The Ingalls-Thomas Bijections

Given a finite acyclic quiver Q with path algebra kQ, Ingalls and Thomas have exhibited a bijection between the set of Morita equivalence classes of support-tilting modules and the set of thick subcategories of mod kQ and they have collected a large number of further bijections with these sets. We add some additional bijections and show that all these bijections hold for arbitrary hereditary artin algebras. The proofs presented here seem to be of interest also in the special case of the path algebra of a quiver.

1. Introduction 1.1. Let Λ be a hereditary artin algebra. We recall that an artin algebra Λ is a k-algebra which is of finite length when considered as a k-module, where k is a commutative artinian ring. An artin algebra is hereditary provided submodules of projective modules are projective. Since this means that the functors Ext i Λ vanish for i ≥ 2, we write Ext(M, M ′ ) instead of Ext 1 Λ (M, M ′ ). A typical example of a hereditary artin algebra is the path algebra of a finite acyclic quiver (if k is an algebraically closed field, any hereditary artin k-algebra is Morita-equivalent to the path algebra of a finite acyclic quiver, but otherwise there are many other hereditary is said to be sincere provided any simple module belongs to the support of M (thus provided the only idempotent e ∈ Λ with eM = 0 is e = 0).

1.2.
The subcategories of mod Λ which we will consider are full subcategories which are closed under direct sums and direct summands. Given a class X of modules, we denote by add X the class of modules which are direct summands of direct sums of modules in X . If X = {X} for a single module X, we write add X instead of add{X}. The modules X, X ′ are said to be Morita equivalent provided add X = add X ′ . Note that multiplicity-free modules which are Morita equivalent are actually isomorphic. On the other hand, every module is Morita equivalent to a multiplicity-free module.

