Quasi-hereditary twisted category algebras

Markus Linckelmann and Michał Stolorz

Abstract We give a sufficient criterion for when a twisted finite category algebra over a field is quasi-hereditary, in terms of the partially ordered set of ℒ -classes in the morphism set of the category. We show that this is a common generalisation of a long list of results in the context of EI-categories, regular monoids, Brauer algebras, Temperley-Lieb algebras, partition algebras.

1 Introduction

Let k be a field and C a small category. Let α be a 2-cocycle in Z^{2}(C; k^{×}); that is, α is a map sending any two morphisms φ, ψ in Mor(C) for which ψ ∘ φ is defined to an element α(ψ, φ) in k^{×} such that for any three morphisms φ, ψ, τ for which the compositions ψ ∘ φ and τ ∘ ψ are defined, we have the 2-cocycle identity α(τ, ψ ∘ φ)α(ψ, φ) = α(τ ∘ ψ, φ)α(τ, ψ). The twisted category algebra k_{α}C is the k-vectorspace having the morphism set Mor(C) as a k-basis, with a k-bilinear multiplication given by ψφ = α(ψ, φ)(ψ ∘φ) if ψ ∘φ is defined, and ψφ = 0, otherwise. The 2-cocycle identity is equivalent to the associativity of this multiplication. The isomorphism class of k_{α}C depends only on the class of α in H^{2}(C; k^{×}), with k^{×} here understood as a constant contravariant functor on C. If α represents the trivial class, then k_{α}C ≅ kC, the category algebra of C over k. For any idempotent endomorphism e of an object X in C we denote by G_{e} the group of all invertible elements in the monoid e ∘ End_{C}(X) ∘ e. As in [7], we consider the wellknown extension to categories of the notion of Green relations in semigroups. In particular, two morphism s, t in C are ℒ -equivalent, if Mor(C) ∘ s = Mor(C) ∘ t. This is an equivalence relation on the set Mor(C), and if L, L′ are two ℒ -classes in Mor(C) we write L′ ≤ L if Mor(C) ∘ s′ ⊆ Mor(C) ∘ s for some (and hence any) morphisms s′ ∈ L′, s ∈ L. A morphism s : X → Y in C is called split if there is a morphism t : Y → X such that s ∘ t ∘ s = s. An ℒ -class L in Mor(C) is called split if it contains a split morphisms. In that case, all morphisms in L are split, and L contains an idempotent endomorphism of some object in C. A (possibly empty) subset I of Mor(C) is called an ideal in Mor(C) if Mor(C) ∘ I ∘ Mor(C) = I.

Theorem 1.1. Let C be a finite category, k a field and α ∈ Z^{2}(C; k^{×}). Suppose that the set of nonsplit morphism in C is an ideal in Mor(C), and suppose that for any ℒ -class L in Mor(C) there is a unique minimal split ℒ -class L′ such that L ≤ L′. Suppose that either char(k) = 0 or that char(k) > 0 does not divide the group orders |G_{e}|, where e runs over the idempotent endomorphisms in C. Then the k-algebra k_{α}C is quasi-hereditary.

The conclusion holds also with the uniqueness hypothesis on minimal split ℒ -classes replaced by the analogous condition on ℛ -classes. This hypothesis is not necessary in the sense that there are examples of categories with quasi-hereditary category algebras but where neither the ℒ -classes nor the ℛ -classes satisfy this uniqueness property - see Example 6.1 below. The hypotheses on C are trivially satisfied if all morphisms in C are split:

Corollary 1.2. Let C be a finite category, k a field and α ∈ Z^{2}(C; k^{×}). Suppose that all morphisms in C are split, and that either char(k) = 0 or that char(k) > 0 does not divide the group orders |G_{e}|, where e runs over the idempotent endomorphisms in C. Then the k-algebra k_{α}C is quasi-hereditary.

This corollary has been proved independently by Boltje and Danz [2]. Specialising 1.2 to regular monoids (that is, monoids in which all elements are split or, following semigroup terminology, von Neumann regular) yields the following:

Corollary 1.3. Let M be a finite regular monoid, k a field and α ∈ Z^{2}(M; k^{×}). Suppose that either char(k) = 0 or that char(k) > 0 does not divide the order of any subgroup of M. Then the algebra k_{α}M is quasi-hereditary.

Further specialised to α = 1 and k = ℂ we obtain a result due to Putcha [10, Theorem 2.1]: if M is a finite regular monoid, then the algebra ℂM is quasi-hereditary.

Many of the combinatorially defined cellular algebras, such as Brauer algebras, Temperley-Lieb algebras, partition algebras, are known to be quasi-hereditary for ‘most’ choices of parameters. Following Wilcox [15], these algebras can be interpreted as twisted monoid algebras, and hence Corollary 1.3 can be used to show that they are quasi-hereditary in many instances. The Brauer algebra B_{r}(δ) on 2r vertices and a nonzero parameter δ in the field k can be viewed as a twisted monoid algebra. By [15, §8], the underlying monoid is regular, and its maximal subgroups are symmetric groups on at most r letter, hence their twisted group algebras are semisimple if char(k) is either zero or greater than r. Thus Corollary 1.3 implies the following result, due to Graham and Lehrer:

Corollary 1.4 ([3, 4.16, 4.17]). Let r be a positive integer and δ a nonzero element in k. Suppose that either char(k) = 0 ar that char(k) > r. Then the Brauer algebra B_{r}(δ) is quasi-hereditary.

The maximal subgroups of the monoid underlying the cyclotomic Brauer algebra Br_{r,m} are wreath products of a cyclic group of order m and symmetric groups on at most r letters. We obtain a part of a result of Rui and Yu (not requiring the hypothesis on k being a splitting field of x^{m} − 1, but requiring that all parameters are nonzero):

Corollary 1.5 ([12, 5.9]). Let r, m be positive integers. Suppose that char(k) = 0 or that char(k) > r and char(k) does not divide m. Then the cyclotomic Brauer algebra B_{r,m} with all parameters nonzero is quasi-hereditary.

