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On the global and \nabla-filtration dimensions of quasi-hereditary algebras

In this paper we consider how the \nabla-, \Delta- and global dimensions of a quasi-hereditary algebra are interrelated. We first consider quasi-hereditary algebras with simple preserving duality and such that if \mu<\lambda then \nabla fd(L(\mu))<\nabla fd(L(\lambda)) where \mu, \lambda are in the poset and L(\mu), L(\lambda) are the corresponding simples. We show that in this case the global dimension of the algebra is twice its \nabla-filtration dimension. We then consider more general quasi-hereditary algebras and look at how these dimensions are affected by the Ringel dual and by two forms of truncation. We restrict again to quasi-hereditary algebras with simple preserving duality and consider various orders on the poset compatible with quasi-hereditary structure and the \nabla-, \Delta- and injective dimensions of the simple and the costandard modules.


Introduction
Quasi-hereditary algebras were first introduced by Scott [19] in order to study highest weight categories in the representation theory of semisimple complex Lie algebras and algebraic groups, and many important results were proved by Cline, Parshall and Scott (see for example [3]). These algebras can be defined in the context of arbitrary finite-dimensional algebras, and they were studied from this point of view by Dlab and Ringel (see for example [5], [6]) and others. In particular, it turns out that quasi-hereditary algebras are quite common.
One important property of quasi-hereditary algebras is that they have finite global dimension.
Furthermore, there is a natural concept of ∇-filtration dimension for representations of quasihereditary algebras. This can be considered as a generalisation of the notion of injective dimenions. This was introduced for algebraic groups by Friedlander and Parshall [10] (where they define the notion of good filtration dimension which equals our notion of ∇-filtration dimension of a module).
Later work [15,17], shows that the understanding of the ∇-filtration dimension gives a strong hold on homological properties.
For Schur algebras, the ∇-filtration dimension of simple modules and of Weyl modules and the projective dimensions of Weyl modules have nice relationships with the partial order of the weights.
Moreover, since there is a duality fixing the simple modules, the combination of ∇-filtration dimension and the dual concept of ∆-filtration dimension, give us an exact relationship between the ∇-filtration dimension and the injective dimension of a module (for the regular blocks).
In this paper, we investigate to what extent the interrelations which were observed for Schur algebras hold for arbitrary quasi-hereditary algebras which have a duality fixing the simple modules.
These include the blocks of the category O defined by Bernšteȋn, Gel'fand and Gel'fand in [1].
As applications, we determine the ∇-filtration dimension and the global dimension of the Ringel duals of Schur algebras S(2, r). Furthermore, we show that a quasi-hereditary algebra with duality for which the ∇-filtration dimension is strictly increasing as a function on the poset has global dimension twice its ∇-filtration dimension. This applies in particular to the regular blocks of Schur algebras S(n, r) with p > n and regular blocks for category O ( [17, theorem 4.7, section 7]).
This proves a particular case of a conjecture of Caenepeel and Zhu [2] and Mazorchuk and Parker [13].

