1. Tilting modules and finitistic dimension Let us forget the stratified structure for a moment. So, let A just be a finite-dimensional, associative, and unital k-algebra. Recall, see [Mi], that an A-module T is called a generalized tilting module if T has fi- nite projective dimension, is ext-self-orthogonal, and its additive closure Add(T ) coresolves AA in a finite number of steps. The generalized cotilt- ing modules are defined dually. Looking at the homomorphisms in Db(A) from T •[i] to the tilting coresolution of AA one easily derives that pd(T ), in fact, equals the length of the shortest tilting coresolution of AA. Here Jo u rn al A lg eb ra D is cr et e M at h .V. Mazorchuk 79 for M ∈ A−mod we denote by M• the complex, whose only non-zero component is M , concentrated in degree zero. The trivial example of a generalized tilting module is P . If gldim(A) < ∞, then I is a generalized tilting module as well. In general I need not be a tilting module, since it may have infinite projective dimension. However, if I is a generalized tilting module then, embedding any M ∈ P<∞(A) into an injective module, and applying HomA(−, L), one derives that fdim(A) = pd(I). Moreover, in this case any M ∈ P<∞(A) can be substituted in Db(A) by its finite projective resolution, which then can be turned into a finite injective complex in Db(A), since I is a tilting module (see for example [MO, Lemma 4]). This implies that any M ∈ P<∞(A) has finite injective coresolution, in particular, P<∞(A) is contravariantly finite in A−mod, see [AR]. 2.2. Using self-dual tilting modules We have seen that finding non-trivial generalized tilting modules in A−mod can give some interesting information about the homological behavior of A−mod. Especially if such modules are self-dual with re- spect to some contravariant exact equivalence on A−mod (usually called a duality). A duality is called simple preserving if it preserve the isomor- phism classes of simple modules. A careful study of the proof of [MO, Theorem 1] shows that what is actually proved there is the following statement: Theorem 1. Let A be a finite-dimensional, associative, and unital k- algebra for which fdim(A) < ∞. Assume that there exists a duality on A−mod, and a generalized tilting A-module T , such that Q? ∼= Q for every indecomposable Q ∈ Add(T ). Then fdim(A) = 2 · pd(T ). Proof. Applying ? to the tilting coresolution of P gives a tilting resolution of I, in particular, pd(I) < ∞. Since fdim(A) < ∞ we can embed any M ∈ A−mod with pd(M) = fdim(A) into an injective module, apply HomA(−, L), and obtain pd(I) = pd(M) = fdim(A). Further, pd(I) is exactly the maximal degree l, for which Extl A(I, P ) does not vanish. The latter can be computed in Db(A) studying homomorphisms from the shifted tilting resolution of I to the tilting coresolution of P . Under our assumptions we can apply [MO, Lemma 1] and the arguments from [MO, Appendix]. The statement of the theorem follows. 2.3. Applications to stratified algebras Assuming A has some sort of stratification makes it in many cases possible to ensure the assumptions of Theorem 1. Indeed, assume that A is an Jo u rn al A lg eb ra D is cr et e M at h .80 On finitistic dimension SSS-algebra having a simple preserving duality (i.e. a duality, which preserves the isomorphism classes of simple modules). Then A is in fact properly stratified, the category F(∆) ∩ F(∇) equals Add(T ) for some generalized tilting module T called the characteristic tilting module, and the category F(∆) ∩ F(∇) equals Add(C) for some generalized tilting module C, called the characteristic cotilting module. Moreover, if T ∼= C, then all indecomposable direct summands of T are self-dual. The condition T ∼= C is satisfied, for example, for quasi-hereditary algebras. Hence we obtain (see [MO, Theorem 1 and Corollary 1]). Corollary 1. Let A be an algebra having a simple preserving duality. 