The Temperley-Lieb algebra T L_{r}(δ) is a subalgebra of B_{r}(δ). Following [15, §8], it can be identified as a twisted monoid algebra, where the underlying monoid is regular and has trivial maximal subgroups. Thus we obtain a result of Westbury:

Corollary 1.6 ([14, §6]). Let r be a positive integer and δ a nonzero element in k. Then the Temperley-Lieb algebra T L_{r}(δ) is quasi-hereditary.

The maximal subgroups of the monoid underlying a cyclotomic Temperley-Lieb algebra T L_{r,m} are direct products of cyclic groups of order m. As before, we obtain a part of a result of Rui and Xi:

Corollary 1.7 ([11, 6.2]). Let r, m be positive integers. Suppose that char(k) = 0 or that char(k) > 0 does not divide m. Then the Temperley-Lieb algebra T L_{r,m} with all parameters nonzero is quasi-hereditary.

Similarly, the partition algebra P_{r}(δ) admits a description as a twisted monoid algebra, where the underlying monoid is regular and has maximal subgroups isomorphic to symmetric groups on at most r letters. This yields a result of Martin [9] and Xi [16]:

Corollary 1.8 ([9, Prop. 3], [16, 4.12]). Let r be a positive integer and δ a nonzero element in k. Suppose that either char(k) = 0 or that char(k) > r. Then the partition algebra P_{r}(δ) is quasi-hereditary.

König and Xi have given in [5] a characterisation of quasi-hereditary cellular algebras; this has been used to determine precisely the possible choices of the field and parameter for which the above diagram algebras are quasi-hereditary.

The fact that Theorem 1.1 is a common generalisation of the results on diagram algebras mentioned above, has been one of the motivations for the present paper. Boltje and Danz showed recently in [1] that certain double Burnside rings are twisted monoid algebras as well. Theorem 1.1 can be generalised further to a statement to also encompass results in [6, 2.4, 4.5] showing that weight algebras of EI-categories (and hence in particular, of fusion systems) are quasihereditary. Following [7, 1.2], the isomorphism classes of simple k_{α}C-modules are parametrised by isomorphism classes of pairs (e, T), where e is an idempotent endomorphism of an object of C and where T is a simple module over the twisted group algebra k_{α}G_{e}, whith α here understood as the restriction of α to G_{e}. The following definition is from [7, 1.4]; the special case of finite EI-categories has been defined in [6, 4.4].

Definition 1.9. Let C be a finite category, k a field, and α ∈ Z^{2}(C; k^{×}). A weight of k_{α}C is a pair (e, T) consisting of an idempotent endomorphism e of an object X in C and a projective simple k_{α}G_{e}-module T. A weight algebra W (k_{α}C) of k_{α}C is a k-algebra of the form W (k_{α}C) = ck_{α}Cc, where c is an idempotent in k_{α}C with the property that cS = S for every simple k_{α}Cmodule S parametrised by a weight, and cS′ = {0} for every simple k_{α}C-module S′ which is not parametrised by a weight.

Remarks 1.10. Let C be a finite category, k a field and α ∈ Z^{2}(C; k^{×}).

(1) An idempotent c ∈ k_{α}C as in the Definition 1.9 is a sum of pairwise orthogonal primitive idempotents i in k_{α}C such that the unique simple quotient of the projective indecomposable k_{α}C-module k_{α}Ci is parametrised by a weight, and such that no primitive idempotent ocurring in a decomposition of 1 − c has this property. This defines the idempotent c uniquely up to conjugation in k_{α}C and hence the algebra W (k_{α}C) is determined uniquely up to isomorphism. The number of isomorphism classes of simple W (k_{α}C)-modules is equal to the number of isomorphism classes of weights of k_{α}C.

(2) If char(k) = 0, or if char(k) > 0 does not divide the order of |G_{e}| for any idempotent endomorphism e in C, then W (k_{α}C) = k_{α}C. Indeed, in that case the twisted group algebras k_{α}G_{e} are semisimple, hence every simple k_{α}C module is parametrised by a weight.

(3) Suppose that k is algebraically closed and that char(k) = p > 0. Let F be a fusion system of a p-block b of a finite group G for which the 2-cocycle gluing problem [6, 4.2] has a solution α. If C is the full subcategory F̄^{c} of the orbit category F̄ of F-centric subgroups of P, then W (k_{α}C) is isomorphic to the algebra F̄(b) defined in [6, 4.4].

Theorem 1.11. Let C be a finite category, k a field, and α ∈ Z^{2}(C; k^{×}). Set A = k_{α}C. Let c be an idempotent in A such that for any simple A-module S we have cS = S if S is parametrised by a weight, and cS = {0}, otherwise. Suppose that the set N of nonsplit morphism in C is an ideal in Mor(C), and suppose that for any ℒ -class L in Mor(C) there is a unique minimal split ℒ -class L′ such that L ≤ L′. Suppose that c commutes with the images in A of all idempotent endomorphisms in C. Then the weight algebra W (k_{α}C) = cAc is quasi-hereditary.

As mentioned above, if the group orders |G_{e}| are invertible in k, then c = 1, and the additional hypotheses on c holds trivially; thus Theorem 1.1 is indeed a special case of Theorem 1.11. The extra hypothesis on c ensures that the chain of ideals constructed in the proof of Theorem 1.1 becomes a hereditary chain upon multiplying the ideals by c on both sides. This hypothesis can be replaced by an a priori weaker but more technical hypothesis - see Remark 6.3 below. The extra hypothesis on c holds if C is a finite EI-category (that is, a category in which all endomorphisms are isomorphisms, a concept introduced by Lück [8]), and hence we obtain the following result:

Corollary 1.12 ([6, 2.4]). Let C be a finite EI-category, k a field, and α ∈ Z^{2}(C; k^{×}). Then the k-algebra W (k_{α}C) is quasi-hereditary.

