Stability Conditions and Maximal Green Sequences in Abelian Categories

In this paper we study the stability functions on abelian categories introduced by Rudakov in \cite{Ru} and their relation with torsion classes and maximal green sequences. Moreover we introduce a new kind of stability function using the wall and chamber structure of the category.


Introduction
The concept of stability condition was introduced in algebraic geometry by Mumford in [18] to study moduli spaces under the action of a group. The success of this new approach motivated the use of these tools in different branches of mathematics. In the case of representation theory of quivers, they were introduced in seminal papers by Schofield [21] and King [17], and the general notion of stability was formalised in the context of abelian categories by Rudakov [19].
We study Rudakov's notion of stability on an abelian length category A, which is given by a function φ on the class Obj * (A) of non-zero objects of A. It assigns to each non-zero object X a phase φ(X), which is an element of a totally ordered set P, satisfying the so-called see-saw condition on short exact sequences (see Definition 2.1). A non-zero object M in A is said to be φ-stable (or φ-semistable) if every non-trivial subobject L ⊂ M satisfies φ(L) < φ(M ) (or φ(L) ≤ φ(M ), respectively). Inspired by [8], but in the more general context of abelian categories allowing infinitely many simple objects, we then define for each phase p a torsion pair (T p , F p ) in A as follows (see Proposition   stability functions. We first review this concept of stability here, and then discuss torsion classes arising from a stability function. 2.1. Stability functions. Throughout this section, we consider an essentially small abelian length category A. Definition 2.1. Let (P, ≤) be a totally ordered set. A function φ : Obj * (A) → P which is constant on isomorphism classes is said to be a stability function if for each short exact sequence 0 → L → M → N → 0 of non-zero objects in A one has the so-called see-saw (or teeter-totter) property, that is, exactly one of the following holds: For a non-zero object x of A, we refer to φ(x) as the phase (or slope) of x. Figure 1. The see-saw (or teeter-totter) property.

Remark 2.2.
Note that the image by φ of the zero object in A would not be well defined if there existed two non-zero objects M and N such that φ(M ) = φ(N ). Indeed, consider the following short exact sequences: Then, applying the see-saw property twice yields φ(0) = φ(M ) and φ(N ) = φ(0), contradicting φ(M ) = φ(N ). Therefore non-constant stability functions can only be defined on the class of non-zero objects of the category.
Note that Rudakov defined stability structures using the notion of a proset, that is, a pre-order ≺ on Obj * (A) satisfying for all L, M in Obj * (A) that L ≺ M or M ≺ L, or both. We can define an equivalence relation on Obj * (A) by setting L ∼ M when both L ≺ M and M ≺ L are satisfied, and denote by P = Obj * (A)/ ∼ the set of equivalence classes. The pre-order ≺ thus turns P into a totally ordered set, whose order relation we denote by ≤. The projection φ : Obj * (A) → P that assigns to each object its equivalence class is then the function from Definition 2.1, and the notion of stability we consider here is equivalent to Rudakov's original formulation.
The stability functions as defined above generalise several notions of stability conditions present in the literature as we can see in the following remarks. In [17], King adapted the notion of stability from geometric invariant theory, introduced by Mumford in [18], to the context of abelian categories with Grothendieck group of finite rank. In [19,Proposition 3.4], Rudakov shows that every stability condition as defined by King induces a stability function in the sense presented here. Remark 2.4. Stability functions are present in the physics literature, and in this case they are induced by a central charge Z. We recall this notion here, following the treatment given in [7]. A linear stability function on an abelian category A is given by a central charge, that is, a group homomorphism Z : K(A) → C on the Grothendieck group K(A) such that for all 0 = M ∈ A the complex number Z(M ) lies in the strict upper half-plane Given such a central charge Z : K(A) → C, the phase of a non-zero object M ∈ A is defined to be φ(M ) = (1/π) arg Z(M ). A simple argument on the sum of vectors in the plane shows that the phase function φ : Obj * (A) → (0, 1] satisfies the see-saw property. The most important feature of a stability function φ is the fact that it creates a distinguished subclass of objects in A called φ-semistables. They are defined as follows. Remark 2.6. Note that, due to the see-saw property, one can equally define the φ-stable (or φ-semistable) objects as those objects M whose non-trivial quotient objects N satisfy φ(N ) > φ(M ) (or φ(N ) ≥ φ(M ), respectively).
The following theorem from [19] implies that morphisms between φ-semistable objects respect the order induced by φ, that is,  Remark 2.9. As observed in [19], Theorem 2.7 implies that φ-stable objects are bricks when A is a Hom-finite k-category over an algebraically closed field k. Here M is called a brick when End(M ) k. This implies in particular that φ-stable objects are indecomposable. In fact, it is easy to see that φ-stable objects are always indecomposable, for any abelian category A.

