Wide subcategories are semistable

For an arbitrary finite dimensional algebra $\Lambda$, we prove that any wide subcategory of $\mathsf{mod} \Lambda$ satisfying a certain finiteness condition is $\theta$-semistable for some stability condition $\theta$. More generally, we show that wide subcategories of $\mathsf{mod} \Lambda$ associated with two-term presilting complexes of $\Lambda$ are semistable. This provides a complement for Ingalls-Thomas-type bijections for finite dimensional algebras.


Introduction
This classification problem of subcategories is a well studied subject in representation theory, algebraic geometry and algebraic topology (e.g. [Hop, N, Th]). Among others, we refer to [B, Hov, IT, KS, MS, Ta] for recent developments on the classification of wide subcategories, which are full subcategories of an abelian category closed under kernels, cokernels and extensions.
Important examples of wide subcategories are given by geometric invariant theory for quiver representations [K]. Recall that a stability condition on mod Λ for a finite dimensional algebra Λ is a linear form θ on K 0 (mod Λ) ⊗ Z R, where K 0 (mod Λ) is the Grothendieck group of mod Λ. We say that M ∈ mod Λ is θ-semistable if θ(M ) = 0 and θ(L) ≤ 0 for any submodule L of M , or equivalently, θ(N ) ≥ 0 for any factor module N of M . The full subcategory of θ-semistable Λ-modules is called the θ-semistable subcategory of mod Λ. It is basic that semistable subcategories of mod Λ are wide.
For quiver representations, Ingalls and Thomas [IT] gave bijections between wide/semistable subcategories and other important objects: For the path algebra kQ of a finite connected acyclic quiver Q over a field k, there are bijections (called Ingalls-Thomas bijections) between the following objects, where we refer to Subsection 3.1 for unexplained terminologies.
(1) Isomorphism classes of basic support tilting modules in mod(kQ).
(4) Functorially finite semistable subcategories of mod(kQ). They also proved that (1)-(4) above correspond bijectively with the clusters in the cluster algebra of Q and the isomorphism classes of basic cluster tilting objects in the cluster category of kQ.
Later, works of Adachi-Iyama-Reiten [AIR] and Marks-Stovicek [MS] gave the following Ingalls-Thomas-type bijections for an arbitrary finite dimensional k-algebra, where we refer to Subsection 3.1 for unexplained terminologies and explicit bijections.
Theorem 1.1. [AIR,Theorem 0.5] [MS,Theorem 3.10] Let Λ be a finite dimensional algebra over a field k. There are bijections between the following objects: (1) Isomorphism classes of basic support τ -tilting modules in mod Λ.
Notice that the statement for semistable subcategories of mod Λ is missing in Theorem 1.1. The aim of this paper is to prove the following complement of Theorem 1.1. Theorem 1.2. For a finite dimensional algebra Λ over a field k, the following objects are the same.
To construct a stability condition θ for a given left finite wide subcategory, we need the following preparation. Let T be a basic two-term silting complex in K b (proj Λ). Then there is a decomposition where add T ′ = add T λ and add T ′′ = add T ρ (see [AI,Proposition 2.24]). Then T corresponds to the left finite wide subcategory via the bijection between (1 ′ ) and (3) in Theorem 1.1 (see Subsection 3.1). Our Theorem 1.2 is a consequence of the following result, where −, − is the Euler form (see (3.2)).
Theorem 1.3. Let Λ be a finite dimensional algebra over a field k. Let T be a basic two-term silting complex in K b (proj Λ). We consider an R-linear form θ defined by where X runs over all indecomposable direct summands of T ρ , and a X is an arbitrary positive real number for each X. Then W T is the θ-semistable subcategory of mod Λ.
We prove Theorem 1.3 in a more general setting. Any basic two-term presilting complex U in K b (proj Λ) gives rise to a wide subcategory of mod Λ as follows: By [AIR,Proposition 2.9] (see also [BPP,Section 5]), there are two torsion pairs is a wide subcategory of mod Λ (e.g. [DIRRT]), which is equivalent to mod C for some explicitly constructed finite dimensional algebra C (see [J,Theorem 1.4]). Our Theorem 1.3 can be deduced from the following result since W T = W Tρ holds for any two-term silting complex T (see Lemma 3.5).
Theorem 1.4. Let U be a basic two-term presilting complex in K b (proj Λ). We consider an R-linear form θ defined by where X runs over all indecomposable direct summands of U , and a X is an arbitrary positive real number for each X. Then W U is the θ-semistable subcategory of mod Λ.
Note that in the context of support τ -tilting modules, Theorem 1.4 was independently obtained by Brüstle-Smith-Treffinger [BST] and Speyer-Thomas [ST].
Notations. Let Λ be a finite dimensional algebra over a field k, and mod Λ (resp., proj Λ, inj Λ) the category of finitely generated right Λ-modules (resp., projective right Λ-modules, injective right Λmodules). For M ∈ mod Λ, let add M (resp., Fac M , Sub M ) be the category of all direct summands (resp., factor modules, submodules) of finite direct sums of copies of M . We denote by D the k-dual Hom k (−, k).
For a full subcategory S of mod Λ, let For an additive (resp., abelian) category A, let K b (A) (resp., D b (A)) be the homotopy (resp., derived) category of bounded complexes over A. We denote by ν the Nakayama functor DΛ ⊗ Λ − :