Proof. The only idempotent endomorphisms in C are the identity morphisms Id_{X}, where X runs over the objects of C. The split morphisms in C are isomorphisms, and the nonisomorphisms in C form an ideal. For s : X → Y a morphism in C, the ℒ -class L = L_{Id_{X}} of Id_{X} is the unique minimal split ℒ -class satisfying L_{s} ≤ L. Any idempotent in a twisted finite category algebra k_{α}C has a conjugate which commutes with all identity morphisms, and hence we may choose c to commute with Id_{X} for all objects X in C. Thus the hypotheses of Theorem 1.11 are satisfied; the result follows.

Applied to fusion systems of blocks, this yields also [6, 4.5].

Remark 1.13. It is possible for a nonunitary semigroup S to have a unitary semigroup algebra kS over some field k (for instance, this is always the case if S is a finite inverse semigroup). In the same vein, the results of this paper can be adapted for finite semicategories (that is, categories without the requirement for the existence of identity morphisms at each object) so long as their twisted category algebras are unitary.

2 On idempotent ideals in algebras

Let A be a finite-dimensional algebra over a field k. We will use without further comment the following standard facts on idempotents in finite-dimensional algebras. The isomorphism class of a simple A-module S is parametrised by a unique conjugacy class of primitive idempotents i in A satisfying iS ≠ {0}, and then Ai is a projective cover of S. The image of a primitive idempotent i in any quotient of A is either zero or a primitive idempotent. A primitive decomposition of an idempotent e in A is a set E of pairwise orthogonal primitive idempotents such that ∑_{i∈E} i = e. Any two primitive decompositions of e are conjugate (this is the algebra theoretic version of the Krull-Schmidt Theorem). This can be used to show that for any two idempotents e, f in A, there is a conjugate f′ of f which commutes with e, and then ef′ = f′e is either zero or an idempotent. It is well-known that an ideal I in A satisfies I^{2} = I if and only if I = AeA for some idempotent e in A. If in addition I = AeA is projective as a left A-module, then every indecomposable direct summand of I as a left A-module is isomorphic to an indecomposable direct summand of Ae; for the convenience of the reader, we include the argument to show this in the proof of the next result. If every indecomposable direct summand of a finitely generated projective A-module U is isomorphic to a direct summand of Ae, then eU is a projective eAe-module.

Proposition 2.1. Let A be a finite-dimensional algebra over a field k, e an idempotent in A and f an idempotent in Z(eAe). Suppose that the ideal AeA is projective as a left A-module. Then the ideal AfA is a direct summand of AeA as a left A-module; in particular, AfA is projective as a left A-module.

Proof. Let X be a k-basis of eA, and set n = |X| = dim_{k}(eA). The inclusions Aex ⊆ AeA, where x ∈ X, induce a surjective homomorphism of left A-modules ⊕_{x∈X} Aex → AeA. Each summand Aex is isomorphic to a quotient of Ae. Thus, as a left A-module, AeA is a quotient of a direct sum (Ae)^{n} of n copies of Ae. If AeA is projective as a left A-module, then AeA is isomorphic to a direct summand of (Ae)^{n} as a left A-module. Thus there is an isomorphism of left A-modules α : AeA ≅ ⊕_{i=1}^{n} Ae_{i}, where each e_{i} is either zero or an idempotent in eAe. Since α is an Ahomomorphism, we have α(fA) ⊆ ⊕_{i=1}^{n} fAe_{i} = ⊕_{i=1}^{n} fAfe_{i}, where the second equality uses the fact that f ∈ Z(eAe). Again, since α is an A-homomorphism, we have α(AfA) ⊆ ⊕_{i=1}^{n} Afe_{i}. Since α is an isomorphism, there exist unique elements y_{i} ∈ AeA such that α(y_{i}) = fe_{i}, for 1 ≤ i ≤ n. Then also α(fy_{i}) = fe_{i}, and hence α(AfA) = ⊕_{i=1}^{n} Afe_{i}, which is a direct summand of α(AeA) as a left A-module. The result follows.

Proposition 2.2. Let A be a finite-dimensional algebra over a field k and e, c idempotents in A. Suppose that c is conjugate to an idempotent c′ in A such that ec′ = c′e ∈ Z(eAe). Then cAec′Ac is an ideal in cAc. Moreover, if AeA is projective as a left A-module, then cAc′eAc is projective as a left cAc-module.

Proof. Two conjugate elements in an algebra generate the same ideal, and hence we may assume that c = c′ commutes with e and that ec ∈ Z(eAe). Then ec is either zero or an idempotent in Z(eAe). Thus 2.1 applied to f = ec implies that AecA is projective as a left A-module. Moreover, every indecomposable direct summand of AecA as a left A-module is isomorphic to a direct summand of Aec, hence of Ac. It follows that AecAc is a projective A-module with this property, and therefore cAecAc is projective as a cAc-module.

Lemma 2.3. Let C be a finite category, k a field, and α ∈ Z^{2}(C, k^{×}). Let e, f be idempotent endomorphisms of some objects in C.

(i) The elements ê = α(e, e)^{−1}e and f̂ = α(f, f)^{−1}f are idempotents in the algebra k_{α}C. (ii) If e and f belong to the same 𝒥 -class, then the idempotents ê and f̂ are conjugate in k_{α}C.

Proof. Set A = k_{α}C. The fact that ê and f̂ are idempotents in the twisted category algebra A follows immediately from the definition of the multiplication of morphisms in A. Suppose that J_{e} = J_{f}. Set M = Mor(C). Then M ∘ e ∘ M = M ∘ f ∘ M. Thus e = u ∘ f ∘ v for some morphisms u ∈ e ∘ M ∘ f and v ∈ f ∘ M ∘ e. Right multiplication by u on Ae = Aê induces an A-homomorphism Aê → Af̂, and right multiplication by v on Af = Af̂ induces an A-homomorphism Af̂ → Aê. The composition of these yields an automorphism of Ae, hence Af has a direct summand isomorphic to Ae. Exchanging the roles of e and f shows that Ae has a direct summand isomorphic to Af, hence both are isomorphic. This implies that ê and f̂ are conjugate.

Remark 2.4. The results in this section remain true with k replaced by a complete discrete valuation ring O. See for instance [13, §3, §4] for background material on idempotents in Oalgebras.

