Silting Objects, Simple-Minded Collections,

t-Structures and Co-t-Structures for

Finite-Dimensional Algebras

Steffen Koenig, Dong Yang

Received: February 19, 2013 Revised: September 13, 2013

Communicated by Wolfgang Soergel

Abstract. Bijective correspondences are established between (1) silting objects, (2) simple-minded collections, (3) bounded t-structures with length heart and (4) bounded co-t-structures. These correspondences are shown to commute with mutations and partial orders. The results are valid for finite-dimensional algebras. A concrete example is given to illustrate how these correspondences help to compute the space of Bridgeland’s stability conditions.

2010 Mathematics Subject Classification: 16E35, 16E45, 18E30 Keywords and Phrases: silting object, simple-minded collection, tstructure, co-t-structure, mutation.

Contents

1. Introduction  404
2. Notations and preliminaries  405
3. The four concepts  407
4. Finite-dimensional non-positive dg algebras  410
5. The maps  415
6. The correspondences are bijections  421
7. Mutations and partial orders  422
8. A concrete example  431
References  435

1. Introduction

Let Λ be a finite-dimensional associative algebra. Fundamental objects of study in the representation theory of Λ are the projective modules, the simple modules and the category of all (finite-dimensional) Λ-modules. Various structural concepts have been introduced that include one of these classes of objects as particular instances. In this article, four such concepts are related by explicit bijections. Moreover, these bijections are shown to commute with the basic operation of mutation and to preserve partial orders. These four concepts may be based on two different general points of view, either considering particular generators of categories ((1) and (2)) or considering structures on categories that identify particular subcategories ((3) and (4)):

(1) Focussing on objects that generate categories, the theory of Morita equivalences has been extended to tilting or derived equivalences. In this way, projective generators are examples of tilting modules, which have been generalised further to silting objects (which are allowed to have negative self-extensions).

(2) Another, and different, natural choice of ‘generators’ of a module category is the set of simple modules (up to isomorphism). In the context of derived or stable equivalences, this set is included in the concept of simple-minded system or simple-minded collection.

(3) Starting with a triangulated category and looking for particular subcategories, t-structures have been defined so as to provide abelian categories as their hearts. The finite-dimensional Λ-modules form the heart of some t-structure in the bounded derived category D^{b}(mod Λ).

(4) Choosing as triangulated category the homotopy category K^{b}(proj Λ), one considers co-t-structures. The additive category proj Λ occurs as the co-heart of some co-t-structure in K^{b}(proj Λ).

The first main result of this article is:

Theorem (6.1). Let Λ be a finite-dimensional algebra over a field K. There are one-to-one correspondences between

(1) equivalence classes of silting objects in K^{b}(proj Λ),

(2) equivalence classes of simple-minded collections in D^{b}(mod Λ),

(3) bounded t-structures of D^{b}(mod Λ) with length heart,

(4) bounded co-t-structures of K^{b}(proj Λ).

Here two sets of objects in a category are equivalent if they additively generate the same subcategory.

A common feature of all four concepts it that they allow for comparisons, often by equivalences. In particular, each of the four structures to be related comes with a basic operation, called mutation, which produces a new such structure from a given one. Moreover, on each of the four structures there is a partial order. All the bijections in Theorem 6.1 enjoy the following naturality properties:

Theorem (7.12). Each of the bijections between the four structures (1), (2),

(3) and (4) commutes with the respective operation of mutation.

Theorem (7.13). Each of the bijections between the four structures (1), (2),

(3) and (4) preserves the respective partial orders.

The four concepts are crucial in representation theory, geometry and topology. They are also closely related to fundamental concepts in cluster theory such as clusters ([20]), c-matrices and g-matrices ([21, 40]) and cluster-tilting objects ([7]). We refer to the survey paper [16] for more details. A concrete example to be given at the end of the article demonstrates one practical use of these bijections and their properties. Finally we give some remarks on the literature. For path algebras of Dynkin quivers, Keller and Vossieck [33] have already given a bijection between bounded t-structures and silting objects. The bijection between silting objects and t-structures with length heart has been established by Keller and Nicolás [32] for homologically smooth non-positive dg algebras, by Assem, Souto Salorio and Trepode [5] and by Vitória [46], who are focussing on piecewise hereditary algebras. An unbounded version of this bijection has been studied by Aihara and Iyama [1]. The bijection between simple-minded collections and bounded t-structures has been established implicitely in Al-Nofayee’s work [3] and explicitely for homologically smooth non-positive dg algebras in Keller and Nicolás’ work [32] and for finite-dimensional algebras in our preprint [37], which has been partly incorporated into the present article, and partially in the work [44] of Rickard and Rouquier. For hereditary algebras, Buan, Reiten and Thomas [17] studied the bijections between silting objects, simple-minded collections (=Hom_{≤0}-configurations in their setting) and bounded t-structures. The correspondence between silting objects and co-t-structures appears implicitly on various levels of generality in the work of Aihara and Iyama [1] and of Bondarko [12] and explicitly in full generality in the work of Mendoza, Sáenz, Santiago and Souto Salorio [39] and of Keller and Nicolás [31]. For homologically smooth non-positive dg algebras, all the bijections are due to Keller and Nicolás [31]. The intersection of our results with those of Keller and Nicolás is the case of finite-dimensional algebras of finite global dimension.

Acknowledgement. The authors would like to thank Paul Balmer, Mark Blume, Martin Kalck, Henning Krause, Qunhua Liu, Yuya Mizuno, David Pauksztello, Pierre-Guy Plamondon, David Ploog, Jorge Vitória and Jie Xiao for inspiring discussions and helpful remarks. The second-named author gratefully acknowledges financial support from Max-Planck-Institut für Mathematik in Bonn and from DFG program SPP 1388 (YA297/1-1). He is deeply grateful to Bernhard Keller for valuable conversations on derived categories of dg algebras.

2. Notations and preliminaries

2.1. Notations. Throughout, K will be a field. All algebras, modules, vector spaces and categories are over the base field K, and D = Hom_{K}(?, K) denotes the K-dual. By abuse of notation, we will denote by Σ the suspension functors of all the triangulated categories. For a category C, we denote by Hom_{C}(X, Y) the morphism space from X to Y, where X and Y are two objects of C. We will omit the subscript and write Hom(X, Y) when it does not cause confusion. For S a set of objects or a subcategory of C, call

⊥S = {X ∈ C | Hom(X, S) = 0 for all S ∈ S}

and

S^{⊥} = {X ∈ C | Hom(S, X) = 0 for all S ∈ S}

the left and right perpendicular category of S, respectively. Let C be an additive category and S a set of objects or a subcategory of C. Let Add(S) and add(S), respectively, denote the smallest full subcategory of C containing all objects of S and stable for taking direct summands and coproducts respectively taking finite coproducts. The category add(S) will be called the additive closure of S. If further C is abelian or triangulated, the extension closure of S is the smallest subcategory of C containing S and stable under taking extensions. Assume that C is triangulated and let thick(S) denote the smallest triangulated subcategory of C containing objects in S and stable under taking direct summands. We say that S is a set of generators of C, or that C is generated by S, when C = thick(S).