2.2.
Harder-Narasimhan filtration and stability functions. From now on, we assume that the abelian category A is a length category, that is, each object M admits a filtration such that the quotients M i /M i−1 are simple. In particular, A is both noetherian and artinian. For a finite dimensional k-algebra A over a field k, the category mod A of finitely generated A-modules is a length category.
We borrow the following terminology from [7]; however, the concept was already used in [19].  We sometimes omit the epimorphism p when referring to a maximally destabilising quotient, and similarly for maximally destabilising subobjects. The following theorem from [19] implies in particular that every non-zero object admits a maximally destabilising quotient and a maximally destabilising subobject. Theorem 2.13 ([19, Theorem 2 and Proposition 1.13]). Let A be an abelian length category with a stability function φ : Obj * A → P, and let M be a non-zero object in A. Up to isomorphism, M admits a unique Harder-Narasimhan filtration, that is, a filtration Moreover, F 1 = M 1 is the maximally destabilising subobject of M and F n = M n /M n−1 is the maximally destabilising quotient of M .
For further use, it is also worthwhile to recall the following weaker version of a result from Rudakov.

Theorem 2.14 ([19, Theorem 3]). Let
A be an abelian length category with a stability function φ : Obj * A → P, and let M be a φ-semistable object in A. There exists a filtration Moreover, the set of quotients {G i } is uniquely determined up to isomorphisms by M and the properties (a) and (b).

Torsion pairs.
The concept of torsion pair in an abelian category was first introduced by Dickson in [13], generalising properties of abelian groups of finite rank. The definition is the following.

Definition 2.15. Let
A be an abelian category. Then the pair (T , F) of full subcategories of A is a torsion pair if the following conditions are satisfied: such that tX ∈ T and X/tX ∈ F. Given a torsion pair (T , F) we say that T is a torsion class and F is a torsion-free class. In this subsection, we show that a stability function φ : Obj * A → P induces a torsion pair (T p , F p ) in A for every p ∈ P, where

It is well known that a subcategory T of
But before doing so, we need to fix some notation. Definition 2.16. Let φ : Obj * (A) → P be a stability function and let p ∈ P. We define A ≥p to be We define in a similar way A ≤p , A >p , A <p and A p .
Given a subcategory X and an object M of A we say that M is filtered by X if there exists a chain of nested subobjects We denote by Filt(X ) the full subcategory of A consisting of all M ∈ A which are filtered by X.
We use the notation Fac(X ) for the class of all objects in A which are a factor object of some X ∈ X . Likewise, Sub(X ) denotes the class of all objects in A which are subobjects of any object X in X .
The following proposition not only shows that T p is a torsion class, but also gives a series of equivalent characterisations. Proposition 2.17. Let φ : Obj * (A) → P be a stability function and consider some phase p ∈ P. Then the class T p defined above satisfies: (a) T p is a torsion class; Proof. (a): We need to show that T p is closed under extensions and quotients.
To show that T p is closed under extensions, suppose that is a short exact sequence in A with L, N ∈ T p . If L or M are zero, then M clearly belongs to T p . Otherwise, let (M , p M ) be the maximally destabilising quotient of M . Then we can construct the following commutative diagram: To show that T p is closed under quotients, suppose that f : M → N is an epimorphism with M and N two non-zero objects and M ∈ T p . Let (M , p M ) and (N , p N ) be the maximally destabilising quotients of M and N , respectively. Then This proves that T p is a torsion class.
It thus remains to show that T p ⊆ Filt(A ≥p ). Let M be a non-zero object of T p , and let M be a maximally destabilising quotient of M . By definition of T p , we have that φ(M ) ≥ p. Therefore we can consider the Harder-Narasimhan filtration of M and Theorem 2.13 implies that M ∈ Filt(A ≥p ). Hence (d): Let M ∈ T p , and suppose that M is its maximally destabilising quotient. By definition of the maximally destabilising quotient, every non- The reverse inclusion is immediate.
The following result is the dual statement for the torsion-free class F p defined above.
Proposition 2.18. Let φ : Obj * (A) → P be a stability function and consider a phase p ∈ P. Then: (a) F p is a torsion-free class; Now we are able to prove the main result of this section.