1. If A is an SSS-algebra and T ∼= C then fdim(A) = 2 · pd(T ). 2. If A is quasi-hereditary then gldim(A) = 2 · pd(T ). 3. Using tilting and various filtration dimensions 3.1. Filtration (co)dimensions Let M be a class of A-modules and F(M) be the full subcategory in A−mod, which consists of all modules having a filtration with subquo- tients, isomorphic to modules from M. For an A-module N we say that N has M-filtration dimension (resp. codimension) l ∈ {0, 1, . . . ,∞} if there exists a resolution (resp. coresolution) of N by modules from F(M) and l is the length of the shortest such resolution. For properly stratified algebras and SSS-algebras the following filtration (co)dimensions appear in a natural way: the Weyl or standard filtration dimension dim∆(N) for M = {∆(λ), λ ∈ Λ}; the proper standard filtration dimension dim∆(N) for M = {∆(λ), λ ∈ Λ}; the good or costandard filtration codimension codim∇(N) for M = {∇(λ), λ ∈ Λ}; and the proper costandard fil- tration codimension codim ∇ (N) for M = {∇(λ), λ ∈ Λ}. If A is an SSS-algebra, then both dim∆(N) and codim ∇ (N) are well-defined for all N ∈ A−mod. In [MP, Lemma 1] it is shown that dim∆(N) = max{l|ExtlA(N,∇) 6= 0}, and codim ∇ (N) = max{l|ExtlA(∆, N) 6= 0}. In particular, codim ∇ (N) ≤ pd(∆) for all N , whereas dim∆(N) < ∞ is obviously equivalent to pd(M) < ∞ as P ∈ F(∆). We define dim∆(A), dim∆(A), codim∇(A), codim ∇ (A), fdim∆(A) and fcodim∇(A) in the nat- ural way and for an SSS-algebra we obtain codim ∇ (A) = pd(∆) = pd(T ) by [MP, Lemma 1]. For properly stratified algebras we dually have dim∆(A) = i. d.(∇) = i. d.(C). Moreover, by [MP, Lemma 2] we also have fdim∆(A) ≤ pd(∇) = pd(C) and fcodim∇(A) ≤ i. d.(∆) = i. d.(T ). These filtration (co)dimensions were reinterpreted in [MO, Subsec- tion 4.3] in terms of tilting complexes. Thus we have that dim∆(N) ≤ l Jo u rn al A lg eb ra D is cr et e M at h .V. Mazorchuk 81 if and only if N• ∈ Db(A) is quasi-isomorphic to a complex, T •, of tilting modules such that T i = 0 for all i < −l. 3.2. An “old” upper bound for fdim(A) The following upper bound for fdim(A) is stated in [MP] for properly stratified algebras. Here we formulate the result for SSS-algebras and present a different proof based on tilting resolutions (see also [MO, Corol- lary 5]). Theorem 2. Let A be an SSS-algebra. Then fdim(A) ≤ fdim∆(A) + pd(T ). Proof. If M ∈ P<∞(A), then pd(M) = max{l|Extl A(M, P ) 6= 0}. We substitute M• ∈ Db(A) by a quasi-isomorphic complex, T •, of tilting modules satisfying T i = 0 for all i < −dim∆(M), and we substitute P by its tilting coresolution of length pd(T ) (see Subsection 2.1). Since for the complexes of tilting modules the homomorphisms in Db(A) can be computed in the homotopic category, it is straightforward that pd(M) ≤ dim∆(M) + pd(T ) and the statement follows. If A is properly stratified, as an immediate consequence we have fdim(A) ≤ i. d.(C) + pd(T ), which is left-right symmetric and hence works for ifdim(A) as well. If A has a duality, everything reduces to fdim(A) ≤ 2 · pd(T ). As we have already seen in Subsection 2.3, the last bound is exact for quite a wide class of quasi-hereditary and stratified algebras, including Schur algebras, algebras associated with the BGG- category O and its parabolic analogues. 3.3. fdim(A) if one can control EndA(T ) Let A be an SSS-algebra. The endomorphism algebra R = EndA(T ) of the characteristic tilting module T is called the Ringel dual of A. The algebra EndA(T )opp is always an SSS-algebra with respect to the opposite order on Λ, see [AHLU2]. However, R does not need to be properly stratified, even in the case when A itself is properly strati- fied. The algebra R comes together with the Ringel duality functor F (−) = HomA(T,−) : A−mod → R−mod, which induces an exact equiv- alence between the category of A-modules having a proper costandard filtration and the category of R-modules having