2.2. Derived categories. For a finite-dimensional algebra Λ, let Mod Λ (respectively, mod Λ, proj Λ, inj Λ) denote the category of right Λ-modules (respectively, finite-dimensional right Λ-modules, finite-dimensional projective, injective right Λ-modules), let K^{b}(proj Λ) (respectively, K^{b}(inj Λ)) denote the homotopy category of bounded complexes of proj Λ (respectively, inj Λ) and let D(Mod Λ) (respectively, D^{b}(mod Λ), D^{−}(mod Λ)) denote the derived category of Mod Λ (respectively, bounded derived category of mod Λ, bounded above derived category of mod Λ). All these categories are triangulated with suspension functor the shift functor. We view D^{−}(mod Λ) and D^{b}(mod Λ) as triangulated subcategories of D(Mod Λ). The categories mod Λ, D^{b}(mod Λ) and K^{b}(proj Λ) are Krull–Schmidt categories. An object M of mod Λ (respectively, D^{b}(mod Λ), K^{b}(proj Λ)) is said to be basic if every indecomposable direct summand of M has multiplicity 1. The finitedimensional algebra Λ is said to be basic if the free module of rank 1 is basic in mod Λ (equivalently, in D^{b}(mod Λ) or K^{b}(proj Λ)). For a differential graded(=dg) algebra A, let C(A) denote the category of (right) dg modules over A and K(A) the homotopy category. Let D(A) denote the derived category of dg A-modules, i.e. the triangle quotient of K(A) by acyclic dg A-modules, cf. [29, 30], and let D_{fd}(A) denote its full subcategory of dg A-modules whose total cohomology is finite-dimensional. The category C(A) is abelian and the other three categories are triangulated with suspension functor the shift functor of complexes. Let per(A) = thick(A_{A}), i.e. the triangulated subcategory of D(A) generated by the free dg A-module of rank 1.

For two dg A-modules M and N, let Hom_{A}(M , N ) denote the complex whose degree n component consists of those A-linear maps from M to N which are homogeneous of degree n, and whose differential takes a homogeneous map f of degree n to d_{N} ∘ f − (−1)^{n}f ∘ d_{M}. Then

(2.1) Hom_{K(A)}(M, N) = H^{0}Hom_{A}(M , N ).

A dg A-module M is said to be K-projective if Hom_{A}(M , N ) is acyclic when N is an acyclic dg A-module. For example, A_{A}, the free dg A-module of rank 1 is K-projective, because Hom_{A}(A, N ) = N . Dually, one defines K-injective dg modules, and D(_{A}A) is K-injective. For two dg A-modules M and N such that M is K-projective or N is K-injective, we have

Hom_{D(A)}(M, N) = Hom_{K(A)}(M, N). (2.2)

Let A and B be two dg algebras. Then a triangle equivalence between D(A) and D(B) restricts to a triangle equivalence between per(A) and per(B) and also to a triangle equivalence between D_{fd}(A) and D_{fd}(B). If A is a finite-dimensional algebra viewed as a dg algebra concentrated in degree 0, then D(A) is exactly D(Mod A), D_{fd}(A) is D^{b}(mod A), per(A) is triangle equivalent to K^{b}(proj A), and thick(D(_{A}A)) is triangle equivalent to K^{b}(inj A).

2.3. The Nakayama functor. Let Λ be a finite-dimensional algebra. The Nakayama functor ν_{mod Λ} is defined as ν_{mod Λ} =? ⊗_{Λ} D(_{Λ}Λ), and the inverse Nakayama functor ν_{mod Λ}^{−1} is its right adjoint ν_{mod Λ}^{−1} = Hom_{Λ}(D(_{Λ}Λ),?). They restrict to quasi-inverse equivalences between proj Λ and inj Λ. The derived functors of ν_{mod Λ} and ν_{mod Λ}^{−1}, denoted by ν and ν^{−1}, restrict to quasi-inverse triangle equivalences between K^{b}(proj Λ) and K^{b}(inj Λ). When Λ is self-injective, they restrict to quasi-inverse triangle auto-equivalences of D^{b}(mod Λ). The Auslander–Reiten formula for M in K^{b}(proj Λ) and N in D(Mod Λ) (cf. [23, Chapter 1, Section 4.6]) provides an isomorphism

D Hom(M, N) ≅ Hom(N, νM),

which is natural in M and N. When K^{b}(proj Λ) coincides with K^{b}(inj Λ) (that is, when Λ is Gorenstein), it has Auslander–Reiten triangles and the Auslander– Reiten translation is τ = ν ∘ Σ^{−1}.

3. The four concepts

In this section we introduce silting objects, simple-minded collections, tstructures and co-t-structure. Let C be a triangulated category with suspension functor Σ.

3.1. Silting objects. A subcategory M of C is called a silting subcategory [33, 1] if it is stable for taking direct summands and generates C (i.e. C = thick(M)) and if Hom(M, Σ^{m}N) = 0 for m > 0 and M, N ∈ M.

Theorem 3.1. ([1, Theorem 2.27]) Assume that C is Krull–Schmidt and has a silting subcategory M. Then the Grothendieck group of C is free and its rank is equal to the cardinality of the set of isomorphism classes of indecomposable objects of M.

An object M of C is called a silting object if add M is a silting subcategory of C. This notion was introduced by Keller and Vossieck in [33] to study tstructures on the bounded derived category of representations over a Dynkin quiver. Recently it has also been studied by Wei [47] (who uses the terminology semi-tilting complexes) from the perspective of classical tilting theory. A tilting object is a silting object M such that Hom(M, Σ^{m}M) = 0 for m < 0. For an algebra Λ, a tilting object in K^{b}(proj Λ) is called a tilting complex in the literature. For example, the free module of rank 1 is a tilting object in K^{b}(proj Λ). Assume that Λ is finite-dimensional. Theorem 3.1 implies that (a) any silting subcategory of K^{b}(proj Λ) is the additive closure of a silting object, and (b) any two basic silting objects have the same number of indecomposable direct summands. We will rederive (b) as a corollary of the existence of a certain derived equivalence (Corollary 5.1).

3.2. Simple-minded collections.

Definition 3.2. A collection X_{1}, …, X_{r} of objects of C is said to be simpleminded (cohomologically Schurian in [3]) if the following conditions hold for i, j = 1, …, r

· Hom(X_{i}, Σ^{m}X_{j}) = 0, ∀ m < 0,

· End(X_{i}) is a division algebra and Hom(X_{i}, X_{j}) vanishes for i ≠ j, · X_{1}, …, X_{r} generate C (i.e. C = thick(X_{1}, …, X_{r})).

Simple-minded collections are variants of simple-minded systems in [36] and were first studied by Rickard [43] in the context of derived equivalences of symmetric algebras. For a finite-dimensional algebra Λ, a complete collection of pairwise non-isomorphic simple modules is a simple-minded collection in D^{b}(mod Λ). A natural question is: do any two simple-minded collections have the same collection of endomorphism algebras?

3.3. t-structures. A t-structure on C ([8]) is a pair (C^{≤0}, C^{≥0}) of strict (that is, closed under isomorphisms) and full subcategories of C such that

· ΣC^{≤0} ⊆ C^{≤0} and Σ^{−1}C^{≥0} ⊆ C^{≥0};

· Hom(M, Σ^{−1}N) = 0 for M ∈ C^{≤0} and N ∈ C^{≥0},

· for each M ∈ C there is a triangle M′ → M → M′′ → ΣM′ in C with

M′ ∈ C^{≤0} and M′′ ∈ Σ^{−1}C^{≥0}.

The two subcategories C^{≤0} and C^{≥0} are often called the aisle and the co-aisle of the t-structure respectively. The heart C^{≤0} ∩ C^{≥0} is always abelian. Moreover, Hom(M, Σ^{m}N) vanishes for any two objects M and N in the heart and for any m < 0. The t-structure (C^{≤0}, C^{≥0}) is said to be bounded if

⋃_{n∈ℤ} Σ^{n}C^{≤0} = C = ⋃_{n∈ℤ} Σ^{n}C^{≥0}.

A bounded t-structure is one of the two ingredients of a Bridgeland stability condition [15]. A typical example of a t-structure is the pair (D^{≤0}, D^{≥0}) for the derived category D(Mod Λ) of an (ordinary) algebra Λ, where D^{≤0} consists of complexes with vanishing cohomologies in positive degrees, and D^{≥0} consists of complexes with vanishing cohomologies in negative degrees. This t-structure restricts to a bounded t-structure of D^{b}(mod Λ) whose heart is mod Λ, which is a length category, i.e. every object in it has finite length. The following lemma is well-known.