Proof. We first show that Hom
For the maximality, suppose for instance that Hom A (T p , N ) = 0 for a non-zero object N of A. If N is the maximally destabilising subobject of N , it follows that Hom A (T p , N ) = 0, and thus φ(N ) < p by definition of T p . Consequently, N ∈ F p . We show in the same way that Hom A (M, F p ) = 0 implies M ∈ T p , which proves maximality.
As a consequence of the previous proposition we have the following result that provides a method to build abelian subcategories of A using stability conditions. Proposition 2.20. Let φ : Obj * (A) → P be a stability function and let p ∈ P be fixed. Then the full subcategory is a wide subcategory of A.

Proof.
A p is a wide subcategory if it is abelian. To show that, we note first that A p = T p ∩ Filt(A ≤p ). Then Proposition 2.17 and its dual imply that A p is the intersection of a torsion class T p and a torsion-free class Filt(A ≤p ). This implies in particular that A p is closed under extensions. Now we show that A p is closed under taking kernels and cokernels. Let f : M → N be a morphism in A p . If f is zero or an isomorphism, the result follows at once. Otherwise, consider the following short exact sequences in A: where all these objects are non-zero. The semistability of M implies that φ(im f ) ≥ φ(M ) = p, while the semistability of N implies that φ(im f ) ≤ φ(N ) = p. Consequently, φ(im f ) = p. The see-saw property applied to the two exact sequences yields φ(ker f ) = p and φ(coker f ) = p.
Moreover, every subobject L of ker f is a subobject of M , thus φ(L) ≤ φ(M ) = φ(ker f ). Therefore ker f is φ-semistable and belongs to A p . Dually we show that coker f also belongs to A p . This finishes the proof.

Remark 2.21.
It is easy to see that the φ-stable objects with phase p are exactly the simple objects of the abelian category A p . Moreover, the proof establishes again the parts (b) and (c) of Theorem 2.7.

Maximal green sequences and stability functions
In the previous section we discussed how a stability function φ : Obj * (A) → P induces a torsion pair (T p , F p ) in A for each phase p ∈ P. Moreover, as noted in [6,Section 3], it is easy to see that if p ≤ q in P, then T p ⊇ T q and F p ⊆ F q . Since P is totally ordered, every stability function φ yields a (possibly infinite) chain of torsion classes in A. In this section we are mainly interested in the different torsion classes induced by φ. We therefore define, for a fixed stability function φ : Obj * (A) → P, an equivalence relation on P by p ∼ q when T p = T q and consider the equivalence classes P/ ∼.
Of particular importance is the case where the chain of equivalence classes P/ ∼ is finite, not refinable, and represented by elements p 0 > · · · > p m ∈ P such that T p0 = {0} and T pm = A. Definition 3.1. A maximal green sequence in A is a finite sequence of torsion classes 0 = X 0 X 1 · · · X n−1 X n = A such that for all i ∈ {0, 1, . . . , n − 1}, the existence of a torsion class X satisfying X i ⊆ X ⊆ X i+1 implies X = X i or X = X i+1 .