3 On ideals in categories

We refer to [7] for notation and well-known background material on Green relations for categories. For t a morphism in a small category C, we denote by L_{t}, R_{t}, J_{t} its ℒ -class, ℛ -class, 𝒥 -class, respectively. That is, the set L_{t} consists of all morphisms t′ satisfying Mor(C) ∘ t = Mor(C) ∘ t′, the set R_{t} consists of all morphisms t′ satisfying t ∘ Mor(C) = t′ ∘ Mor(C), and the set J_{t} consists of all morphisms t′ satisfying Mor(C) ∘ t ∘ Mor(C) = Mor(C) ∘ t′ ∘ Mor(C). For any two ℒ -classes L, L′ we write L ≤ L′ if Mor(C) ∘ L ⊆ Mor(C) ∘ L′; this defines a partial order on the set of ℒ -classes. Similarly, we have partial orders on the sets of ℛ -classes and 𝒥 -calsses given for any two ℛ -classes R, R′ by by R ≤ R′ if R ∘ Mor(C) ⊆ R′ ∘ Mor(C), and for any two 𝒥 -classes J, J′ by J ≤ J′ if Mor(C) ∘ J ∘ Mor(C) ⊆ Mor(C) ∘ J′ ∘ Mor(C). Any ideal I in Mor(C) is a disjoint union of 𝒥 -classes, and the k-span k_{α}I is an ideal in k_{α}C, where α ∈ Z^{2}(C; k^{×}). We adopt the convention that the empty ideal in Mor(C) corresponds to the zero ideal in k_{α}C.

Proposition 3.1. Let C be a finite category and let t, t′ ∈ Mor(C). If L_{t} < L_{t′} or R_{t} < R_{t′}, then J_{t} < J_{t′}.

Proof. Set M = Mor(C). Arguing by contradiction, suppose that L_{t} < L_{t′} and J_{t} = J_{t′}. That is, M ∘ t is a proper subset of M ∘ t′, and M ∘ t ∘ M = M ∘ t′ ∘ M. In particular, t′ = u ∘ t ∘ v for some morphisms u, v. Thus there is a surjective map M ∘ u ∘ t → M ∘ t′ sending a morphism w to w ∘ v. In particular, |M ∘ u ∘ t| ≥ |M ∘ t′|. However, M ∘ u ∘ t is a subset of M ∘ t, hence |M ∘ t| ≥ |M ∘ t′|. This contradicts the inequality |M ∘ t| < |M ∘ t′|, and shows that if L_{t} < L_{t′}, then J_{t} < J_{t′}. A similar argument yields the statement for ℛ -classes instead of ℒ -classes.

If a 𝒥 -class J of a morphism in a category C contains a split morphism, then all morphisms in J are split.

Corollary 3.2. Let C be a finite category, I a proper ideal in Mor(C), and t a split morphism in Mor(C) ∖ I. The following are equivalent:

(i) The 𝒥 -class J_{t} is minimal in the set of split 𝒥 -classes not contained in I. (ii) The ℒ -class L_{t} is minimal in the set of split ℒ -classes not contained in I. (iii) The ℛ -class R_{t} is minimal in the set of split ℛ -classes not contained in I.

The next Proposition describes some basic properties of finite categories in which the nonsplit morphisms form an ideal.

Proposition 3.3. Let C be a finite category such that the set of nonsplit morphisms is an ideal in Mor(C). Then the following hold.

(i) Every endomorphism of an object in C is split. (ii) The set of split morphisms in Mor(C) is closed under composition of morphisms. (iii) If X, Y are objects in C such that Hom_{C}(X, Y) contains a nonsplit morphism, then every morphism in Hom_{C}(X, Y) is nonsplit, and Hom_{C}(Y, X) = ∅. (iv) There is a positive integer i such that N^{i} = ∅.

(v) If t is a split morphism in Mor(C) then there are idempotents e, f in Mor(C) such that L_{t} = L_{e} and R_{t} = R_{f}.

Proof. Let s be an endomorphism of an object in C. Since C is finite, there is a positive integer n such that s^{n} is an idempotent endomorphism; in particular, s^{n} is split. Since the set N of nonsplit morphisms is an ideal it follows that s is split, whence (i). Let now s, t be split morphisms in C such that t∘s is defined. Since s, t are split, there are morphisms s′, t′ satisfying s = s ∘ s′ ∘ s and t = t ∘ t′ ∘ t. Then s′ ∘ t′ ∘ t ∘ s is an endomorphism of an object in C, hence split by (i). Since N is an ideal it follows that t ∘ s is split. This proves (ii). Let u : X → Y be a morphism in C. If there is a morphism v : Y → X, then v ∘ u is an endomorphism of X, hence split by (i), and thus u is split because N is an ideal. This implies (iii). It follows from (iii) that if

\xymatrix{X_{1} \ar[r]^{u_{1}} & X_{2} \ar[r]^{u_{2}} \ar[r] & ⋯ \ar[r]^{u_{r−1}} & X_{r}}

is a sequence of nonsplit morphisms u_{j}, then the objects X_{j} in this sequence are pairwise different, and hence the integer r is bounded by the number of objects in C, which proves (iv). To prove (v), one can take e = t′ ∘ t and f = t ∘ t′, where t′ satisfies t ∘ t′ ∘ t = t. This completes the proof.

Lemma 3.4. Let C be a finite category such that the set N of nonsplit morphisms is an ideal in M = Mor(C). Let e be an idempotent endomorphism in M. We have

M ∘ e ∘ M = (∪_{e′} M ∘ e′) ∪ (M ∘ e ∘ N) ,

where the first union runs over the set of idempotent endomorphisms e′ contained in M ∘ e ∘ M.

Proof. Clearly the union on the right side is contained in the left side. For the converse inclusion, let s, t ∈ M such that s ∘ e ∘ t is defined. If t ∈ N, then s ∘ e ∘ t ∈ M ∘ e ∘ N, which is a subset of the right side. If t ∉ N, then t is split and by 3.3 (ii) and (v), e ∘ t is split and L_{e∘t} = L_{e′} for some idempotent endomorphism e′. Then clearly e′ ∈ M ∘ e ∘ M and s ∘ e ∘ t ∈ M ∘ e′ as required.