Lemma 3.3. Let (C^{≤0}, C^{≥0}) be a bounded t-structure on C with heart A.

(a) The embedding A → C induces an isomorphism K_{0}(A) → K_{0}(C) of Grothendieck groups.

(b) C^{≤0} respectively C^{≥0} is the extension closure of Σ^{m}A for m ≥ 0 respectively for m ≤ 0.

(c) C = thick(A). Assume further A is a length category with simple objects {S_{i} | i ∈ I}.

(d) C^{≤0} respectively C^{≥0} is the extension closure of Σ^{m}{S_{i} | i ∈ I} for m ≥ 0 respectively for m ≤ 0.

(e) C = thick(S_{i}, i ∈ I).

(f) If I is finite, then {S_{i} | i ∈ I} is a simple-minded collection.

3.4. Co-t-structures. According to [41], a co-t-structure on C (or weight structure in [12]) is a pair (C_{≥0}, C_{≤0}) of strict and full subcategories of C such that

· both C_{≥0} and C_{≤0} are additive and closed under taking direct summands, · Σ^{−1}C_{≥0} ⊆ C_{≥0} and ΣC_{≤0} ⊆ C_{≤0}; · Hom(M, ΣN) = 0 for M ∈ C_{≥0} and N ∈ C_{≤0}, · for each M ∈ C there is a triangle M′ → M → M′′ → ΣM′ in C with M′ ∈ C_{≥0} and M′′ ∈ ΣC_{≤0}.

The co-heart is defined as the intersection C_{≥0} ∩ C_{≤0}. This is usually not an abelian category. For any two objects M and N in the co-heart, the morphism space Hom(M, Σ^{m}N) vanishes for any m > 0. The co-t-structure (C^{≤0}, C^{≥0}) is said to be bounded [12] if

⋃_{n∈ℤ} Σ^{n}C_{≤0} = C = ⋃_{n∈ℤ} Σ^{n}C_{≥0}.

A bounded co-t-structure is one of the two ingredients of a Jørgensen– Pauksztello costability condition [27]. A typical example of a co-t-structure is the pair (K_{≥0}, K_{≤0}) for the homotopy category K^{b}(proj Λ) of a finitedimensional algebra Λ, where K_{≥0} consists of complexes which are homotopy equivalent to a complex bounded below at 0, and K_{≤0} consists of complexes which are homotopy equivalent to a complex bounded above at 0. The co-heart of this co-t-structure is proj Λ.

Lemma 3.4. ([39, Theorem 4.10 (a)]) Let (C_{≥0}, C_{≤0}) be a bounded co-t-structure on C with co-heart A. Then A is a silting subcategory of C.

Proof. For the convenience of the reader we give a proof. It suffices to show that C = thick(A). Let M be an object of C. Since the co-t-structure is bounded, there are integers m ≥ n such that M ∈ Σ^{m}C_{≥0} ∩ Σ^{n}C_{≤0}. Up to suspension and cosuspension we may assume that m = 0. If n = 0, then M ∈ A. Suppose n < 0. There exists a triangle

\xymatrix{M′ \ar[r] & M \ar[r] & M′′ \ar[r] & ΣM′}

with M′ ∈ Σ^{−1}C_{≥0} and M′′ ∈ C_{≤0}. In fact, M′′ ∈ A, see [12, Proposition 1.3.3.6]. Moreover, ΣM′ ∈ Σ^{n+1}C_{≤0} due to the triangle

\xymatrix{M′′ \ar[r] & ΣM′ \ar[r] & ΣM \ar[r] & ΣM′′}

since both M′′ and ΣM belong to Σ^{n+1}C_{≤0} and C_{≤0} is extension closed (see [12, Proposition 1.3.3.3]). So ΣM′ ∈ C_{≥0} ∪ Σ^{n+1}C_{≤0}. We finish the proof by induction on n. √

Proposition 3.5. ([1, Proposition 2.22], [12, (proof of) Theorem 4.3.2], [39, Theorem 5.5] and [31]) Let A be a silting subcategory of C. Let C_{≤0} respectively C_{≥0} be the extension closure of Σ^{m}A for m ≥ 0 respectively for m ≤ 0. Then (C_{≥0}, C_{≤0}) is a bounded co-t-structure on C with co-heart A.

4. Finite-dimensional non-positive dg algebras

In this section we study derived categories of non-positive dg algebras, i.e. dg algebras A = ⨁_{i∈ℤ} A^{i} with A^{i} = 0 for i > 0, especially finite-dimensional nonpositive dg algebras, i.e. , non-positive dg algebras which, as vector spaces, are finite-dimensional. These results will be used in Sections 5.1 and 5.4.

Non-positive dg algebras are closely related to silting objects. A triangulated category is said to be algebraic if it is triangle equivalent to the stable category of a Frobenius category.

Lemma 4.1. (a) Let A be a non-positive dg algebra. The free dg A-module

of rank 1 is a silting object of per(A).

(b) Let C be an algebraic triangulated category with split idempotents and let M ∈ C be a silting object. Then there is a non-positive dg algebra A together with a triangle equivalence per(A) →^{∼} C which takes A to M.

Proof. (a) This is because Hom_{per(A)}(A, Σ^{i}A) = H^{i}(A) vanishes for i > 0.

(b) By [30, Theorem 3.8 b)] (which is a ‘classically generated’ version of [29, Theorem 4.3]), there is a dg algebra A′ together with a triangle equivalence per(A′) →^{∼} C. In particular, there are isomorphisms Hom_{per(A′)}(A′, Σ^{i}A′) ≅ Hom_{C}(M, Σ^{i}M) for all i ∈ ℤ. Since M is a silting object, A′ has vanishing cohomologies in positive degrees. Therefore, if A = τ_{≤0}A′ is the standard truncation at position 0, then the embedding A ↪ A′ is a quasi-isomorphism. It follows that there is a composite triangle equivalence

\xymatrix{per(A) \ar[r]^{∼} & per(A′) \ar[r]^{∼} & C}

which takes A to M. √

In the sequel of this section we assume that A is a finite-dimensional nonpositive dg algebra. The 0-th cohomology Ā = H^{0}(A) of A is a finitedimensional K-algebra. Let Mod Ā and mod Ā denote the category of (right) modules over Ā and its subcategory consisting of those finite-dimensional modules. Let π : A → Ā be the canonical projection. We view Mod Ā as a subcategory of C(A) via π. The total cohomology H^{∗}(A) of A is a finite-dimensional graded algebra with multiplication induced from the multiplication of A. Let M be a dg A-module. Then the total cohomology H^{∗}(M) carries a graded H^{∗}(A)-module structure, and hence a graded Ā = H^{0}(A)-module structure. In particular, a stalk dg A-module concentrated in degree 0 is an Ā-module.

4.1. The standard t-structure. We follow [22, 4, 34], where the dg algebra is not necessarily finite-dimensional.

Let M = … → M^{i−1} →^{d^{i−1}} M^{i} →^{d^{i}} M^{i+1} → … be a dg A-module. Consider the

standard truncation functors τ_{≤0} and τ_{>0}:

\begin{matrix}τ_{≤0}M = \\ τ_{>0}M = \end{matrix} \xymatrix{… \ar[r] & M^{−2} \ar[r]^{d^{−2}} & M^{−1} \ar[r]^{d^{−1}} & kerd^{0} \ar[r] & 0 \ar[r] & 0 \ar[r] & 0 \ar[r] & … \\ … \ar[r] & 0 \ar[r] & 0 \ar[r] & M^{0}/kerd^{0} \ar[r]^{d^{0}} & M^{1} \ar[r]^{d^{1}} & M^{2} \ar[r]^{d^{2}} & M^{3} \ar[r] & …}

Since A is non-positive, τ_{≤0}M is a dg A-submodule of M and τ_{>0}M is the corresponding quotient dg A-module. Hence there is a distinguished triangle in D(A)

τ_{≤0}M → M → τ_{>0}M → Στ_{≤0}M.