Remark 3.2.
Note that this definition is not the original definition of maximal green sequence as given by Keller in [16] and studied in [9]. However, the equivalence between both definitions follows directly from [11,Proposition 4.9].
Our aim is to establish conditions under which the chain of torsion classes induced by a stability function is a maximal green sequence. Observe first that if φ : Obj * (A) → P is a stability function and the totally ordered set P has a maximal element p, then T p is the minimal element in the chain of torsion classes induced by φ.  (b) By assumption, the chain of torsion classes induced by φ is finite, say T p0 T p1 · · · T pn .
If T p0 = {0}, choose a non-zero object M in T p0 . Let M be the maximally destabilising quotient of M , thus M ∈ T p0 and φ(M ) ≥ p 0 . Since P does not have a maximal element, there exists a p ∈ P with p > φ(M ). It follows that M / ∈ T p , while T p ⊆ T p0 , contradicting the minimality of T p0 . Thus T p0 = {0} and the statement follows from Proposition 2.17.
Following Engenhorst [14], we call a stability function φ : A → P discrete at p if two φ-stable objects M 1 , M 2 satisfy φ(M 1 ) = φ(M 2 ) = p precisely when M 1 and M 2 are isomorphic in A. Moreover, we say that φ is discrete if φ is discrete at p for every p ∈ P. Proposition 3.4. Let φ : Obj * (A) → P be a stability function and let p, q ∈ P be such that T p T q . Then the following statements are equivalent: (a) There is no r ∈ P such that T p T r T q , and φ is discrete at every q with q ∼ q. Consider a φ-stable object X in A ≥φ(M ) . In particular, X ∈ T φ(M ) = T q . If φ(X) = φ(M ), then X is isomorphic to M by the discreteness, and X ∈ T . Else φ(X) > φ(M ), and M ∈ T φ(M ) \ T φ(X) . Therefore, T φ(X) T φ(M ) = T q , which implies, by assumption, that T φ(X) ⊆ T p ⊆ T . In particular, X ∈ T . Since T is a torsion class, this implies that A ≥φ(M ) ⊆ T , and furthermore This shows that T q = T .
(b) implies (a): The fact that there is no r ∈ P such that T p T r T q is immediate. To show that φ is discrete, assume that there exist two non-isomorphic φ-stable objects M and N such that φ(M ) = φ(N ) = q , with q ∼ q. Consider the set T = Filt(A ≥p ∪ {N }). We will show that T is a torsion class such that T p T T q , a contradiction to our hypothesis. First, because T p T q = T q , we have q < p. Since T p = Filt(A ≥p ), we have N / ∈ T p , so T p T . On the other hand, M and N are non-isomorphic φ-stable objects in A q , so M is not filtered by N , by Theorem 2.14. Moreover, Proposition 2.17 implies that M is not in Filt(A ≥q ). Hence, M does not belong to T . Since M ∈ T q , this shows that T T q . Thus T p T T q . We now show that T = Filt(A ≥p ∪ {N }) is a torsion class, that is, T is closed under extensions and quotients. By definition, T is closed under extensions. To show that T is closed under quotients, suppose that is an exact sequence in A and T ∈ T .
If T ∈ T p , then T ∈ T p since T p is a torsion class and therefore T ∈ T . Else, T ∈ T \ T p . Let Q be the maximally destabilising quotient of T . Since T / ∈ T p , we have φ(Q) < p. Moreover, φ(Q) ≥ q since T ∈ T T q . Consequently, q ≤ φ(Q) < p, and it follows from our hypothesis that φ(Q) = q (otherwise T p T φ(Q) T q ). So Q ∈ T q = T q . This shows in particular that q = q . Indeed, if q < q , then the fact that Q is φ-semistable leads to Q ∈ T q / ∈ T q , a contradiction. Similarly, if q < q, then N ∈ T q / ∈ T q , again a contradiction. So q = q , and consequently Q, N ∈ A q . Now, suppose that is a Harder-Narasimhan filtration of T , as in Theorem 2.13. In particular, Q ∼ = T /T n−1 and semistable object since it is the maximally destabilising quotient of T . Therefore Theorem 2.14 implies that Q ∈ Filt(A q ). But, at the same time, we have by hypothesis that the only possible composition factor of T in A q is N , which implies that Q ∈ Filt({N }) ⊂ T . Now, let Q be the maximally destabilising quotient of T . Since Q is the maximally destabilising quotient of T , we have , and it follows from the fact that Q is the maximally destabilising quotient of T that the epimorphism from T to Q factors through Q, and thus there exists an epimorphism f : Q → Q in A, and thus in A q .