Example
Before proving our results, we give an example. Let Q be the quiver Table 1 gives a complete list of two-term silting complexes, support τ -tiling Λ-modules, functorially finite torsion classes and left finite wide subcategories in mod Λ. The objects in each row correspond to each other under the bijections of Theorem 1.1. For T ∈ 2-siltΛ, we write the class of indecomposable direct summands of T in K 0 (proj Λ). Moreover, indecomposable direct summands X of T ρ and H 0 (X) are colored in blue.
For a basic two-term presilting complex Since Λ is τ -tilting finite, we have a decomposition [DIJ] where U runs over isomorphism classes of basic two-term presilting complexes in K b (proj Λ) (see Figure  1). By Theorem 1.4, any θ in the cone C(U ) gives rise to the wide/semistable subcategory W U of mod Λ.
• sτ -tiltΛ : the set of isomorphism classes of basic support τ -tilting modules in mod Λ.
• f-torsΛ : the set of functorially finite torsion classes in mod Λ.
• f L -wideΛ : the set of left finite wide subcategories of mod Λ.

2-siltΛ
sτ -tiltΛ f-torsΛ f L -wideΛ be an R-linear form, where a U1 > 0 and a U2 > 0. We can calculate the values of θ for indecomposable Λ-modules as follows: Then the θ-semistable subcategory of mod Λ is add 2 3 . Thus W U = add 2 3 is wide and semistable.
Then there is a triangle in K b (proj Λ). Thus T λ = U 3 and T ρ = U 1 ⊕ U 2 = U . By Theorem 1.3, W T is the θ-semistable subcategory of mod Λ for the above θ. In particular, we have W T = W U = add 2 3 , as the second row of the right column in Table 1 shows.