These two functors define a t-structure (D^{≤0}, D^{≥0}) on D(A), where D^{≤0} is the subcategory of D(A) consisting of dg A-modules with vanishing cohomology in positive degrees, and D^{≥0} is the subcategory of D(A) consisting of dg Amodules with vanishing cohomology in negative degrees. By the definition of the t-structure (D^{≤0}, D^{≥0}), the heart H = D^{≤0} ∩ D^{≥0} consists of those dg A-modules whose cohomology is concentrated in degree 0. Thus the functor H^{0} induces an equivalence

H^{0} : H ⟶ Mod Ā.
M ↦ H^{0}(M)

See also [26, Theorem 1.3]. The t-structure (D^{≤0}, D^{≥0}) on D(A) restricts to a bounded t-structure on D_{fd}(A) with heart equivalent to mod Ā.

4.2. Morita reduction. Let d be the differential of A. Then d(A^{0}) = 0. Let e be an idempotent of A. For degree reasons, e must belong to A^{0}, and the graded subspace eA of A is a dg submodule: d(ea) = d(e)a + ed(a) = ed(a). Therefore for each decomposition 1 = e_{1} + … + e_{n} of the unity into a sum of primitive orthogonal idempotents, there is a direct sum decomposition A = e_{1}A ⊕ … ⊕ e_{n}A of A into indecomposable dg A-modules. Moreover, if e and e′ are two idempotents of A such that eA ≅ e′A as ordinary modules over the ordinary algebra A, then this isomorphism is also an isomorphism of dg modules. Indeed, there are two elements of A such that fg = e and gf = e′. Again for degree reasons, f and g belong to A^{0}. So they induce isomorphisms of dg A-modules: eA → e′A, a ↦ ga and e′A → eA, a ↦ fa. It follows that the above decomposition of A into a direct sum of indecomposable dg modules is essentially unique. Namely, if 1 = e′_{1} + … + e′_{n} is another decomposition of the unity into a sum of primitive orthogonal idempotents, then m = n and up to reordering, e_{1}A ≅ e′_{1}A, …, e_{n}A ≅ e′_{n}A.

4.3. The perfect derived category. Since A is finite-dimensional (and thus has finite-dimensional total cohomology), per(A) is a triangulated subcategory of D_{fd}(A). We assume, as we may, that A is basic. Let 1 = e_{1}+…+e_{n} be a decomposition of 1 in A into a sum of primitive orthogonal idempotents. Since d(x) = λ_{1}e_{i_{1}} + … + λ_{s}e_{i_{s}} implies that d(e_{i_{j}}x) = λ_{j}e_{i_{j}}, the intersection of the space spanned by e_{1}, …, e_{n} with the image of the differential d has a basis consisting of some e_{i}’s, say e_{r+1}, …, e_{n}. So, e_{r+1}A, …, e_{n}A are homotopic to zero. We say that a dg A-module M is strictly perfect if its underlying graded module is of the form ⨁_{j=1}^{N} R_{j}, where R_{j} belongs to add(Σ^{t_{j}}A) for some t_{j} with t_{1} < t_{2} < … < t_{N}, and if its differential is of the form d_{int} + δ, where d_{int} is the direct sum of the differential of the R_{j}’s, and δ, as a degree 1 map from ⨁_{j=1}^{N} R_{j} to itself, is a strictly upper triangular matrix whose entries are in A. It is minimal if in addition no shifted copy of e_{r+1}A, …, e_{n}A belongs to add(R_{1}, …, R_{j}), and the entries of δ are in the radical of A, cf. [42, Section 2.8]. Strictly perfect dg modules are K-projective. If A is an ordinary algebra, then strictly perfect dg modules are precisely bounded complexes of finitely generated projective modules.

Lemma 4.2. Let M be a dg A-module belonging to per(A). Then M is quasiisomorphic to a minimal strictly perfect dg A-module.

Proof. Bearing in mind that e_{1}A, …, e_{r}A have local endomorphism algebras and e_{r+1}A, …, e_{n}A are homotopic to zero, we prove the assertion as in [42, Lemma 2.14]. √

4.4. Simple modules. Assume that A is basic. According to the preceding subsection, we may assume that there is a decomposition 1 = e_{1} + … + e_{r} + e_{r+1} +…+e_{n} of the unity of A into a sum of primitive orthogonal idempotents such that 1 = ē_{1} + … + ē_{r} is a decomposition of 1 in Ā into a sum of primitive orthogonal idempotents.

Let S_{1}, …, S_{r} be a complete set of pairwise non-isomorphic simple Ā-modules and let R_{1}, …, R_{r} be their endomorphism algebras. Then

Hom_{A}(e_{i}A, S_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j, \\ 0 & otherwise. \end{cases}

Therefore, by (2.1) and (2.2),

Hom_{D(A)}(e_{i}A, Σ^{m}S_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

Moreover, {e_{1}A, …, e_{r}A} and {S_{1}, …, S_{r}} characterise each other by this property. On the one hand, if M is a dg A-module such that for some integer 1 ≤ j ≤ r

Hom_{D(A)}(e_{i}A, Σ^{m}M) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise, \end{cases}

then M is isomorphic in D(A) to S_{j}. On the other hand, let M be an object of per(A) such that for some integer 1 ≤ i ≤ r

Hom_{D(A)}(M, Σ^{m}S_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

Then by replacing M by its minimal perfect resolution (Lemma 4.2), we see that M is isomorphic in D(A) to e_{i}A. Further, recall from Section 4.1 that D_{fd}(A) admits a standard t-structure whose heart is equivalent to mod Ā. This implies that the simple modules S_{1}, …, S_{r} form a simple-minded collection in D_{fd}(A).

4.5. The Nakayama functor. For a complex M of K-vector spaces, we define its dual as D(M) = Hom_{K}(M, K), where K in the second argument is considered as a complex concentrated in degree 0. One checks that D defines a duality between finite-dimensional dg A-modules and finite-dimensional dg A^{op}-modules. Let e be an idempotent of A and M a dg A-module. Then there is a canonical isomorphism

Hom_{A}(eA, M) ≅ Me.

If in addition each component of M is finite-dimensional , there are canonical isomorphisms

Hom_{A}(eA, M) ≅ Me ≅ DHom_{A}(M, D(Ae)).

Let C(A) denote the category of dg A-modules. The Nakayama functor ν : C(A) → C(A) is defined by ν(M) = DHom_{A}(M, A) [29, Section 10]. There are canonical isomorphisms

DHom_{A}(M , N ) ≅ Hom_{A}(N , νM )

for any strictly perfect dg A-module M and any dg A-module N. Then ν(eA) = D(Ae) for an idempotent e of A, and the functor ν induces a triangle equivalences between the subcategories per(A) and thick(D(A)) of D(A) with quasi-inverse given by ν^{−1}(M) = Hom_{A}(D(A), M). Moreover, we have the Auslander–Reiten formula

D Hom(M, N) ≅ Hom(N, νM),

which is natural in M ∈ per(A) and N ∈ D(A). Let e_{1}, …, e_{r}, S_{1}, …, S_{r} and R_{1}, …, R_{r} be as in the preceding subsection. Then

Hom_{A}(S_{j}, D(Ae_{i})) ≅ DHom_{A}(e_{i}A, S_{j} ) = \begin{cases}(R_{j})_{R_{j}} & if i = j, \\ 0 & otherwise. \end{cases}

Therefore, by (2.1) and (2.2),

Hom_{D(A)}(S_{j}, Σ^{m}D(Ae_{i})) = \begin{cases}(R_{j})_{R_{j}} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

Moreover, {D(Ae_{1}), …, D(Ae_{r})} and {S_{1}, …, S_{r}} characterise each other in D(A) by this property. This follows from the arguments in the preceding subsection by applying the functors ν and ν^{−1}.