Support-tilting modules. Following earlier considerations of Brenner and
Butler, tilting modules have been defined in [7]. We say that a module M has no self-extensions, provided Ext(M, M ) = 0. In the present setting, a module T without self-extensions is said to be a tilting module provided it has precisely n isomorphism classes of indecomposable direct summands (where n is the rank of the artin algebra Λ), or, equivalently, provided Λ Λ is the kernel of a surjective map in add T (or an injective cogenerator is the cokernel of an injective map in add T ).
A module M is said to be support-tilting provided M considered as a Λ(M )-module is a tilting module.
Here is one of the sets we are interested in: the set of Morita equivalence classes of support-tilting modules. 1.4. Thick subcategories with a cover. A subcategory A of mod Λ is called a thick (or wide) subcategory provided it is closed under kernels, cokernels and extensions. Note that a thick subcategory is an abelian category, and the inclusion functor A → mod Λ is exact.
A module X is said to generate a module Y provided Y is a factor module of a direct sum of copies of X. Dually, a module X cogenerates a module Y provided Y is a submodule of a direct sum of copies of X (since the modules considered here are of finite length, it is sufficient to look at direct sums of copies of X; for general modules one would have to use products). Given a class X of modules, let G(X ) be the subcategory of all modules which are generated by modules in add X , and let H(X ) be the subcategory of all modules which are cogenerated by modules in add X . If C is a subcategory and C ∈ C, then C is said to be a cover of C provided C ⊆ G(C), and C is said to be a cocover of C provided C ⊆ H(C). This is the second set of interest: the set of thick subcategories of mod Λ with covers.
1.5. If Λ is the path algebra of a finite acyclic quiver, Ingalls and Thomas have exhibited a bijection between the set of Morita equivalence classes of support-tilting modules and the set of thick subcategories of mod Λ with covers. The aim of this paper is to provide a proof of the Ingalls-Thomas bijection for arbitrary hereditary artin algebras. Our proof draws attention to three additional sets which are in bijection with the set of Morita equivalence classes of support-tilting modules: the set of isomorphism classes of exceptional antichains in mod Λ, as well as the set of isomorphism classes of normal or of conormal modules without self-extensions.
Here are the definitions.
1.6. Exceptional antichains. Given an additive category C, a brick is C is an object whose endomorphism ring is a division ring. Bricks A 1 , A 2 are said to be orthog- in C is a set of pairwise orthogonal bricks (antichains are called discrete subsets in [6] and Hom-free subsets in [8], see also the remark 7.3). Antichains A = {A 1 , . . . , A t } and A ′ = {A ′ 1 , . . . , A ′ t ′ } are said to be isomorphic, provided the objects i A i and j A ′ j are isomorphic. Given an antichain A = {A 1 , . . . , A t } in mod Λ, its Ext-quiver Q A has as vertices the elements A i and there is an arrow A i → A j provided Ext(A i , A j ) = 0 (one may endow this quiver with a valuation, taking into account the size of the Ext-groups, but this is not needed here). We say that an antichain A is exceptional, provided its Ext-quiver Q A is acyclic, thus provided we may index the elements of A in such a way that Ext(A i , A j ) = 0 for all pairs i ≥ j.
There is the following well-known fact (see, for example [14]): A sincere module without self-extensions is faithful, thus any module M without self-extensions is a faithful Λ(M )-module.
1.8. Since its introduction, tilting theory concerns the study of suitable torsion pairs in mod Λ. It seems worthwhile to include this aspect in our considerations.
Recall that a torsion class in mod Λ is a class of modules which is closed under factor modules and extensions. A torsionfree class in mod Λ is a class of modules which is closed under submodules and extensions.
It was the decisive idea of Ingalls and Thomas [9] to relate the support-tilting modules to thick subcategories and to exhibit in this way a number of bijections.
They were dealing with path algebras of finite acyclic quivers, here we consider the case of an arbitrary hereditary artin algebra. • (4) Morita equivalence classes of support-tilting modules.
We have separated the five sets in Theorem 1.1 into two groups, since there is a great affinity between (1), (2) and (3) on the one hand, and (4) and (5) on the other hand. The essential bijection concerns the sets (2) and (4). As we have mentioned, such a bijection was exhibited by Ingalls-Thomas [9] in case Λ is the path algebra of a finite acyclic quiver. A bijection between (4) and (5) has been known for a long time. A bijection between (1) and (2) was exhibited already in 1976, see [13].
For a bijection between (1) and (3), one may refer to [5], as we will see below.
1.9. Outline of the paper. Sections 2 to 4 provide the required bijections in detail, and an outline of the corresponding proofs. Section 5 is devoted to duality.
Whereas the sets of the form (1), (2) and (4) are preserved under duality, this is not the case for the sets (3) and (5), thus, using duality, we obtain bijections with two further sets: the set (6) of isomorphism classes of conormal modules without self-extensions, and the set (7) of the torsionfree classes with a cocover. In the final section 6 we deal with the support of the various modules and subcategories.
As a supplement of the theorem, we have mentioned that for Λ representationfinite, certain conditions are always satisfied. First of all, if Λ is representationfinite, then any subcategory of mod Λ has both a cover and a cocover. And second, it is well-known that for an antichain A which is not exceptional, the class F (A) of all modules with a filtration with factors in A contains infinitely many isomorphism classes of indecomposable Λ-modules, thus Λ cannot be representation-finite.
1.10. The case of Λ being representation-finite is studied in more detail in our paper [11]. Such an artin algebra Λ is called a Dynkin algebra, since the underlying graph of its valuated quiver is the disjoint union of Dynkin diagrams. There, we will discuss the number of tilting and support-tilting modules for these algebras.
For the Dynkin cases A, we obtain the Catalan triangle, for the cases B and C we obtain the increasing part of the Pascal triangle, and finally for the cases D we obtain an expansion of the increasing part of the Lucas triangle. For a further study of the Ingalls-Thomas bijections in general, we also may refer to the forthcoming survey [16].

The bijections between (1), (2) and (3)
From (1)  For the step (1) to (2), we also may refer to [5]. Namely, an exceptional antichain A is a standardizable set as considered in [5] and the proof of Theorem 2 in [5] asserts that there is a quasi-hereditary algebra B such that the subcategory F (A) is equivalent to the category of ∆-filtered B-modules. Since the standardizable set A consists of pairwise orthogonal modules, the same is true for the ∆-modules of B, and consequently the ∆-modules of B are just the simple B-modules. This shows that the category of ∆-filtered B-modules is the whole category mod B. (1): If A is a thick subcategory with a cover, let S(A) be the set of simple objects in A, one from each isomorphism class. Then S(A) is an exceptional antichain in mod Λ. Namely, a thick subcategory with a cover is equivalent, as a category, to the module category mod Λ ′ of an artin algebra Λ ′ . Such an equivalence identifies the quiver Q S(A) with the quiver of the artin algebra Λ ′ (the quiver of an artin algebra is just the Ext-quiver of the simple Λ ′ -modules). It is well-known (and easy to see) that the quiver of an artin algebra is acyclic.