Proposition 3.5. Let C be a finite category such that the set N of nonsplit morphisms is an ideal in M = Mor(C). Let I be a proper ideal in M, and let J be a split 𝒥 -class which is minimal in the set of split 𝒥 -classes not contained in I. Suppose that J ∘ N ⊆ I. We have

M ∘ J ∘ M ⊆ M ∘ J ∪ I .

Proof. By Lemma 3.4 we have M ∘ e ∘ M = ∪_{e′} M ∘ e′ ∪ M ∘ e ∘ N, with e′ running over the idempotent endomorphisms contained in M ∘ e ∘ M. The term M ∘ e ∘ N as well as the terms M ∘ e′ with J_{e′} < J_{e} are contained in I by the assumptions. The remaining terms are contained in M ∘ J, whence the result.

Proposition 3.6. Let C be a finite category such that the set N of nonsplit morphisms is an ideal in Mor(C). Let I be a proper ideal in Mor(C). Then there exists a 𝒥 -class J in Mor(C) with the following properties.

(i) J is minimal in the set of split 𝒥 -classes not contained in I. (ii) J ∘ N ⊆ I. Moreover, if J is a 𝒥 -class satisfying (i) and (ii), then every ℒ -class L contained in J is minimal in the set of split ℒ -classes not contained in I, and satisfies L ∘ N ⊆ I.

Proof. Since I is a proper ideal in Mor(C), it follows that Mor(C) ∖ I contains a split morphism. Indeed, otherwise I would contain the identity morphisms Id_{X} for each object X in C hence I would be equal to Mor(C). Choose a minimal split 𝒥 -class J not contained in I such that N^{i} ∘ J ≠ ∅, with i ≥ 0 maximal possible with the convention N^{0} = Mor(C). The integer i is well-defined by 3.3 (iv). Suppose that s ∘ N is not contained in I for some morphism s : Y → Z in J. That is, there is a morphism u : X → Y in N such that s ∘ u ∉ I. Then u, and hence Id_{X}, are not contained in the ideal I. Let J′ be a minimal split 𝒥 -class not contained in I such that J′ ≤ J_{Id_{X}}. Note that every morphism in J′ factors through Id_{X}. Since J′ is split, J′ contains an endomorphism s′ of X. Thus the composition s ∘ u ∘ s′ is defined. Since N^{i} ∘ s is nonempty and u ∈ N, hence s ∘ u ∈ N, it follows that N^{i+1} ∘ s′ is nonempty. This contradicts the initial choice of J, and shows that J ∘ N ⊆ I. If J satisfies (i) and (ii), it follows from 3.2 that every ℒ -class L contained in J is minimal in the set of split ℒ -classes not contained in I, whence the last statement.

Proposition 3.7. Let C be a small category. Set M = Mor(C). Suppose that for any ℒ -class L in M there is a unique minimal split ℒ -class L′ such that L ≤ L′. Let I be a proper ideal in M, and let L_{1}, L_{2} be two different minimal split ℒ -classes not contained in I. Then

M ∘ L_{1} ∩ M ∘ L_{2} ⊆ I .

Proof. Let L be an ℒ -class contained in M ∘ L_{1} ∩ M ∘ L_{2}. Then L ≤ L_{1} and L ≤ L_{2}. By the assumptions, there is a unique minimal split ℒ -class L′ such that L ≤ L′. The uniquenss of L′ implies that L′ ≤ L_{1} and L′ ≤ L_{2}. Since L_{1} ≠ L_{2}, this implies L′ < L_{1} and L′ < L_{2}. The minimality of L_{1}, L_{2} forces L′ ⊆ I. Since I is an ideal and since L ≤ L′, this implies that L ⊆ I.

Proposition 3.8. Let C be a finite category. Suppose that the set of nonsplit morphisms N in C is an ideal in M = Mor(C). Suppose that for any ℒ -class L in M there is a unique minimal split ℒ -class L′ such that L ≤ L′. Let I be a proper ideal in in M, and let J, J′ be two different minimal split 𝒥 -classes not contained in I. Suppose that J ∘ N ⊆ I and J′ ∘ N ⊆ I. Then M ∘ J ∘ M ∩ M ∘ J′ ∘ M ⊆ I.

Proof. By 3.5 we have M ∘ J ∘ M ∩ M ∘ J′ ∘ M ⊆ (M ∘ J ∪ I) ∩ (M ∘ J ∪ I) = (M ∘ J ∩ M ∘ J′) ∪ I. Since the 𝒥 -classes J, J′ are different, they contain no common ℒ -class. Moreover, by 3.6, any ℒ -class L contained in J or in J′ is a minimal split ℒ -class not contained in I and satisfies L ∘ N ⊆ I. It follows from 3.7 that the intersection M ∘ J ∩ M ∘ J′ is contained in I, whence the result.

Proposition 3.9. Let C be a finite category, k a field, and α ∈ Z^{2}(C; k^{×}). Suppose that the set of nonsplit morphisms N in C is an ideal in M = Mor(C). Suppose that for any ℒ -class L in M there is a unique minimal split ℒ -class L′ such that L ≤ L′. Let I be a proper ideal in M, and let J be a minimal split 𝒥 -class not contained in I and satisfying J ∘ N ⊆ I. Then k_{α}[M ∘ J ∘ M]/k_{α}[M ∘ J ∘ M ∩ I] is projective as a left k_{α}C/k_{α}I-module.

Proof. By 3.5, we have M ∘ J ∘ M ⊆ M ∘ J ∪ I. Thus

k_{α}[M ∘ J ∘ M]/k_{α}[M ∘ J ∘ M ∩ I] = k_{α}[M ∘ J]/k_{α}[M ∘ J ∩ I]

The set M ∘ J is the union of the sets M ∘ e, where e runs over a set of representatives X of the ℒ -classes contained in J. By Proposition 3.3(v) we may chose X which contains only idempotent endomorphisms. This yields the following sum of left k_{α}C-modules

k[M ∘ J] = ∑_{e∈X} k[M ∘ e] = ∑_{e∈X} k_{α}Cê .