4.6. The standard co-t-structure. Let P_{≤0} (respectively, P_{≥0}) be the smallest full subcategory of per(A) containing {Σ^{m}A | m ≥ 0} (respectively, {Σ^{m}A | m ≤ 0}) and closed under taking extensions and direct summands. The following lemma is a special case of Proposition 3.5. For the convenience of the reader we include a proof.

Lemma 4.3. The pair (P_{≥0}, P_{≤0}) is a co-t-structure on per(A). Moreover, its co-heart is add(A_{A}).

Proof. Since Hom(A, Σ^{m}A) = 0 for m ≥ 0, it follows that Hom(X, ΣY) = 0 for M ∈ P_{≥0} and N ∈ P_{≤0}. It remains to show that any object M in per(A) fits into a triangle whose outer terms belong to P_{≥0} and P_{≤0}, respectively. By Lemma 4.2, we may assume that M is minimal perfect. Write M = (⨁_{j=1}^{N} R_{j}, d_{int} + δ) as in Section 4.3. Let N′ ∈ {1, …, N} be the unique

integer such that t_{N′} ≥ 0 but t_{N′+1} < 0. Let M′ be the graded module ⨁_{j=1}^{N′} R_{j} endowed with the differential restricted from d_{int} +δ. Because d_{int} +δ is upper triangular, M′ is a dg submodule of M. Clearly M′ belongs to P_{≥0} and the quotient M′′ = M/M′ belongs to ΣP_{≤0}. Thus we obtain the desired triangle

\xymatrix{M′ \ar[r] & M \ar[r] & M′′ \ar[r] & ΣM′}

with M′ in P_{≥0} and M′′ in ΣP_{≤0}. √

5. The maps

Let Λ be a finite-dimensional basic K-algebra. This section is devoted to defining the maps in the following diagram.

bounded co-t-structures on K^{b}(proj Λ) equivalence classes of silting objects in K^{b}(proj Λ)

equivalence classes of simple-minded collections in D^{b}(mod Λ) bounded t-structures on D^{b}(mod Λ) with length heart

ϕ_{41}

ϕ_{14}

ϕ_{32}

ϕ_{23}

ϕ_{34} ϕ_{12} ϕ_{21}

[figure]

ϕ_{31}

5.1. Silting objects induce derived equivalences. Let M be a basic silting object of the category K^{b}(proj Λ). By definition, M is a bounded complex of finitely generated projective Λ-modules such that Hom_{K^{b}(proj Λ)}(M, Σ^{m}M) vanishes for all m > 0. By Lemma 4.1, there is a non-positive dg algebra whose perfect derived category is triangle equivalent to K^{b}(proj Λ). This equivalence sends the free dg module of rank 1 to M. Below we explicitly construct such a dg algebra. Consider Hom_{Λ}(M , M ). Recall that the degree n component of Hom_{Λ}(M , M ) consists of those Λ-linear maps from M to itself which are homogeneous of degree n. The differential of Hom_{Λ}(M , M ) takes a homogeneous map f of degree n to d ∘ f − (−1)^{n}f ∘ d, where d is the differential of M. This differential and the composition of maps makes Hom_{Λ}(M , M ) into a dg algebra. Therefore Hom_{Λ}(M , M ) is a finite-dimensional dg algebra. Moreover, H^{m}(Hom_{Λ}(M , M )) = Hom_{D(Λ)}(M , Σ^{m}M ) for any integer m, by (2.1) and (2.2). Because M is a silting object, Hom_{Λ}(M , M ) has cohomology concentrated in non-positive degrees. Take the truncated dg algebra Γ̃ = τ_{≤0}Hom_{Λ}(M , M ), where τ_{≤0} is the standard truncation at position 0. Then the embedding Γ̃ → Hom_{Λ}(M , M ) is a quasi-isomorphism of dg algebras, and hence Γ̃ is a finite-dimensional non-positive dg algebra. Therefore, the derived category D(Γ̃) carries a natural t-structure (D^{≤0}, D^{≥0}) with heart D^{≤0} ∩ D^{≥0} equivalent to Mod Γ, where Γ = H^{0}(Γ̃) = End_{D(A)}(M). This t-structure restricts to a t-structure on D_{fd}(Γ̃), denoted by (D_{fd}^{≤0}, D_{fd}^{≥0}), whose heart is equivalent to mod Γ. Moreover, there is a standard co-t-structure (P_{≥0}, P_{≤0}) on per(Γ̃), see Section 4.

The object M has a natural dg Γ̃-Λ-bimodule structure. Moreover, since it generates K^{b}(proj Λ), it follows from [29, Lemma 6.1 (a)] that there are triangle equivalences

[figure: F =? ⊗^{L}Γ̃ M :; D(Γ̃); ∼; D(Λ); D(Mod Λ); D_{fd}(Γ̃); ∼; D_{fd}(Λ); D^{b}(mod Λ); per(Γ̃); ∼; per(Λ); K^{b}(proj Λ)]

These equivalences take Γ̃ to M. The following special case of Theorem 3.1 is a consequence.

Corollary 5.1. The number of indecomposable direct summands of M equals the rank of the Grothendieck group of K^{b}(proj Λ). In particular, any two basic silting objects of K^{b}(proj Λ) have the same number of indecomposable direct summands.

Proof. The number of indecomposable direct summands of M equals the rank of the Grothendieck group of mod Γ, which equals the rank of the Grothendieck group of D_{fd}(Γ̃) ≅ D^{b}(mod Λ) since mod Γ is the heart of a bounded t-structure (Lemma 3.3). √

Write M = M_{1} ⊕ … ⊕ M_{r} with M_{i} indecomposable. Suppose that X_{1}, …, X_{r} are objects in D^{b}(mod Λ) such that their endomorphism algebras R_{1}, …, R_{r} are division algebras and that the following formula holds for i, j = 1, …, r and m ∈ ℤ

Hom(M_{i}, Σ^{m}X_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

Then up to isomorphism, the objects X_{1}, …, X_{r} are sent by the derived equiv-

alence ? ⊗^{L}Γ̃ M to a complete set of pairwise non-isomorphic simple Γ-modules, see Section 4.4.

Lemma 5.2. (a) Let X′_{1}, …, X′_{r} be objects of D^{b}(mod Λ) such that the following formula holds for 1 ≤ i, j ≤ r and m ∈ ℤ

Hom(M_{i}, Σ^{m}X′_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

Then X_{i} ≅ X′_{i} for any i = 1, …, r.

(b) Let M′_{1}, …, M′_{r} be objects of K^{b}(proj Λ) such that the following formula holds for 1 ≤ i, j ≤ r and m ∈ ℤ

Hom(M′_{i}, Σ^{m}X_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

Then M_{i} ≅ M′_{i} for any i = 1, …, r.

Proof. This follows from the corresponding result in D(Γ̃), see Section 4.4. √

5.2. From co-t-structures to silting objects. Let (C_{≥0}, C_{≤0}) be a bounded co-t-structure of K^{b}(proj Λ). By Lemma 3.4, the co-heart A = C_{≥0} ∩ C_{≤0} is a silting subcategory of K^{b}(proj Λ). Since Λ is a silting object of K^{b}(proj Λ), it follows from Theorem 3.1 that A has an additive generator, say M, i.e. A = add(M). Then M is a silting object in K^{b}(proj Λ). Define

ϕ_{14}(C_{≥0}, C_{≤0}) = M.