Preliminaries
Definition 1.1.1. Suppose S is a finite-dimensional algebra over a field k. Let L(λ) for λ ∈ Λ + be a full set of irreducible S-modules, and let P (λ) be the projective cover of L(λ). We fix a partial order (Λ + , ≤). We then define the standard module ∆(λ) to be the largest quotient of P (λ) with composition factors L(µ) such that µ ≤ λ.
Recall that S is quasi-hereditary if for each λ ∈ Λ + , (i) the simple module L(λ) occurs only once as a composition factor of ∆(λ), and (ii) the projective P (λ) has a filtration by standard modules where ∆(λ) occurs once, and if ∆(µ) occurs then µ ≥ λ.
The costandard modules ∇(λ), are defined dually by replacing projective by injective modules and quotients by submodules.
We work with finite-dimensional S-modules. We write F (∆) for the class of S-modules which have a filtration where the sections are ∆(µ) for various µ, and similarly we write F (∇) for the class of S-modules which have a filtration where the sections are ∇(µ) for various µ.
We henceforth assume that S is a quasihereditary algebra with poset (Λ + , ≤). Note that by the definition of a quasi-hereditary algebra all the projective S-modules belong to F (∆) and all the injective S-modules belong to F (∇). There are other ways of defining quasi-hereditary algebras, but they turn out to be equivalent. See [12] or [8, appendix] for a reasonably self-contained introduction to quasi-hereditary algebras.
We define Ext i S (−, −) in the usual way (using projective resolutions) on the category of S-modules. We will drop the subscript if it is clear which category we are working in. Definition 1.1.2. Any S-module X has a ∇-resolution, that is, there is an exact sequence following two equivalent conditions hold: (i) X has a ∇-resolution of length d but no ∇-resolution of length smaller than d; (ii) Ext i S (∆(λ), X) = 0 for all i > d and all λ ∈ Λ + , but there exists λ ∈ Λ + such that Ext d S (∆(λ), X) = 0.
(See [10, proposition 3.4] for a proof of the equivalence of (i) and (ii) where this property is known as the good filtration dimension of X.).
Dually we have the notion of ∆-filtration dimension. This is denoted as ∆. f. d.(X). We also define for a quasi-hereditary algebra S, But note that S S considered as a left S-module is projective and hence we have ∆. We have the following important lemma. It is possible that different partial orders on the set Λ + lead to the same quasi-hereditary structure.
(I.e. different partial orders may lead to the same standard and costandard modules.) Once we have a given quasi-hereditary structure (i.e. we are given the standard and costandard modules) we can replace the given partial order by a different one which gives the same standard and costandard modules but is which more labels would be incomparable.
That is if λ < µ and λ and µ are adjacent in the order (that is there is no ν ∈ Λ + such that µ < ν < λ), but L(λ) is not composition factor of ∆(µ) nor of ∇(µ) (and hence ∆(λ) is not a section of I(µ) by Brauer-Humphreys reciprocity), then we may safely remove this relation without affecting the standards or the costandards, since we still get the same modules by Definition 1.1.1.
We may continue removing relations in this fashion until we obtain some minimal partial order which still gives the original standards and costandards. Thus, we may assume that if µ < λ and µ and λ are adjacent in the order then L(µ) is a composition factor of ∆(λ) or of ∇(λ).
Essentially we have replaced the original parital order by one that is generated by the preorder µ < λ if L(µ) occurs as a composition factor of ∇(λ) or of ∆(λ).
In this paper we will often assume that S has a duality • fixing the simple modules. (Such a duality is sometimes known as strong duality.) For such an algebra, it then follows that the dual of the costandard module ∇(λ) • is isomorphic to ∆(λ). It is also clear that Ext . We ask whether equality holds. (This was orginally conjectured for Schur algebras in [16] and for more general S in [2] and [13].) For most of this section we will be assuming that S satisfies a particular property which we will call strong property A. (We will weaken this condition slightly in section 5). That is: Regular blocks of the Schur algebra satisfy this property as well as the regular blocks of category O. [17].
In the following we write S ∈ S n if S is a quasi-hereditary algebra with duality fixing the simples, with an ordering on the simples such that strong property A is satisfied and ∇. f. d.(S) = n.
Proof. We first note that the presence of strong property A gives us that if λ 1 and λ 2 are both in Λ + i for i ∈ {0, 1} then λ 1 and λ 2 are incomparable and hence Ext 1 Case (1): Assume first that the socle of Q has only L(λ i ) with λ i ∈ Λ + 1 . Then we have a (non-split) injective hull This latter Ext group is non-zero (consider the above exact sequence).
Case (2): Now suppose Q is arbitrary, then we have an exact sequence [15, lemma 2.5]. Using case (1) we know that Ext 2 S (Q • ,Q) = 0. We will show that there is an epimorphism from and this will be enough to show that the first Ext group is non-zero.
Apply Hom S (Q • , −) to the exact sequence for Q, this gives an exact sequence Now apply Hom S (−,Q) to the exact sequence This gives an exact sequence The composite of these two maps gives the desired epimorphism.
2.2. Assume now that S is a quasi-hereditary algebra in S n . We know an algebra S 1 in S 1 has global dimension 2. Moreover for every Proof. We have already proved this for n = 1. We now assume inductively that an algebra S n−1 ∈ S n−1 has global dimension 2(n − 1) and that for every and ∇. Applying Hom S (L(λ), −) to the above exact sequence and then Hom S (−, Q) to its • -dual, gives us (For the last equality see section 4.1).
We must show that Ext 2n S (Q • , Q) = 0. We note that the modules ∇(λ i ) with ∇. f. d.(L(λ i )) = n must be injective. Also Q must have at least one L(λ) as a composition factor with ∇. f. d.(L(λ)) = n.
Assume first that the socle of Q is a direct sum of L(λ i )'s with ∇. f. d.(L(λ i )) = n. Then we have the injective hull We also know that R is an S n−1 -module, (S n−1 as before) by construction. So by the inductive hypothesis we know that Ext 2(n−1) as before as I is injective.
We now consider the general case. Let U ⊂ Q be the largest submodule with no composition factors (L(λ i )) = n, and let V = Q/U . Then the socle of V has only composition factors L(λ i ), note also that V = 0. Consider the exact sequence Apply Hom S (Q • , −) to the exact sequence ( * ), this gives The last term is zero since S is known to have global dimension 2n.
Next, apply Hom S (−, V ) to the exact sequence The composite ψ • φ gives the required epimorphism.
2.3. The previous section proves a special case of the conjecture of [2,13] that the global dimension of any quasi-hereditary algebra S with simple preserving duality is twice its ∇-filtration dimension.
We suspect that a stronger property may be true. That is that one of the equivalent conditions of the following lemma hold.
We denote the images of the map By dimension shifting and duality we have that In a similar vein we have: Indeed, we have proved that the first condition of this lemma holds for our special case in the previous section. We will give another example of a quasi-hereditary algebra for which the first condition of this lemma holds in example 5.5.1.