Proofs of our results
3.1. Preliminary. We recall unexplained terminologies and the bijections of Theorem 1.1 from [Ai, AI, AIR, ASS, KV]. Let S be a full subcategory of mod Λ. We call S a torsion class (resp., torsion free class) if it is closed under extensions and quotients (resp., extensions and submodules) [ASS]. For subcategories T and F of mod Λ, a pair (T , F ) is called a torsion pair if T = ⊥ F and F = T ⊥ . Then T is a torsion class and F is a torsion free class. Conversely, any torsion class (resp., torsion free class) gives rise to a torsion pair. We call S functorially finite if any Λ-module admits both a left and a right S-approximation. More precisely, for any M ∈ mod Λ, there are morphisms g 1 : M → S 1 and g 2 : S 2 → M with S 1 , S 2 ∈ S such that Hom Λ (g 1 , S) and Hom Λ (S, g 2 ) are surjective for any S ∈ S. Then g 1 is called a left S-approximation of M and g 2 is called a right S-approximation of M . We call S left finite if the minimal torsion class containing S is functorially finite (see [As]).
Let T ∈ mod Λ. We call T τ -rigid if Hom Λ (T, τ T ) = 0, where τ is the Auslander-Reiten translation of mod Λ [AIR]. We call T support τ -tilting if T is τ -rigid and |T | = |Λ/ e | for some idempotent e of Λ such that eT = 0, where |T | is the number of non-isomorphic indecomposable direct summands of T .
Let P ∈ K b (proj Λ). We call P presilting if Hom K b (proj Λ) (P, P [i]) = 0 for any i > 0 [Ai, AI, KV]. We call P silting if P is presilting and satisfies thick P = K b (proj Λ), where thick P is the smallest subcategory of K b (proj Λ) containing P which is closed under shifts, cones and direct summands. We say that P = (P i , d i ) is two-term if P i = 0 for all i = 0, −1. We denote by 2-presiltΛ (resp., 2-siltΛ) the set of isomorphism classes of basic two-term presilting (resp., silting) complexes in K b (proj Λ).
The bijections of Theorem 1.1 are given in the following way [AIR,Theorem 2.7,3.2] [MS,Theorem 3.10] [S, Theorem]: where H i (T ) is the i-th cohomology of T . Recall that we have for T ∈ 2-siltΛ [AIR,Proposition 2.16].
3.2. Linear forms on Grothendieck groups. Let Λ be a finite dimensional algebra over a field k. Let K 0 (mod Λ) and K 0 (proj Λ) be the Grothendieck groups of the abelian category mod Λ and the exact category proj Λ with only split short exact sequences, respectively. Then we have natural has a basis consisting of the isomorphism classes S i of simple Λ-modules, and K 0 (proj Λ) has a basis consisting of the isomorphism classes P i of indecomposable projective Λ-modules, where topP i = S i . The Euler form is a non-degenerate pairing between K 0 (proj Λ) and K 0 (mod Λ) given by for any P ∈ K b (proj Λ) and M ∈ D b (mod Λ). Then {P i } and {S i } are dual bases of each other. In particular, we have a Z-linear form P, − : K 0 (mod Λ) → Z for P ∈ K b (proj Λ).
Recall that there is a Serre duality, that is, a bifunctorial isomorphism for P ∈ K b (proj Λ) and M ∈ D b (mod Λ). The following observation is basic.
Using R-linear forms on K 0 (mod Λ)⊗ Z R, we have the following properties of torsion pairs (T + U , F + U ) and (T − U , F − U ) for U ∈ 2-presiltΛ. Proposition 3.3. Let U ∈ 2-presiltΛ and θ the corresponding R-linear form on K 0 (mod Λ) ⊗ Z R defined in Theorem 1.4. For M ∈ mod Λ, the following assertions hold. ( where X runs over all indecomposable direct summands of U . Since a X > 0 holds for any X, (a) holds. Similarly, (b) holds by (3.5) and (3.7). Now, we are ready to prove Theorem 1.4.
Proof of Theorem 1.4. Let M ∈ W U = T + U ∩ F − U . Then θ(M ) = 0 holds by Proposition 3.3. Since F − U is a torsion free class, any submodule L of M is also belongs to F − U . Thus θ(L) ≤ 0 holds by Proposition 3.3(b). Therefore, M is θ-semistable.
Conversely, assume that M ∈ mod Λ is θ-semistable. Since (T − U , F − U ) is a torsion pair, there is an exact sequence 0 → L → M → N → 0, where L ∈ T − U and N ∈ F − U . Since θ(L) ≤ 0 holds, we have L = 0 by Proposition 3.3(a). Thus M = N ∈ F − U . Similarly, taking a canonical sequence of M with respect to the torsion pair (T + U , F + U ), we have M ∈ T + U . Thus M ∈ T + U ∩ F − U = W U holds. Next, we make preparations to prove Theorem 1.3. For T ∈ 2-siltΛ, we have the following characterization of the corresponding torsion pairs (T + T , F + T ) and (T − T , F − T ) in mod Λ. Lemma 3.4. Let T = T λ ⊕ T ρ ∈ 2-siltΛ as in (1.1). The following equalities hold.
Proof of Theorem 1.3. The assertion immediately follows from Lemma 3.5 and Theorem 1.4.