a proper standard filtra- tion. The Ringel dual R is properly stratified if and only if the module T has a filtration with subquotients isomorphic to N(λ) = T (λ)/TraceT <λ(T (λ)), where T<λ = ⊕µ<λT (µ) (see [FM]). In the case when R is properly Jo u rn al A lg eb ra D is cr et e M at h .82 On finitistic dimension stratified we denote by H(λ), λ ∈ Λ, the preimage under F of the inde- composable tilting R-module corresponding to λ, and by H the preimage under F of the characteristic tilting R-module T (R). The module H is called the two-step tilting module for A (since it is a tilting module for the Ringel dual of A). The following properties of H were obtained in [FM]: Theorem 3. Assume that R is properly stratified and H is the two-step tilting module for A. Then 1. H is a generalized tilting module; 2. pd(H) = fdim(A); 3. P<∞(A) coincides with the category of A-modules, which admit a finite coresolution by modules from Add(H), in particular, P<∞(A) is contravariantly finite. In particular, the module H is a good test module for fdim(A) and it completely describes P<∞(A) in the homological sense. It is also shown in [FM] that the existence of H makes it possible to relate fdim(A) with the projective dimension of the characteristic tilting module: Theorem 4. Let A be a properly stratified algebra having a simple pre- serving duality. Assume R is properly stratified. Then 1. fdim(A) = 2 · pd(T (R)). 2. fdim(A) = 2 · pd(T ), in particular, pd(T ) = pd(T (R)), if R has a simple preserving duality itself. 3.4. A new lower bound for fdim(A) Carefully combining the results of [MO] and [FM] one can deduce the following lower bound for the finitistic dimension of properly stratified algebras having a simple preserving duality. Theorem 5. Let A be properly stratified with a simple preserving duality ?. Then we have fdim(A) ≥ 2 · fdim∆(A). Proof. We have to produce a module from P<∞(A) of projective dimen- sion at least 2·fdim∆(A). For this it is enough to show that any A-module M , such that dim∆(M) = fdim∆(A), satisfies pd(M) ≥ 2 · fdim∆(A). Set k = fdim∆(A). By [MO, Lemma 6], M• is quasi-isomorphic to a finite complex, T •, of tilting modules, satisfying T i = 0 for all i < −k. Apply- ing ? gives a finite complex, C•, of cotilting modules satisfying Ci = 0 for Jo u rn al A lg eb ra D is cr et e M at h .V. Mazorchuk 83 all i > k. Using [FM, Lemma 11] one finds a (possibly infinite) complex, Q•, of tilting modules, which is quasi-isomorphic to C•, and which satis- fies Qi = 0 for all i > k. Moreover, using [FM, Lemma 12] one can also guarantee that T −k is non-trivial and is a direct summand of Qk. Using the arguments as in [MO, Section 3] one shows that there is a non-zero morphism from T •[−2k] to Q•, implying pd(M) ≥ 2k. It is interesting to compare the bound, given in Theorem 5, with the results, described in Subsection 3.3. For this we will need the following lemma: Lemma 1. Let A be an SSS-algebra and M ∈ F(∇) such that pd(M) < ∞. Then dim∆(M) = pd(F (M)). Proof. Taking the minimal projective resolution P• of M and applying [MO, Lemma 4.1] we obtain a finite complex, T •, of tilting modules, which is quasi-isomorphic to M• ∈ Db(A). Using the arguments from the proof of [MO, Lemma 5] one even shows that T • is quasi-isomorphic to a finite minimal (in the sense of [MO]) complex, Q•, of tilting modules satisfying Qi = 0, i > 0. In other words, the module M admits a finite tilting resolution. Applying F gives a projective resolution of F (M) and we see that the length of the minimal tilting resolution of M is exactly pd(F (M)). From [MO, Lemma 6] it also follows that the length of the minimal tilting resolution of M equals dim∆(M), completing the proof. Corollary 2. Let A be properly stratified and assume that R is also properly stratified. Then pd(T (R)) = fdim∆(A). Proof. By Lemma 1 we have pd(T (R)) = dim∆(H). Further, let M ∈ P<∞(A) be such