∇-filtration and Global dimensions for Ringel duals
In this section we investigate the relationship between the ∇-and ∆-filtration dimensions for a quasi-hereditary algebra and its Ringel dual (as defined in [18]).

3.1.
A tilting module is a module with both a ∇-filtration and a ∆-filtration. There is a unique indecomposable tilting module T (λ) for each λ ∈ Λ + such that L(λ) occurs only once and any other composition factor L(µ) of T (λ) has µ < λ. Every tilting module is a direct sum of T (µ i ) for some µ i ∈ Λ + . A full tilting module T is a tilting module for which for all µ ∈ Λ + , T (µ) is a direct summand. We take a full tilting module T and form a Ringel dual S ′ = End S (T ) op .
A Ringel dual is also a quasi-hereditary algebra with poset (Λ + , ′ ), where ′ is the opposite ordering to on Λ + . We distinguish the standards, costandards etc. for a Ringel dual from that of the starting algebra by a prime. Different T lead to different 'Ringel duals' but it is unique up to Morita equivalence. So we often say the Ringel dual. There is a left exact functor F : S → S ′ which takes a module M to Hom S (T, M ) regarded as an S ′ -module in the usual manner.
Proof. (i). We take a minimal length tilting resolution for ∇(λ) (ii). We similarly take a minimal length injective resolution for ∇(λ)) and apply F to get a minimal length tilting resolution for ∆ ′ (λ). By a similar argument to that above we know that the length of a minimal tilting resolution for ∆ ′ (λ) is the same as its ∇-filtration dimension.
(iii) and (iv) follow by applying (i) and (ii) to the modules for S ′ and using the fact that S and S ′′ are Morita equivalent as quasi-hereditary algebas. Proof. We prove the first statement, the second is similar. Since Recall that the simples for the Schur algebra, S(n, r), as defined in [11], are indexed by the set of partitions of r into less than or equal to n parts, Λ + (n, r). A partition λ = (λ 1 , λ 2 , . . . , λ n ) We say a block is regular if all the partitions in a block are regular. We define for λ ∈ Λ + (n, r)

Truncation properties
In this section we investigate the behaviour of ∇-filtration dimensions and injective dimensions under two forms of truncation. Let S be a quasi-hereditary algebra with poset Λ + . We fix a saturated subset Π of Λ + , that is Π is a subset of Λ + with the property that if λ ∈ Π and µ ∈ Λ + then µ < λ implies µ ∈ Π. We write Γ for Λ + \Π. Let e Γ := λ∈Γ e λ , an idempotent; where we use a fixed decomposition of 1 into a sum of orthogonal primitive idempotents, and where e λ S is a projective module isomorphic to P (λ). Note however that if M is an S(Π)-module then the injective and projective dimensions as a module for S are usually larger than those as a module for S(Π).

4.2.
Let e = e Γ as before, then the algebra eSe is also quasi-hereditary, with respect to (Γ, ≤), and with standard modules e∆(λ) and costandard modules e∇(λ), for λ ∈ Γ (see [8, A3.11] or [9, §1.6]). We note that e∆(µ) = 0 if and only if µ ∈ Γ. If µ is in Γ and N is an S-module  We assume that the partial order < is minimal in the sense of section 1, and that S consists of one block (for C).