Recall from Proposition 2.20 that A q is an abelian category whose φ-stable objects coincide with the simple objects by Remark 2.21. Consequently, it follows from the existence of the epimorphism f : Q → Q and the fact that Q is filtered by the φ-stable object N that Q ∈ Filt({N }). Let be the Harder-Narasimhan filtration of T . Then Q ∼ = T /T m−1 and This finishes the proof.
We are now able to characterise the stability functions inducing maximal green sequences in A. Theorem 3.5. Let φ : Obj * (A) → P be a stability function. Suppose that P has no maximal element, or that the maximal element of P is not in φ(A). Then φ induces a maximal green sequence if and only if φ is a discrete stability function inducing finitely many equivalence classes on P.
Proof. Suppose that φ induces a maximal green sequence, say In particular, the set of equivalence classes on P is finite. Moreover, it follows from Proposition 3.4 that φ is discrete.
Conversely, suppose that φ is a discrete stability function inducing finitely many equivalence classes on P. So we get a (complete) chain of torsion classes T p0 T p1 · · · T pn induced by φ. The discreteness of φ implies by Proposition 3.4 that this chain of torsion classes is maximal. Moreover, Lemma 3.3 shows that T p0 = {0}: If P has no maximal element this follows from part (b), and if the maximal element of P is not in φ(A) this follows from part (a) of the Lemma. It remains to show that T pn = A. If M ∈ A but M / ∈ T pn , then the maximally destabilising quotient M of M satisfies φ(M ) < p n . Since M ∈ T φ(M ) , it follows that T pn T φ(M ) , a contradiction to the maximality of T pn . So T pn = A.
As an immediate corollary we have the following result, which is of importance for the study of the representation theory of the so-called τ -tilting finite algebras. Corollary 3.6. Let A be an abelian category having only finitely many torsion classes. Then every discrete stability function φ : Obj * (A) → P induces a maximal green sequence. Example 3.7. We illustrate by the following example that non-linear stability functions sometimes allow one to describe all torsion classes, which would not have been possible using linear stability conditions. Consider the Kronecker quiver It is well known that the indecomposable representations of Q are parametrised by two discrete families P n and I n , for n ∈ N, of dimension vectors (n, n + 1) and (n + 1, n), respectively, together with a P 1 (k)-family of representations R λ,n of dimension vector (n, n), with λ ∈ P 1 (k), n ∈ N, for an algebraically closed field k. We order the indecomposables by their slope and thus obtain a stability function It is known that one obtains all functorially finite torsion classes of rep Q in the form T p for some p ∈ R ∪ {∞}. Moreover, note that every indecomposable object in rep Q is φ-semistable. For more details on this, see [1,12]. However, there are lots of torsion classes for rep Q that are not functorially finite; they are given by selection of indecomposables as follows (see [2, Example 6.9]): Let S be any subset of P 1 (k); then the additive hull of all indecomposables R λ,n and I n , for n ∈ N and λ ∈ S, forms a torsion class which we denote by T S . Every non-functorially finite torsion class of rep Q is of this form for some set S, and we can certainly not obtain these classes by a linear stability function, since the elements R λ,n , where λ is in S, share the same dimension vector with those where λ does not lie in S.
We therefore define a set P = R ∪ {∞} ∪ {1 * }, where we add a new element, 1 * , as a double of 1, at the same order relative to the other elements x = 1, but we agree on setting 1 * < 1. Thus P is totally ordered, and we define a stability function φ * : rep Q → P by the following values on the indecomposables: if dim V = (n 1 , n 2 ) and n 1 = n 2 , 1 if V = R λ,n and λ ∈ S, 1 * if V = R λ,n and λ ∈ S.
Using this setting, one obtains the torsion class T S as T 1 with respect to the element p = 1 ∈ P.