that l = dim∆(M) = fdim∆(A). By Theorem 3, we have a short exact sequence M ↪→ H1 � K, where H1 ∈ Add(H) and K ∈ P<∞(A). In particular, dim∆(H1) and dim∆(K) do not exceed dim∆(M). Applying HomA(−,∇) we obtain that Extl A(H1,∇) surjects onto Extl A(M,∇) 6= 0 and hence dim∆(H1) = l by [MP, Lemma 1]. This implies dim∆(H) = l and completes the proof. An immediate corollary of Theorem 4 and Corollary 2 is: Corollary 3. Let A be properly stratified having a simple preserving duality. Assume that R is also properly stratified. Then fdim(A) = 2 · fdim∆(A). Jo u rn al A lg eb ra D is cr et e M at h .84 On finitistic dimension 4. A counterexample In [MP, Conjecture 1] it was conjectured that the finitistic dimension of a properly stratified algebra having a simple preserving duality always equals twice the projective dimension of the characteristic tilting module. As we saw above this is true under assumptions that R is properly strati- fied and has a simple preserving duality, which includes, in particular, the cases of quasi-hereditary algebras, and properly stratified algebras whose tilting modules are also cotilting. Unfortunately, in the full generality the statement of the conjecture is wrong. As a counter example one can consider the following algebra (the first counter example was constructed by the author, computed by Birge Huisgen-Zimmermann, and simplified by Steffen König). Let A be the path algebra of the quiver •1 α '' x 77 •2 β gg y gg modulo the relations αβ = x2 = y2 = xβ = αx = 0. The map α 7→ β, β 7→ α extends to an anti-involution on A and hence gives rise to a duality on A−mod. The radical filtrations of the projective, standard, and proper stan- dard modules look as follows: Jo u rn al A lg eb ra D is cr et e M at h . P (1) 1 α ��? ?? ??x ���� �� � 1 2 y ��? ?? ??β ���� �� � 1 2 β ���� �� � 1 P (2) = ∆(2) 2 y ��? ?? ??β ���� �� � 1 2 β ���� �� � 1 ∆(1) 1 x �� 1 ∆(2) 2 β �� 1 ∆(1) 1 . It follows that A is properly stratified. Since A has a duality, the socle filtrations of injective, costandard and proper costandard modules are duals of the radical filtrations of the corresponding projective, stan- dard and proper standard modules above. The indecomposable tilting Jo u rn al A lg eb ra D is cr et e M at h .V. Mazorchuk 85 A-modules have the following radical filtration: T (2) 1 x ���� �� α ��? ?? ? 1 2 β ���� �� y ��? ?? ? 1 α ���� �� x ��? ?? ? 1 2 β ���� �� 1 1 T (1) = ∆(1) 1 x �� 1 . Neither T (2) nor ∆(1) are projective, which implies that fdim(A) ≥ 1. Further it is easy to see that there are the following minimal projective resolutions of tilting modules: 0 → P (2) → P (1) → T (1) → 0 and 0 → P (2) → P (1) ⊕ P (1) → T (2) → 0, and hence pd(T ) = 1. It is also easy to see that any injection between tilting modules is an isomorphism. This and [MO, Lemma 6] implies fdim∆(A) = 0. Hence, by Theorem 2 we obtain fdim(A) ≤ fdim∆(A) + pd(T ) = 1 and thus fdim(A) = 1. In particular, in this example we have 2 · fdim∆(A) = 0 < fdim(A) = 1 < 2 · pd(T ) = 2. The example together with Theorem 2 motivates to make the follow- ing correction to [MP, Conjecture 1]: Corrected conjecture. Let A be a properly stratified algebra with a simple preserving duality. Then fdim(A) = fdim∆(A) + pd(T ). Remark that for algebras having a simple preserving duality we always have fdim∆(A) ≤ pd(T ) = codim ∇ (A), see [MP]. 5. A bound for ifdim(A) in the case of an SSS-algebra Up to this point all the results mentioned were about the projectively defined finitistic dimension. A natural question is: what can one say about the injectively defined version? In the case of an algebra having a (simple preserving) duality the answer is very easy: the injectively and the projectively defined finitistic dimensions obviously coincide. But what can be said in the general case? This question is still more or less open, see Section 6. Here we just present an easy upper bound for the case, when we have enough information about the Ringel dual of the algebra. Jo u rn al A lg eb ra D is cr et e M at h .86 On finitistic dimension Theorem 6. Let A be and SSS-algebra and assume that R is properly stratified. Then ifdim(A) ≤ fdim(A) ≤ fdim∆(A) + pd(T ). Proof. Because of Theorem 2 it is enough to prove that ifdim(A) ≤ fdim(A). To do this we will show that for any M with i. d.