5.2.
Comparisons. First we list some easy observations.
• We have A implies D; this follows by induction on ≤, using [15, 2.5]. Moreover, if D holds then A and B are equivalent. So A (and D) imply B.
• A and C imply E.
• If C holds then B and E are equivalent.  We know that A, B, C, D and E all hold for the blocks of the Schur algebra consisting of regular weights [17]. We also know that in the case n = 3 (the first value of n for which there are primitive non-regular weights) that A, B and D hold but that C and E do not hold in general for the non-regular blocks.
As the representation theory for the Schur algebra is controlled by that of the Special linear group we now give an example for SL 3 where condition E fails (and necessarily C fails as well). We use the standard notation and terminology of algebraic groups as in [8].
We claim now that I(3, 0) is isomorphic to the tilting module T (4, 1) and hence is also projective. Now the tilting module T (2, 1) is isomorphic to T (1, 0)⊗ St and hence is isomorphic to the injective hull of L(1, 0) as a G 1 -module. Thus it is indecomposable as a G 1 module so by [7, proposition 2.1], the tilting module T (4, 1) is isomorphic to T (2, 1) ⊗ T (1, 0) F . The module T (4, 1) has simple socle L(3, 0) and its injective hull is I(3, 0) and it follows that T (4, 1) ∼ = I(3, 0) since both have the same ∇-quotients. Note that since B holds but E does not, this means that in particular that S(3, 6) is not isomorphic to its Ringel dual. Also note that the Ringel dual of S(3, 6) has property E but not B. Let S be the algebra kQ/I where Q is the quiver with relations (composing on the right): This is quasi-hereditary, with respect to the natural order on The composition factors of the ∇(i) are as follows Let S be the algebra kQ/I where Q is the quiver with the following relations: We take the natural order on the index set Λ + = {0, 1, 2, 3}. The structure of the standard modules is as follows.  Proof. (1) From the projective cover for ∆(1) (see above) we get Ext i (∆(1), ∆(2)).
Thus any non-split extension is isomorphic to ∇(1) (by dimensions).
Clearly Hom(∆(0), ∆(2)) = k, so we can now use the fact that It remains to consider i = 0, we substitute the dimensions proved in the Lemma and get an exact To complete the proof we will show that the map β * is zero.
Take an element in Ext 1 (∆(0), ∆(3)), say η, which is represented by then β * (η) is represented by the push-out of β. Suppose this is non-zero, then the middle term of the sequence β * (η) must be uniserial with a simple top L(0) and therefore, the top of V must also be simple, isomorphic to L(0). So there is an epimorphism ψ : e 0 S = P (0) → V . Now we will use the relation δ 1 γ 1 = ǫ 0 β 0 to derive a contradiction.

5.4.
We expect that C is independent of any of the other conditions. It certainly is not implied by any of them. But to construct an example with C but not A seems to need rather a lot of technical detail and would require many simple modules.
In summary B, D and E are independent of each other and none of these imply A or C, while A implies B and D but not C or E.
5.5. We now consider an algebra satisfying property A and show that the first condition of lemma 2.3.2 holds for this algebra. Since L(2) is not a composition factor of R and all the top composition factors of I • are L(2) we deduce that Hom S2 (I • , R) = 0 and hence the required Ext-space is isomorphic to Hom S2 (R • , R).
This is certainly non-zero.
General Case Let U be the largest submodule of Q which does not have L(2) as a composition factor, and let V be the quotient. Then the socle of V has only L(2) as composition factors.
Consider the exact sequence of S 2 -modules So each term has ∇-filtration dimension at most one. Now ∇. f. d.(Q) = 1, therefore at least one of U and V has ∇-filtration dimension equal to one.
If ∇. f. d.(V ) = 1 then proceed as in the previous proof, to show that there is a surjection the latter Ext group being non-zero by Case 2.
So assume now that ∇. f. d.(V ) = 0. Then actually V is a direct sum of ∇(2)'s and hence is injective. Using this we show that But U is as in Case 1 and hence this is non-zero.
Thus if property A holds for a quasi-hereditary algebra with simple preserving duality then if the global dimension is not twice the ∇-filtration dimension then we must have at least five simples.