From (2) to (3): If
A is a thick subcategory with a cover, let P be a minimal projective generator of A. Then P is a normal module without self-extensions.
If we start with (1), say with an exceptional antichain A, and use [5] in order to find an equivalence η : F (A) → mod B, the proof of Theorem 2 in [5] first constructs indecomposable objects in F (A) which correspond under η to the indecomposable projective B-modules. In this way, one constructs a minimal projective generator for the abelian category F (A).
From (3) to (1). Let N be a normal module without self-extensions. Write is normal, the map u i cannot be surjective. Since Λ is hereditary, it follows that u i is injective and we denote by p i : N i → ∆(i) the cokernel of u i . Since u i is not surjective, we see that ∆(i) = 0. We claim that the modules ∆(i) are pairwise orthogonal bricks. Let h : N j → ∆(i) be a map, and form the induced exact First of all, consider the case j = i. Let g be any endomorphism of ∆(i) and look at the map h = gp i : N i → ∆(i). We see that there is an endomorphism g ′ : N i → N i with gp i = p i g ′ . Since all non-zero endomorphisms of N i are invertible, the same is true for ∆(i). In this way, we see that ∆(i) is a brick.
Second, let g : ∆(j) → ∆(i) be a homomorphism with j = i and consider left N i -approximation, it follows that g ′ = u i g ′′ for some g ′′ : N j → U i . But then gp j = p i g ′ = p i u i g ′′ = 0 and therefore g = 0.
In this way, we have shown that ∆ = {∆(i) | i} is an antichain. Using induction on the length |N i | of N i , we see that N i belongs to F (∆). Namely, if N i is of length 1, then U i = 0 since ∆(i) = 0. If |N i | ≥ 2, then U i is a direct sum of modules of the form N j with |N j | < |N i |, thus by induction U i belongs to F (∆) and therefore also N i belongs to F (∆).
The Starting with an exceptional antichain A in (1), and going via (2) to (3), we obtain a minimal projective generator P of F (A). Going from (3) to (1), we attach to P the antichain ∆ whose elements are just the simple objects in F (A), but these are just the elements of A. Conversely, starting in (3) say with a normal module N without self-extensions, then going to (1), we attach to it the antichain ∆. Going via (2) to (3), we form a minimal projective generator in F (∆). But N is up to isomorphism the only minimal projective generator in F (∆). (3) and (4) From (4) to (3): If T is a support-tilting module, let ν(T ) be its normalization.

The bijection between
This clearly is a normal module without self-extensions. Here we use that any module M can be written in the form M = M ′ ⊕ M ′′ where M ′ is normal and generates M ′′ (this of course is trivial), and that such a decomposition is unique up to isomorphism (this is not so obvious); the module M ′ is called a normalization of the module M . The uniqueness was first shown by Roiter [17] and then also by Auslander-Smalø [4], see also [15]. The uniqueness shows that the map ν going from (4) to (3) is well-defined.
Let us show that ν is injective when we are dealing with support-tilting modules.
We claim the following: if T, T ′ are support-tilting modules with ν(T ) = ν(T ′ ), then T and T ′ are Morita equivalent. For the proof, we may replace Λ by the support algebra Λ(T ) = Λ(T ′ ), thus we may assume that T, T ′ are tilting modules. Now, T ′ is generated by ν(T ′ ) = ν(T ), thus by T . Since T generates T ′ , it follows from Ext(T, T ) = 0 that Ext(T, T ′ ) = 0. Similarly, T ′ generates T and therefore Ext(T ′ , T ) = 0. Altogether we see that Ext(T ⊕ T ′ , T ⊕ T ′ ) = 0. Since T is a tilting module, this implies that T ′ belongs to add T . Similarly, since T ′ is a tilting module, we see that T belongs to add T ′ .
In order to see that ν is also surjective, we need to find for any normal module N without self-extensions a support-tilting module T with ν(T ) = N. This we will show next.
From (3) to (4): If N is a module without self-extensions, there is a module Y , with the following properties: first, Y is generated by N , and second, N ⊕ Y is a support-tilting module; we call Y a factor complement for N (this is the dual version of forming a Bongartz complement, see for example [14]).
Here is the construction of a factor complement Y of a module without selfextensions (we follow [14]). Let Λ(N ) be the support algebra for N and Z an injective cogenerator for mod Λ(N ). We claim that there exists an epimorphism