(M) = k < ∞ we have Extk A(H, M) 6= 0. From the definition of H(λ) it follows that there is an injection ∇(λ) ↪→ H(λ), and hence there is an injection L ↪→ H with cokernel, K say. Applying HomA(−, M) to the short exact sequence L ↪→ H � K and using i. d.(M) = k we get a surjection of Extk A(H, M) onto Extk A(L, M) 6= 0. This completes the proof. 6. Some comments and questions Summarizing the results of the paper we can say the following: if A is a properly stratified algebra having a simple preserving duality, then we have the following bounds for fdim(A): 2 · fdim∆(A) ≤ fdim(A) ≤ fdim∆(A) + pd(T ) ≤ 2 · pd(T ). (1) Most of the components of (1) have equivalent reformulations in other homological terms for A or for the Ringel dual R of A. In many cases, for example for quasi-hereditary algebras, we know that all inequalities in (1) are in fact equalities. We also know that the first and the third inequalities can be strict. This gives rise to the following question: 1. Let A be an SSS-algebra. How different can pd(T ) and fdim∆(A) be? The same for properly stratified algebras and for properly stratified algebras with duality. 2. Describe the class of SSS-algebras with properly stratified Ringel duals, satisfying pd(T ) = pd(T (R)) (remark that the last condition immediately makes all inequalities of (1) into equalities). The same for properly stratified algebras and for properly stratified algebras with duality. We saw that the module H, which appears in the case when R is properly stratified, can be used as a test module for fdim(A). It was shown in [FM] that EndA(H)opp is always an SSS-algebra. Hence, very natural questions are: 3. Find, in terms of A−mod, necessary and sufficient conditions for EndA(H) to be properly stratified. 4. Is there any relation between fdim(A) and fdim(EndA(H))? Jo u rn al A lg eb ra D is cr et e M at h .V. Mazorchuk 87 As we have already mentioned, much less is know about the injectively defined finitistic dimension, so even the following very general question is not answered: 5. Let A be an SSS-algebra or a properly stratified algebra. Can one use tilting modules to estimate or compute ifdim(A)? Acknowledgments The author acknowledges the supports of The Swedish Research Council, The Royal Swedish Academy of Sciences, and The Swedish Foundation for International Cooperation in Research and Higher Education (STINT). The author thanks Anders Frisk and Catharina Stroppel for their com- ments on the paper. References [AHLU1] I.Ágoston, D.Happel, E.Lukács, L.Unger, Finitistic dimension of standardly stratified algebras. Comm. Algebra 28 (2000), no. 6, 2745–2752. [AHLU2] I.Ágoston, D.Happel, E.Lukács, L.Unger, Standardly stratified algebras and tilting. J. Algebra 226 (2000), no. 1, 144–160. [AR] M.Auslander, I.Reiten, Applications of contravariantly finite subcategories. Adv. Math. 86 (1991), no. 1, 111–152. [CPS] E.Cline, B.Parshall, L.Scott, Stratifying endomorphism algebras. Mem. Amer. Math. Soc. 124 (1996), no. 591, viii+119 pp. [Dl1] V.Dlab, Properly stratified algebras. C. R. Acad. Sci. Paris Ser. 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Z. 193 (1986), no. 1, 113–146. Jo u rn al A lg eb ra D is cr et e M at h .88 On finitistic dimension Contact information V. Mazorchuk Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, SWEDEN E-Mail: mazor@math.uu.se URL: www.math.uu.se/∼mazor Received by the editors: 19.04.2004 and final form in 27.09.2004.

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