5.3. From t-structures to simple-minded collections. Let (C^{≤0}, C^{≥0}) be a bounded t-structure of D^{b}(mod Λ) with length heart A. Boundedness implies that the Grothendieck group of A is isomorphic to the Grothendieck group of D^{b}(mod Λ), which is free, say, of rank r. Therefore, A has precisely r isomorphism classes of simple objects, say X_{1}, …, X_{r}. By Lemma 3.3 (f), X_{1}, …, X_{r} is a simple-minded collection in D^{b}(mod Λ). Define

ϕ_{23}(C^{≤0}, C^{≥0}) = {X_{1}, …, X_{r}}.

5.4. From silting objects to simple-minded collections, tstructures and co-t-structures. Let M be a silting object of K^{b}(proj Λ). Define full subcategories of C

C^{≤0} = {N ∈ D^{b}(mod Λ) | Hom(M, Σ^{m}N) = 0, ∀ m > 0},
C^{≥0} = {N ∈ D^{b}(mod Λ) | Hom(M, Σ^{m}N) = 0, ∀ m < 0},
C_{≤0} = the additive closure of the extension closure
of Σ^{m}M, m ≥ 0 in K^{b}(proj Λ),
C_{≥0} = the additive closure of the extension closure
of Σ^{m}M, m ≤ 0 in K^{b}(proj Λ).

Lemma 5.3. (a) The pair (C^{≤0}, C^{≥0}) is a bounded t-structure on D^{b}(mod Λ) whose heart is equivalent to mod Γ for Γ = End(M). Write M = M_{1} ⊕ … ⊕ M_{r} and let X_{1}, …, X_{r} be the corresponding simple objects of the heart with endomorphism algebras R_{1}, …, R_{r} respectively. Then the following formula holds for 1 ≤ i, j ≤ r and m ∈ ℤ

Hom(M_{i}, Σ^{m}X_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

(b) The pair (C_{≥0}, C_{≤0}) is a bounded co-t-structure on K^{b}(proj Λ) whose co-heart is add(M).

The first statement of part (a) is proved by Keller and Vossieck [33] in the case when Λ is the path algebra of a Dynkin quiver and by Assem, Souto and Trepode [5] in the case when Λ is hereditary.

Proof. Let Γ̃ be the truncated dg endomorphism algebra of M, see Section 5.1. Then per(Γ̃) has a standard bounded co-t-structure (P_{≥0}, P_{≤0}) and D_{fd}(Γ̃) has a standard bounded t-structure (D_{fd}^{≤0}, D_{fd}^{≥0}) with heart equivalent to mod Γ.

One checks that the triangle equivalence ?⊗^{L}Γ̃ M takes (P_{≥0}, P_{≤0}) to (C_{≥0}, C_{≤0}) and it takes (D_{fd}^{≤0}, D_{fd}^{≥0}) to (C^{≤0}, C^{≥0}). √

Define

ϕ_{31}(M) = (C^{≤0}, C^{≥0}),
ϕ_{41}(M) = (C_{≥0}, C_{≤0}),
ϕ_{21}(M) = {X_{1}, …, X_{r}}.

5.5. From simple-minded collections to t-structures. Let X_{1}, …, X_{r} be a simple-minded collection of D^{b}(mod Λ). Let C^{≤0} (respectively, C^{≥0}) be the extension closure of {Σ^{m}X_{i} | i = 1, …, r, m ≥ 0} (respectively, {Σ^{m}X_{i} | i = 1, …, r, m ≤ 0}) in D^{b}(mod Λ).

Proposition 5.4. The pair (C^{≤0}, C^{≥0}) is a bounded t-structure on D^{b}(mod Λ). Moreover, the heart of this t-structure is a length category with simple objects X_{1}, …, X_{r}. The same results hold true with D^{b}(mod Λ) replaced by a Homfinite Krull–Schmidt triangulated category C.

Proof. The first two statements are [3, Corollary 3 and Proposition 4]. The proof there still works if we replace D^{b}(mod Λ) by C. √

Define

ϕ_{32}(X_{1}, …, X_{r}) = (C^{≤0}, C^{≥0}).

Later we will show that the heart of this t-structure always is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra (Corollary 6.2). This was proved by Al-Nofayee for self-injective algebras Λ, see [3, Theorem 7].

Corollary 5.5. Any two simple-minded collections in D^{b}(mod Λ) have the same cardinality.

Proof. By Proposition 5.4, the cardinality of a simple-minded collection equals the rank of the Grothendieck group of D^{b}(mod Λ). The assertion follows. √

5.6. From simple-minded collections to silting objects. Let X_{1}, …, X_{r} be a simple-minded collection in D^{b}(mod Λ). We will construct a silting object ν^{−1}T of K^{b}(proj Λ) following a method of Rickard [43]. Then we define

ϕ_{12}(X_{1}, …, X_{r}) = ν^{−1}T.

The same construction is studied by Keller and Nicolás [32] in the context of positive dg algebras. In the case of Λ being hereditary, Buan, Reiten and Thomas [17] give an elegant construction of ν^{−1}(T) using the Braid group action on exceptional sequences. Unfortunately, their construction cannot be generalised. Let R_{1}, …, R_{r} be the endomorphism algebras of X_{1}, …, X_{r}, respectively. Set X_{i}^{(0)} = X_{i}. Suppose X_{i}^{(n−1)} is constructed. For i, j = 1, …, r and m < 0, let B(j, m, i) be a basis of Hom(Σ^{m}X_{j}, X_{i}^{(n−1)}) over R_{j}. Put

Z_{i}^{(n−1)} = ⨁_{m<0} ⨁_{j} ⨁_{B(j,m,i)} Σ^{m}X_{j}

and let α_{i}^{(n−1)} : Z_{i}^{(n−1)} → X_{i}^{(n−1)} be the map whose component corresponding to f ∈ B(j, m, i) is exactly f. Let X_{i}^{(n)} be a cone of α_{i}^{(n−1)} and form the corresponding triangle

\xymatrix{Z_{i}^{(n−1)} \ar[r]^{α_{i}^{(n−1)}} & X_{i}^{(n−1)} \ar[r]^{β_{i}^{(n−1)}} & X_{i}^{(n)} \ar[r] & ΣZ_{i}^{(n−1)}.}

Inductively, a sequence of morphisms in D(Mod Λ) is constructed:

\xymatrix{X_{i}^{(0)} \ar[r]^{β_{i}^{(0)}} & X_{i}^{(1)} \ar[r] & … \ar[r] & X_{i}^{(n−1)} \ar[r]^{β_{i}^{(n−1)}} & X_{i}^{(n)} \ar[r] & ….}

Let T_{i} be the homotopy colimit of this sequence. That is, up to isomorphism, T_{i} is defined by the following triangle

\xymatrix{⨁_{n≥0} X_{i}^{(n)} \ar[r]^{id−β} & ⨁_{n≥0} X_{i}^{(n)} \ar[r] & T_{i} \ar[r] & Σ ⨁_{n≥0} X_{i}^{(n)}} .

Here β = (β_{mn}) is the square matrix with rows and columns labeled by nonnegative integers and with entries β_{mn} = β_{i}^{(n)} if n + 1 = m and 0 otherwise. These properties of T_{i}’s were proved by Rickard in [43] for symmetric algebras Λ over algebraically closed fields. Rickard remarked that they hold for arbitrary fields, see [43, Section 8]. In fact, his proofs verbatim carry over to general finite-dimensional algebras.

Lemma 5.6. (a) ([43, Lemma 5.4]) For 1 ≤ i, j ≤ r, and m ∈ ℤ,

Hom(X_{j}, Σ^{m}T_{i}) = \begin{cases}(R_{j})_{R_{j}} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

(b) ([43, Lemma 5.5]) For each 1 ≤ i ≤ r, T_{i} is quasi-isomorphic to a bounded complex of finitely generated injective Λ-modules.