For different e, e′ in X, the intersection M ∘ e ∩ M ∘ e′ is contained in I, thanks to 3.7. Thus this sum becomes a direct sum upon taking the quotient by the ideal k_{α}I, and each summand k_{α}[M ∘ e]/k_{α}[M ∘ e ∩ I] is isomorphic to k_{α}C/k_{α}Iē, where ē is the image of the idempotent ê = α(e, e)^{−1}e in the quotient k_{α}C/k_{α}I. The statement follows.

4 Proof of Theorem 1.1

Let C be a finite category satisfying the assumptions of Theorem 1.1. We set M = Mor(C) and denote by N the ideal of nonsplit morphisms in M. Let k be a field and α ∈ Z^{2}(C; k^{×}). Set A = k_{α}C. We denote by rad(A) the Jacobson radical of A.

If e is an idempotent in A and S a simple A-module, then either eS is zero, or eS is a simple eAe-module. By Green [4, 6.2], the correspondence sending S to eS induces a bijection between isomorphism classes of simple A-modules not annihilated by e and isomorphism classes of simple eAe-modules. Moreover, the inverse of this correspondence can be described as follows: if T is a simple eAe-module, then the A-module Ae ⊗_{eAe} T has a unique maximal submodule, hence a unique simple quotient S, and then eS ≅ T. In conjunction with the description of a projective cover of a simple A-module S, it follows that if e is an idempotent satisfying eS ≠ {0}, then there is a primitive idempotent i such that i = ie = ei and such that Ai is a projective cover of S. It follows further that if R is a set of isomorphism classes of simple A-modules, then there is an idempotent e such that eS = S for every simple A-module whose isomorphism class belongs to R, and eS = {0} for any simple A-module whose isomorphism class does not belong to R. The idempotent e is uniquely determined up to conjugation by the set R, adopting the convention e = 0 if R is empty.

We will further use the parametrisation of isomorphism classes of simple A-modules from [7]. More precisely, the isomorphism class of a simple A-module S is parametrised by the isomorphism class of a pair (e, T), where e is a minimal idempotent endomorphism of an object in C such that eS ≠ {0}, and where T ≅ eS is a simple eAe-module. As mentioned in 2.3, the image in A of an idempotent endomorphism in C need not be an idempotent, but the scalar multiple ê = α(e, e)^{−1}e is an idempotent in A, and hence eAe = êAê is a unitary algebra with unit element ê. If a simple A-module S is parametrised by a pair (e, T) as before, then T is annihilated by the ideal of noninvertible morphisms in e ∘ M ∘ e, hence T can be viewed as a simple k_{α}G_{e}-module, where G_{e} is the maximal subgroup of the monoid e ∘ M ∘ e.

Let J be a split 𝒥 -class J. Choose an idempotent endomorphism e in J; thus G_{e} is the maximal subgroup of the monoid e ∘ M ∘ e. By [7, 2.6], this monoid is a disjoint union

e ∘ M ∘ e = G_{e} ∪ M_{e} ,

where M_{e} consists of all morphisms in e ∘ M ∘ e whose 𝒥 -class is strictly smaller than J. Moreover, M_{e} is an ideal in e∘M ∘e. Since endomorphisms of objects in C are all split, it follows that M_{e} consists of split morphisms. We denote by

σ_{e} : k_{α}(e ∘ M ∘ e) → k_{α}G_{e}

the canonical surjective k-algebra homomorphism with kernel k_{α}M_{e}.

We construct a chain of ideals in A which will be shown to be a hereditary chain. We start by defining a chain of ideals in M as follows. Set I_{0} = ∅. For n ≥ 0, if I_{n} is already defined, define I_{n+1} by

I_{n+1} = I_{n} ∪ (∪_{J} M ∘ J ∘ M) ,

where J runs over the split 𝒥 -classes which are minimal in the set of split 𝒥 -classes not contained in I_{n}, and which satisfy J ∘ N ⊆ I_{n}. Since the ideals I_{n} in M are generated by split 𝒥 -classes, hence by idempotent endomorphisms, we have I_{n}^{2} = I_{n} for n ≥ 0. It follows from 3.6 that if I_{n} is a proper ideal, then I_{n+1} is strictly bigger than I_{n}, and hence for n sufficiently large, we have I_{n} = M. For n ≥ 0 we set

H_{n} = k_{α}I_{n} ;

that is, H_{n} is the k-subspace of A spanned by the image of the set I_{n} in A. Since I_{n} is an ideal in the morphism set M satisfying I_{n} ∘ I_{n} = I_{n}, it follows that H_{n} is an ideal in the algebra A satisfying

H_{n}^{2} = H_{n} ,

for n ≥ 0. Let n ≥ 0 such that I_{n} is a proper ideal in M, or equivalently, such that H_{n} is a proper ideal in A. If J, J′ are two different split 𝒥 -classes contained in I_{n+1} ∖ I_{n}, then M ∘ J ∘ M ∩ M ∘ J′ ∘ M is contained in I_{n} by 3.8. It follows that H_{n+1}/H_{n} is the direct sum of the A/H_{n}-modules

k_{α}[M ∘ J ∘ M]/k_{α}[M ∘ J ∘ M ∩ I_{n}] ,

with J running over the split 𝒥 -classes in I_{n+1} ∖ I_{n}. By 3.9, each of these summands is projective as an A/H_{n}-module. It follows that H_{n+1}/H_{n} is projective as an A/H_{n}-module.