h ′ h
Since Ext(N, N ) = 0, the lower sequence splits, thus N ′′ belongs to add N . Since h is surjective, also h ′ is surjective, thus Y is generated by N .
It remains to be seen that N ⊕ Y is support-tilting. Since N generates Y , it follows from Ext(N ⊕ Y, N ) = 0 that Ext(N ⊕ Y, Y ) = 0. In this way, we see that N ⊕ Y has no self-extensions. The exact sequence 0 → i N → Y → Z → 0 shows that Z is the cokernel of an injective map in add(N ⊕ Y ), thus N ⊕ Y is a support-tilting module. This completes the proof that Y is a factor complement for N .
If we choose a minimal direct summand φ(N ) of Y such that N ⊕ φ(N ) is a support-tilting module, then φ(N ) is uniquely determined by N and may be called a minimal factor complement for N . Thus, going from (3) to (4), we may attach to a normal module N without self-extension the multiplicity-free support-tilting Of course, if N is normal, then N is the normalization of N ⊕ Y . Thus starting with a normal module N without self-extensions, then going from (3) to (4) and back to (3), we obtain N . On the other hand, let T be support-tilting. From (4) to (3) we take ν(T ). From (3) to (4), we add to ν(T ) a factor complement, say N ′ .
But T and T ′ = ν(T ) ⊕ N ′ both are support-tilting modules with ν(T ) = ν(T ′ ) and generated by this module ν(T ), thus they are Morita equivalent. From (4) to (5): If T is a module without self-extensions, let G(T ) be the class of modules generated by T . Then it is well-known (and easy to see) that T is a torsion class. Of course, T is a cover for G(T ). From (5) to (4): If C is a torsion class with a cover C, then we attach to it a module T such that add T is the class of Ext-projective modules in G. In order to do so, we need to know that the class E of Ext-projective modules in C is finite, say E = add T for some module T . We also have to show that T is support-tilting.

The bijection between (4) and (5)
Along with C, its normalization ν(C) is also a cover. A normal cover of a torsion class has no self-extension (see Proposition 1 of [15]). Let B be a factor complement for ν(C). As we have seen, T = ν(C) ⊕ B is a support-tilting module. Since B is generated by ν(C), we have G(T ) = G(ν(C)) = G(C) = C. But we have shown already that add T is the class of Ext-projective modules in G(T ).
From (4) to (5) to (4): Let us start with a support-tilting module T and attach to it G = G(T ). As we have seen, the class of Ext-projectives in G is add T . We choose T ′ with add T ′ = add T . But this just means that T, T ′ are Morita equivalent.
From (5) to (4) to (5). We start with a torsion class C with a cover, we choose a support-tilting module T with C = G(T ), thus we are back at C.

Duality
By definition, given an artin algebra Λ, there is a commutative artinian ring k such that Λ is a k-algebra and is of finite length when considered as a k-module. If Λ is an artin algebra, also the opposite algebra Λ op is an artin algebra. If we denote by E a minimal injective cogenerator for mod k, the functor D = Hom k (−, E) provides an equivalence between mod Λ and (mod Λ op ) op . We can use this duality in order to exhibit further bijections.
Using duality, the sets (1), (2) and (4) are preserved. Of course, the dual concept of a thick subcategory with a cover is a thick subcategory with a cocover. An abelian k-category with finitely many simple objects and such that the Hom and Ext-groups are k-modules of finite length, has a cover if and only if it has a cocover.
Remark. The bijections between the set (2) of thick subcategories A and the sets (1), (3) and (6) of isomorphism classes of suitable modules can be reformulated as follows: In an abelian category we may look at the semi-simple, the projective and the injective objects: the set of simple objects in A is an antichain in mod Λ, a minimal projective generator in A is a normal module without self-extensions, a minimal injective cogenerator is a conormal module without self-extensions. These are the procedures to obtain from a thick subcategory the corresponding antichain, as well as a normal or conormal module without self-extensions.
Conversely, let us start with (1), (3) or (6). It has been mentioned already that starting with an antichain A, we take the full subcategory F (A) of all modules with a filtration with factors in A. Starting with a normal module P without selfextensions, the corresponding thick subcategory A consists of all modules which arise as the cokernel of a map in add P (in this way, we specify projective presentations of the objects in A). Dually, starting with a conormal module I without self-extensions, the corresponding thick subcategory A consists of all modules which arise as the kernel of a map in add I (in this way, we specify injective presentations of the objects in A).
6. The support of a module, sincere modules and subcategories Proposition 6.1. The bijections which we have constructed preserve the support.
Specializing the Ingalls-Thomas bijections to sincere modules, it follows from the proposition that we get bijections between: • (1) Isomorphism classes of exceptional sincere antichains.
Of course, conversely this special case implies the general case.