(c) ([43, Lemma 5.8]) Let C be an object of D^{−}(mod Λ). If Hom(C, Σ^{m}T_{i}) = 0 for all m ∈ ℤ and all 1 ≤ i ≤ r, then C = 0.

From now on we assume that T_{i} is a bounded complex of finitely generated injective Λ-modules. Recall from Section 2.3 that the Nakayama functor ν and the inverse Nakayama functor ν^{−1} are quasi-inverse triangle equivalences between K^{b}(proj Λ) and K^{b}(inj Λ) The following is a consequence of Lemma 5.6 and the Auslander–Reiten formula.

Lemma 5.7. (a) For 1 ≤ i, j ≤ r, and m ∈ ℤ,

Hom(ν^{−1}T_{i}, Σ^{m}X_{j}) = \begin{cases}_{R_{j}}R_{j} & if i = j and m = 0, \\ 0 & otherwise. \end{cases}

(b) For each 1 ≤ i ≤ r, ν^{−1}T_{i} is a bounded complex of finitely generated projective Λ-modules.

(c) Let C be an object of D^{−}(mod Λ). If Hom(ν^{−1}T_{i}, Σ^{m}C) = 0 for all m ∈ ℤ and all 1 ≤ i ≤ r, then C = 0.

Put T = ⨁_{i=1}^{r} T_{i} and ν^{−1}T = ⨁_{i=1}^{r} ν^{−1}T_{i}.

Lemma 5.8. We have Hom(ν^{−1}T, Σ^{m}T) = 0 for m < 0. Equivalently, Hom(ν^{−1}T, Σ^{m}ν^{−1}T) = Hom(T, Σ^{m}T) = 0 for m > 0.

Proof. Same as the proof of [43, Lemma 5.7], with the T_{i} in the first entry of Hom there replaced by ν^{−1}T_{i}. √

It follows from Lemma 5.7 (c) that ν^{−1}T generates K^{b}(proj Λ). Combining this with Lemma 5.8 implies

Proposition 5.9. ν^{−1}T is a silting object of K^{b}(proj Λ).

Rickard’s construction was originally motivated by constructing tilting complexes over symmetric algebras which yield certain derived equivalences, see [43, Theorem 5.1]. His work was later generalised by Al-Nofayee to self-injective algebras, see [2, Theorem 4].

5.7. From co-t-structures to t-structures. Let (C_{≥0}, C_{≤0}) be a bounded co-t-structure of K^{b}(proj Λ). Let

C^{≤0} = {N ∈ D^{b}(mod Λ) | Hom(M, N) = 0, ∀ M ∈ Σ^{−1}C_{≥0}}
C^{≥0} = {N ∈ D^{b}(mod Λ) | Hom(M, N) = 0, ∀ M ∈ ΣC_{≤0}}.

Lemma 5.10. The pair (C^{≤0}, C^{≥0}) is a bounded t-structure on D^{b}(mod Λ) with length heart.

Proof. Because (C^{≤0}, C^{≥0}) = ϕ_{31} ∘ ϕ_{14}(C_{≥0}, C_{≤0}). √

By definition (C^{≤0}, C^{≥0}) is right orthogonal to the given co-t-structure in the sense of Bondarko [11, Definition 2.5.1]. Define

ϕ_{34}(C_{≥0}, C_{≤0}) = (C^{≤0}, C^{≥0}).

If Λ has finite global dimension, then K^{b}(proj Λ) is identified with D^{b}(mod Λ). As a consequence, C_{≤0} = C^{≤0} and C^{≥0} = νC_{≥0}. Thus the t-structure (C^{≤0}, C^{≥0}) is right adjacent to the given co-t-structure (C_{≥0}, C_{≤0}) in the sense of Bondarko [12, Definition 4.4.1].

5.8. Some remarks. Some of the maps ϕ_{ij} are defined in more general setups:

– ϕ_{14} and ϕ_{41} are defined for all triangulated categories, with silting objects replaced by silting subcategories, by Proposition 3.5 and Lemma 3.4, see also [12, 31, 39].

– ϕ_{23} is defined for all triangulated categories, with simple-minded collections allowed to contain infinitely many objects (Lemma 3.3).

– ϕ_{32} is defined for all algebraic triangulated categories (see [32]) and for Hom-finite Krull–Schmidt triangulated categories (see Proposition 5.4).

– ϕ_{21} and ϕ_{31} are defined for all algebraic triangulated categories (replacing K^{b}(proj Λ)), with D^{b}(mod Λ) replaced by a suitable triangulated category; then we may follow the arguments in Sections 4.1 and 5.4.

– ϕ_{34} is defined for all algebraic triangulated categories (replacing K^{b}(proj Λ)), with D^{b}(mod Λ) replaced by a suitable triangulated category. Then we may follow the argument in Section 5.7.

– ϕ_{12} is defined for finite-dimensional non-positive dg algebras, since these dg algebras behave like finite-dimensional algebras from the perspective of derived categories. Similarly, ϕ_{12} is defined for homologically smooth non-positive dg algebras, see [31].

6. The correspondences are bijections

Let Λ be a finite-dimensional K-algebra. In the preceding section we defined the maps ϕ_{ij}. In this section we will show that they are bijections. See [5, 46] for related work, focussing on piecewise hereditary algebras.

Theorem 6.1. The ϕ_{ij}’s defined in Section 5 are bijective. In particular, there are one-to-one correspondences between

(1) equivalence classes of silting objects in K^{b}(proj Λ),

(2) equivalence classes of simple-minded collections in D^{b}(mod Λ),

(3) bounded t-structures on D^{b}(mod Λ) with length heart,

(4) bounded co-t-structures on K^{b}(proj Λ).

There is an immediate consequence:

Corollary 6.2. Let A be the heart of a bounded t-structure on D^{b}(mod Λ). If A is a length category, then A is equivalent to mod Γ for some finitedimensional algebra Γ.

Proof. By Theorem 6.1, such a t-structure is of the form ϕ_{31}(M) for some silting object M of K^{b}(proj Λ). The result then follows from Lemma 5.3 (a). √

The proof of the theorem is divided into several lemmas, which are consequences of the material collected in the previous sections.

Lemma 6.3. The maps ϕ_{14} and ϕ_{41} are inverse to each other.

Proof. Let M be a basic silting object. The definitions of ϕ_{14} and ϕ_{41} and Lemma 5.3 (b) imply that ϕ_{14} ∘ ϕ_{41}(M) ≅ M. Let (C_{≥0}, C_{≤0}) be a bounded co-t-structure on K^{b}(proj Λ). It follows from Lemma 3.4 that ϕ_{41} ∘ ϕ_{14}(C_{≥0}, C_{≤0}) = (C_{≥0}, C_{≤0}). √ Recall from Section 5.8 that ϕ_{14} and ϕ_{41} are defined in full generality. Lemma 6.3 holds in full generality as well, see [39, Corollary 5.8] and [31].

Lemma 6.4. The maps ϕ_{21} and ϕ_{12} are inverse to each other.

Proof. This follows from the Hom-duality: Lemma 5.7 (a), Lemma 5.3 (a) and Lemma 5.2. √

Lemma 6.5. The maps ϕ_{23} and ϕ_{32} are inverse to each other.

Proof. Let X_{1}, …, X_{r} be a simple-minded collection in D^{b}(mod Λ). It follows from Proposition 5.4 that ϕ_{23} ∘ ϕ_{32}(X_{1}, …, X_{r}) = {X_{1}, …, X_{r}}. Let (C^{≤0}, C^{≥0}) be a bounded t-structure on D^{b}(mod Λ) with length heart. It follows from Lemma 3.3 that ϕ_{32} ∘ ϕ_{23}(C^{≤0}, C^{≥0}) = (C^{≤0}, C^{≥0}). √

Lemma 6.6. For a triple i, j, k such that ϕ_{ij}, ϕ_{jk} and ϕ_{ik} are defined, there is the equality ϕ_{ij} ∘ ϕ_{jk} = ϕ_{ik}. In particular, ϕ_{31} and ϕ_{34} are bijective.