In order to show that A is quasi-hereditary, it remains to show that H_{n+1}rad(A)H_{n+1} ⊆ H_{n}. Since H_{n+1} is generated, as an ideal, by H_{n} and by idempotent endomorphisms in I_{n+1} ∖ I_{n}, it suffices to show that erad(A)e′ ⊆ H_{n}, where e, e′ are idempotents in I_{n+1} ∖ I_{n}. If e, e′ belong to different 𝒥 -classes, then eAe′ is spanned by e ∘ M ∘ e′, and it follows again from 3.8, that this set is contained in I_{n}, whence eAe′ ⊆ H_{n}. If e, e′ belong to the same 𝒥 -class, then the idempotents ê, ê′ in A are conjugate by 2.3. Thus we may assume that e′ = e. Then

eAe = k_{α}G_{e} ⊕ k_{α}M_{e} ,

where M_{e} = e ∘ M ∘ e ∖ G_{e}. In particular, since all morphisms in M_{e} are split and belong to 𝒥 -classes strictly smaller than J, we have M_{e} ⊆ I_{n}. The hypothesis on the characteristic of k implies that rad(k_{α}G_{e}) = {0}, and hence erad(A)e = rad(eAe) ⊆ k_{α}M_{e} ⊆ H_{n} as required. This completes the proof of Theorem 1.1.

5 Proof of Theorem 1.11

We use the notation and hypotheses from Theorem 1.11. In particular, we have A = k_{α}C, and c is an idempotent in A with the property that cS = S for any simple A-module S which is parametrised by a weight, and cS = {0} for any simple A-module S which is not parametrised by a weight. This determines c up to conjugation by an element in A^{×}, and we have cAc ≅ W (k_{α}C). As in the previous section, for n ≥ 0, we denote by I_{n} the ideal in M = Mor(C) constructed inductively by I_{0} = ∅ and I_{n+1} = I_{n} ∪ (∪_{J}M ∘ J ∘ M), with J running over the minimal split 𝒥 -classes not contained in I_{n} which satisfy J ∘ N ⊆ I_{n}. For any idempotent endomorphism e in M we denote by M_{e} the complement of the group G_{e} in e ∘ M ∘ e and by σ_{e} : eAe → k_{α}G_{e} the split surjective algebra homomorphism with kernel k_{α}M_{e}. By the assumptions, c commutes with e, hence the product ec is an idempotent in eAe. By the construction of c, the element z_{e} = σ_{e}(ec) is either zero or it is the central idempotent in k_{α}G_{e} which is the sum of all block idempotents of simple block algebras of k_{α}G_{e}. Equivalently, k_{α}G_{e}z_{e} is the largest semisimple direct factor of the twisted group algebra k_{α}G_{e}. Combining the above parametrisations of simple A-modules yields the following result.

Lemma 5.1. Let S be a simple module, parametrised by the pair (e, T) for some idempotent endomorphism e and a simple k_{α}G_{e}-module T. Then there is a primitive idempotent i in eAe with the following properties.

(i) The projective indecomposable A-module Ai is a projective cover of S. (ii) We have σ_{e}(i) ≠ 0, and σ_{e}(i) is a primitive idempotent in k_{α}G_{e}. (iii) The projective indecomposable k_{α}G_{e}-module k_{α}G_{e}σ_{e}(i) is a projective cover of T. (iv) If (e, T) is a weight, then T ≅ k_{α}G_{e}σ_{e}(i).

Proof. Since e does not annihilate S, there is a primitive idempotent i in eAe which does not annihilate S, whence (i). Since split morphisms in 𝒥 -classes strictly smaller than J_{e} annihilate S, it follows that ker(σ_{e}) annihilates S, and hence σ_{e}(i) ≠ 0. Since σ_{e} is surjective, it follows that σ_{e}(i) is a primitive idempotent in k_{α}G_{e} which does not annihilate T. This implies (ii) and (iii). If (e, T) is a weight, then T is a projective k_{α}G_{e}-module, hence isomorphic to its projective covers. This completes the proof.

We define ideals H_{n} in cAc by setting

H_{n} = ck_{α}I_{n}c = k_{α}I_{n} ∩ cAc

for any n ≥ 0, with the convention H_{0} = {0}. We have H_{n} = cAc for n sufficiently large. In order to show that the ideals H_{n} form a hereditary chain of ideals in cAc, we need to show that H_{n}^{2} = H_{n}, that H_{n+1}/H_{n} is projective as a left cAc/H_{n}-module, and that H_{n+1}rad(cAc)H_{n+1} ⊆ H_{n}, for n ≥ 0.

Lemma 5.2. Let n be a nonnegative integer, and e an idempotent endomorphism in I_{n+1} ∖ I_{n}. Then ker(σ_{e}) ⊆ k_{α}I_{n}.

Proof. The kernel of σ_{e} is spanned, as a k-vector space, by the image of the complement M_{e} of G_{e} in e ∘ M ∘ e. Since the morphisms in M_{e} have split 𝒥 -classes strictly smaller than J_{e}, this space is contained in k_{α}I_{n}.

Lemma 5.3. Let n be a nonnegative integer, and e an idempotent endomorphism in I_{n+1} ∖ I_{n}. We have

cAeAc ⊆ cAecAc + ck_{α}I_{n}c .

Proof. By the construction of I_{n} and I_{n+1} we have e ∘ N ⊆ I_{n}, and we have e′ ∈ I_{n} for any idempotent e′ in M ∘ e ∘ M such that J_{e′} < J_{e}. It follows from Lemma 3.4 that cAeAc ⊆ ∑_{e′} cAe′c + k_{α}I_{n}, where e′ runs over the idempotents in J_{e}. For any such e′, the idempotents ê and ê′ are conjugate in A, hence so are the idempotents êc and ê′c. This implies AecA = Ae′cA for any idempotent e′ ∈ J_{e}, hence cAe′c ⊆ cAe′cAc = cAecAc. The result follows.

Lemma 5.4. Let n be a nonnegative integer and let E be a set of representatives of the split 𝒥 -classes contained in I_{n+1} ∖ I_{n} such that E consists of idempotent endomorphisms. We have H_{n+1} = H_{n} + ∑_{e∈E} cAecAc ; in particular, we have H_{n}^{2} = H_{n}. Moreover, we have a direct sum decomposition of cAc-cAc-bimodules

H_{n+1}/H_{n} = ⊕_{e∈E} cAecAc/(H_{n} ∩ cAecAc) .