Final remarks
7.1. The aim of our discussion was to extend results of Ingalls and Thomas which were established for path algebras of finite acyclic quivers to arbitrary hereditary artin algebras. Experts may not be surprised that results concerning path algebras of finite acyclic quivers can be extended in this way: after all, there is a general feeling that such generalizations are always possible. But the paper [12] may serve as a warning. The paper provides a description of the cofinite quotient-closed subcategories of mod Λ, where Λ is the path algebra of a finite acyclic quiver. In section 9 of [12], the author discuss the problem of extending the result to finitedimensional hereditary k-algebras, but they are able to provide a solution only in the case of k being a finite field.
On the other hand, one may ask whether the setting may be further enlarged to deal with hereditary artinian or even hereditary semi-primary rings, and not just with hereditary artin algebras. Note that our considerations use duality arguments and finiteness conditions which rely on the artin algebra assumption.

7.2.
A further possible generalization has been stressed by the referee: to drop the condition on Λ to be hereditary, thus to deal with an arbitrary artin algebra. For any finite-dimensional k-algebra Λ, with k an algebraically closed field, the paper [1] by Adachi, Iyama and Reiten provides a bijection between support τ -tilting modules in mod Λ and torsion classes with covers, extending in this way the corresponding Ingalls-Thomas bijection (for a hereditary artin algebra, the τ -tilting modules are just the tilting modules). Also, let us remark that the relationship between torsion classes and thick subcategories in mod Λ has been discussed by Marks and Stovicek [10].

7.3.
Our presentation of the Ingalls-Thomas bijections is centered around the notion of antichains in additive categories. Let us motivate the definition. Given a poset P , a chain in P is a subset of pairwise comparable elements, whereas an antichain in P is a subset of pairwise incomparable elements. Now consider the linearization kP of P , were k is a field: this is an additive k-category whose indecomposable objects are the elements of P such that Hom kP (x, y) = k provided x ≤ y in P and Hom kP (x, y) = 0 otherwise, such that the composition of maps in kP is given by the multiplication in k, and, finally, such that any object in kP is the direct sum of indecomposable objects. Of course, a subset A of P is an antichain in P if and only if A (considered as a set of objects in kP ) consists of pairwise orthogonal bricks (thus, is an antichain in the additive category kP ). As we see, antichains in additive categories have to be considered as a direct generalization of antichains in posets.
The reader should be aware that starting with a Dynkin diagram ∆ and its set Φ + (∆) of positive roots, several kinds of (different, but related) antichains have to be distinguished: First of all, Φ + (∆) is in an intrinsic way a poset, called the root poset of type ∆, and we may consider the set A(∆) of antichains in this root poset Φ + (∆). Second, choosing an orientation Ω of the Dynkin diagram (or, equivalently, a Coxeter element in the corresponding Weyl group), we may identify the elements of Φ + (∆) with the indecomposable Λ-modules, thus with the indecomposable objects in the additive category mod Λ. The set of antichains in mod Λ only depends on ∆ and Ω (and not on the choice of Λ), thus we may denote it by A(∆, Ω). It is known for a long time that the set A(∆) of antichains in the root poset Φ + (∆) and the set A(∆, Ω) of antichains in mod Λ have the same enumeration (for a uniform proof, see [3]), but a fully satisfactory explanation is still missing. In the case of the quiver A n with linear orientation, this concerns the quite obvious bijection between non-nesting and non-crossing partitions. Note that if Ω and Ω ′ are orientations of ∆, it is easy to construct a natural bijection between A(∆, Ω) and A(∆, Ω ′ ). For a detailed discussion of the sets A(∆) and A(∆, Ω), we may refer to [16].