Paths in the wall and chamber structure
In this section we focus on abelian length categories A with finitely many simple objects, that is, rk(K 0 (A)) = n for some n ∈ N. We provide a construction of stability functions on A that conjecturally induce all its maximal green sequences. These stability functions are induced by certain curves, called red paths, in the wall and chamber structure of A, described in [11] when A is the module category of an algebra. In particular, we show that red paths give a non-trivial compatibility between the stability conditions introduced by King in [17] and the stability functions introduced by Rudakov in [19]. As a consequence, we show that the wall and chamber structure of an algebra can be recovered using red paths. In this section the canonical inner product of R n is denoted by −, − . That is, given two vectors v, w ∈ R n , we have that v, w = n i=1 v i w i . 4.1. The wall and chamber structure of an abelian category. One of the main motivations of Rudakov for introducing stability functions was to generalise the stability condition introduced by King in [17]. The definition given by King is the following.  Note that not every θ ∈ R n belongs to the stability space D(M ) for some nonzero object M . For instance, it is easy to see that θ = (1, 1, . . . , 1) is an example of such a vector for every A. This leads to the following definition.

Remark 4.5.
In [11,Section 4] the notion of D-generic paths in the wall and chamber of an algebra A is studied. In particular, every wall crossing of a Dgeneric path is either green or red. We use the name red paths here because every wall crossing is red in the sense of [11].
Remark 4.6. Note that, by definition, red paths can pass through the intersection of walls, which is not allowed in the definition of Bridgeland's D-generic paths (see [8,Definition 2.7]) nor in that of Engenhorst's discrete paths (see [14]).
Another key difference between red paths and the other paths cited above is that red paths take account of crossing of all hyperplanes, not only the walls. In the next proposition we show that we can recover the information of crossings from the stability structure induced by the path. Proof. This is a direct consequence of the definition of red path and the fact that the function ρ M induced by γ and M is continuous.
The following result shows that each red path γ : [0, 1] → R n yields a stability function φ γ : Obj * (A) → [0, 1] keeping track of the walls that are crossed by γ. Proof. Let γ be a red path in R n . First, note that φ γ is a well defined function by the definition of red path. We want to show that φ γ induces a stability structure on A. It follows from Lemma 4.7 that γ(t), [ Therefore M is φ γ -semistable.
As a consequence of the previous theorem we get the following result, in which we use the notation of Subsection 2.3 with P = [0, 1].  Proof. It follows directly from Theorem 3.5 that the set S γ of φ γ -stable objects has the properties indicated in the statement if γ induces a maximal green sequence. Now we show the other implication. Since S γ is finite, we can write it as S γ = {M 1 , . . . , M n }. Without loss of generality we can suppose that t Mi ≤ t Mj if i < j. It is easy to see that the finiteness of S γ implies that the chain of torsion classes induced by γ is finite. Moreover, Proposition 4.9 implies that T 0 = A and T 1 = {0}. Finally, we have that t Mi < t Mj whenever i < j, and we conclude that φ γ is discrete. Therefore Theorem 3.5 implies that γ induces a maximal green sequence.
These remarks allow us to define D-generic paths equivalently as follows. On the other hand, one of the main results in [11] says that every maximal green sequence is induced by a D-generic path. This leads us to the following conjecture. Conjecture 4.14. Let A be an abelian length category of finite rank. Then every maximal green sequence in A is induced by a red path in the wall and chamber structure of A.