Proof. By 5.3 we have cAeAc ⊆ cAecAc + ck_{α}I_{n}c = cAecAc + H_{n}, where e ∈ E. This shows the first statement. In particular, H_{n} is generated, as an ideal, by idempotents in cAc, and hence H_{n}^{2} = H_{n}. The fact that upon dividing by H_{n} the first sum becomes a direct sum decomposition follows from 3.8. Indeed, for different e, e′ ∈ E we have M ∘ e ∘ M ∩ M ∘ e′ ∘ M ⊆ I_{n}, whence the result.

Lemma 5.5. Let n be a nonnegative integer. Then H_{n+1}/H_{n} is projective as a left cAc/H_{n}- module.

Proof. It follows from Lemma 5.4 that it suffices to show that the left cAc/H_{n}-module U = cAecAc/(H_{n} ∩ cAecAc) is projective, where e is an idempotent endomorphism in I_{n+1} ∖ I_{n}. Set B = cAc/H_{n}. By construction, we have U ≅ Bēc̄B, where ēc̄ is the image of ec in B. By 5.2 we have ēBē ≅ k_{α}G_{e}z_{e}. In particular, ēc̄ is a nonzero scalar multiple of a central idempotent in ēBē. By 3.9, the quotient AeA/(AeA ∩ k_{α}I_{n}) is projective as an A/k_{α}I_{n}-module. Equivalently, ĀēĀ is projective as a left Ā-module, where Ā = A/k_{α}I_{n}, and ē is identified to its image in Ā. It follows from 2.2 that U is projective as a left B-module.

Lemma 5.6. Let n be a nonnegative integer. We have H_{n+1} rad(cAc) H_{n+1} ⊆ H_{n}.

Proof. Let e, e′ be idempotent endomorphisms in I_{n+1} ∖ I_{n}. It suffices to show that we have ecArad(cAc)Ae′c ⊆ k_{α}I_{n}. If J_{e} ≠ J_{e′}, this follows from 3.8. If J_{e} = J_{e′}, then êc and êc′ are conjugate idempotents. Thus we may assume that e = e′. Since σ_{e}(ecAec) = k_{α}G_{e}z_{e} is semisimple, it follows that ecArad(cAc)Aec ⊆ rad(ecAec) ⊆ ker(σ_{e}). By Lemma 5.2, we have ker(σ_{e}) ⊆ k_{α}I_{n}. The result follows.

Combining the Lemmas 5.4, 5.5, and 5.6 completes the proof of Theorem 1.11.

6 Examples and further remarks

Example 6.1. Let k be a field. Let C be a category with two objects X and Y and seven morphisms {i, e, e′, j, f, f′, t} such that i and j are the identity morphisms of X and Y, respectively, and where the remaining morphisms are as follows. The morphism t is the unique morphism from X to Y, and this is the unique nonsplit morphism in C. The morphisms e, e′ are idempotent endomorphisms of X satisfying e ∘ e′ = e′ and e′ ∘ e = e; the morphisms f, f′ are idempotent endomorphisms satisfying f ∘ f′ = f and f′ ∘ f = f′. Note that the monoids End_{C}(X) and End_{C}(Y) are opposite to each other. The ℒ -classes of e, e′ are different, and they satisfy L_{t} ≤ L_{e} and L_{t} ≤ L_{e′}. Thus the minimality condition on ℒ -classes in Theorem 1.1 does not hold. Similarly, the ℛ -classes of f, f′ are different and satisfy R_{t} ≤ R_{f} and R_{t} ≤ R_{f′}. Thus C does not satisfy the corresponding minimality condition for ℛ -classes either. The algebra A = kC is, however, quasi-hereditary. This can be seen as follows. The 𝒥 -classes of e and e′ coincide, hence e and e′ are conjugate in A. Similarly, the idempotents f and f′ are conjugate in A. The idempotents e, 1 − e, f, 1 − f are easily seen to be primitive, pairwise orthogonal and pairwise nonconjugate in A. Thus A is basic and has four isomorphism classes of simple modules. Denote by S, T, U, V the simple quotients of the projective indecomposable A-modules Ae, A(i − e), Af, A(j − f), respectively. Then V ≅ A(j − f). The remaining projective indecomposable modules Ae, A(i − e), Af have dimension 2, and their composition series are {S, U}, {T, S}, {U, V }, respectively. The ideals generated by the idempotents j − f, j, e + j, i + j = 1 are easily seen to form a hereditary chain in A. By contrast, the ideals I_{n} from the proof of Theorem 1.1 do not yield a hereditary chain: we have I_{1} = {e, e′, t}, but kI_{1} is not projective as a left A-module. This suggests where to look for improvements of Theorem 1.1: one needs to consider ideals generated not just by idempotents in C but by idempotents in k_{α}C.

Example 6.2. It is less clear whether one can avoid the hypothesis in Theorem 1.1 on the set of nonsplit morphisms to be an ideal - the ‘smallest’ cases which do not satisfy this hypothesis do not yield quasi-hereditary algebras. Let k be a field, and let M = {1, a, e} be a monoid consisting of three elements, with identity element 1, such that the product of any two nonidentity elements is e. Then M is abelian, e is an idempotent in M, and a is the unique nonsplit element in M. In particular, the set of nonsplit elements in M is not an ideal. The algebra kM is not quasi-hereditary: an easy calculation shows that kM ≅ kMe × kM(1 − e), and kMe ≅ k, while kM(1 − e) is a local 2-dimensional algebra (with basis {1 − e, a − e}). In particular, kM is a symmetric non-semisimple k-algebra, hence not quasi-hereditary.

Remark 6.3. With the notation of Theorem 1.11, one can replace the hypothesis that c commutes with idempotent endomorphism by the following hypothesis: for any idempotent endomorphism e in C there is a conjugate c_{e} of c in A which commutes with the image of e in A and which satisfies cAeAc = cAec_{e}Ac + cAek_{α}Nc. This is easily seen to hold with c_{e} = c, if c commutes in A with all idempotent endomorphisms e in C. This a priori weaker hypothesis ensures that the statements and proofs of the Lemmas 5.3, 5.4, 5.5, and 5.6 remain correct, with ec replaced in the statements and proofs (as appropriate) by ec_{e}. The verdict on what would be the weakest hypotheses for the weight algebra of a twisted finite category algebra over a field to be quasi-hereditary is still out.

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