Proof. In view of the preceding three lemmas, it suffices to prove ϕ_{23}∘ϕ_{31} = ϕ_{21} and ϕ_{31} ∘ ϕ_{14} = ϕ_{34}, which is clear from the definitions. √

7. Mutations and partial orders

In this section we introduce mutations and partial orders on the four concepts in Section 3, and we show that the maps defined in Section 5 commute with mutations and preserve the partial orders. Let C be a Hom-finite Krull–Schmidt triangulated category with suspension functor Σ.

7.1. Silting objects. We follow [1, 18] to define silting mutation. Let M be a silting object in C. We assume that M is basic and M = M_{1} ⊕ … ⊕ M_{r} is a decomposition into indecomposable objects. Let i = 1, …, r. The left mutation of M at the direct summand M_{i} is the object μ_{i}^{+}(M) = M′_{i} ⊕ ⨁_{j≠i} M_{j} where M′_{i} is the cone of the minimal left add(⨁_{j≠i} M_{j})-approximation of M_{i}

\xymatrix{M_{i} \ar[r] & E.}

Similarly one can define the right mutation μ_{i}^{−}(M).

Theorem 7.1. ([1, Theorem 2.31 and Proposition 2.33]) The objects μ_{i}^{+}(M) and μ_{i}^{−}(M) are silting objects. Moreover, μ_{i}^{+} ∘ μ_{i}^{−}(M) ≅ M ≅ μ_{i}^{−} ∘ μ_{i}^{+}(M).

Let silt C be the set of isomorphism classes of basic tilting objects of C. The silting quiver of C has the elements in silt C as vertices. For P, P′ ∈ silt C, there are arrows from P to P′ if and only if P′ is obtained from P by a left mutation, in which case there is precisely one arrow. See [1, Section 2.6]. For P, P′ ∈ silt C, define P ≥ P′ if Hom(P, Σ^{m}P′) = 0 for any m > 0. According to [1, Theorem 2.11], ≥ is a partial order on silt C.

Theorem 7.2. ([1, Theorem 2.35]) The Hasse diagram of (silt C, ≥) is the silting quiver of C.

Next we define (a generalisation of) the Brenner–Butler tilting module for a finite-dimensional algebra, and show that it is a left mutation of the free module of rank 1. The corresponding right mutation is the Okuyama–Rickard complex, see [1, Section 2.7]. Let Λ be a finite-dimensional basic algebra and 1 = e_{1} + … + e_{n} be a decomposition of the unity into the sum of primitive idempotents and Λ = P_{1} ⊕ … ⊕ P_{n} the corresponding decomposition of the free module of rank 1. Fix i = 1, …, n and let S_{i} be the corresponding simple module and let S_{i}^{+} = D(Λ/Λ(1 − e_{i})Λ). Assume that

· S_{i}^{+} is not injective,

· the projective dimension of τ_{mod Λ}^{−1}S_{i}^{+} is at most 1.

Definition 7.3. Define the BB tilting module with respect to i by

T = τ_{mod Λ}^{−1}S_{i}^{+} ⊕ ⨁_{j≠i} P_{j}.

We call it the APR tilting module if Λ/Λ(1 − e_{i})Λ is projective as a Λ-module.

When Λ/Λ(1 − e_{i})Λ is a division algebra (i.e. there are no loops in the quiver of Λ at the vertex i), this specialises to the ‘classical’ BB tilting module [13] and APR tilting module [6]. The following proposition generalises [1, Theorem 2.53].

Proposition 7.4. (a) T is isomorphic to the left mutation μ_{i}^{+}(Λ) of Λ. (b) T is a tilting Λ-module of projective dimension at most 1.

Proof. We modify the proof in [1]. Take a minimal injective copresentation of S_{i}^{+}:

\xymatrix{0 \ar[r] & S_{i}^{+} \ar[r] & D(e_{i}Λ) \ar[r]^{f} & I.}

Since Ext_{Λ}^{1}(S_{i}, S_{i}^{+}) = Ext_{Λ/Λ(1−e_{i})Λ}^{1}(S_{i}, S_{i}^{+}) = 0, it follows that the injective module I belongs to add D((1 − e_{i})Λ). Applying the inverse Nakayama functor ν_{mod Λ}^{−1} yields an exact sequence

\xymatrix{P_{i} \ar[r]^{ν_{mod Λ}^{−1}f} & ν_{mod Λ}^{−1}I \ar[r] & τ_{mod Λ}^{−1}S_{i}^{+} \ar[r] & 0.}

Moreover, ν_{mod Λ}^{−1}f is a minimal left approximation of P_{i} in add(P_{j}, j ≠ i). Since the projective dimension of τ_{mod Λ}^{−1}S_{i}^{+} is at most 1, it follows that ν_{mod Λ}^{−1}f is injective. This completes the proof for (a).

(b) follows from [1, Theorem 2.32]. √

7.2. Simple-minded collections. Let X_{1}, …, X_{r} be a simple-minded collection in C and fix i = 1, …, r. Let X_{i} denote the extension closure of X_{i} in C. Assume that for any j the object Σ^{−1}X_{j} admits a minimal left approximation g_{j} : Σ^{−1}X_{j} → X_{ij} in X_{i}.

Definition 7.5. The left mutation μ_{i}^{+}(X_{1}, …, X_{r}) of X_{1}, …, X_{r} at X_{i} is a new collection X′_{1}, …, X′_{r} such that X′_{i} = ΣX_{i} and X′_{j} (j ≠ i) is the cone of the above left approximation

\xymatrix{Σ^{−1}X_{j} \ar[r]^{g_{j}} & X_{ij}.}

Similarly one defines the right mutation μ_{i}^{−}(X_{1}, …, X_{r}).

This generalises Kontsevich–Soibelman’s mutation of spherical collections [38, Section 8.1] and appeared in [35] in the case of derived categories of acyclic quivers.

Proposition 7.6. (a) μ_{i}^{+} ∘ μ_{i}^{−}(X_{1}, …, X_{r}) ≅ (X_{1}, …, X_{r}) ≅ μ_{i}^{−} ∘

μ_{i}^{+}(X_{1}, …, X_{r}).

(b) Assume that · for any j ≠ i the object Σ^{−1}X_{j} admits a minimal left approximation g_{j} : Σ^{−1}X_{j} → X_{ij} in X_{i}, · the induced map Hom(g_{j}, X_{i}) : Hom(X_{ij}, X_{i}) → Hom(Σ^{−1}X_{j}, X_{i}) is injective, · the induced map Hom(g_{j}, ΣX_{i}) : Hom(X_{ij}, ΣX_{i}) → Hom(Σ^{−1}X_{j}, ΣX_{i}) is injective.

Then the collection μ_{i}^{+}(X_{1}, …, X_{r}) is simple-minded.

(c) Assume that · for any j ≠ i the object X_{j} admits a minimal right approximation g_{j}^{−} : Σ^{−1}X_{ij}^{−} → X_{j} in Σ^{−1}X_{i}, · the induced map Hom(X_{i}, Σg_{j}^{−}) : Hom(X_{i}, X_{ij}^{−}) → Hom(X_{i}, ΣX_{j}) is injective, · the induced map Hom(X_{i}, Σ^{2}g_{j}^{−},) : Hom(X_{i}, ΣX_{ij}^{−}) → Hom(X_{i}, Σ^{2}X_{j}) is injective.

Then the collection μ_{i}^{−}(X_{1}, …, X_{r}) is simple-minded.

Proof. (a) Because in the triangle

\xymatrix{Σ^{−1}X_{j} \ar[r]^{g_{j}} & X_{ij} \ar[r]^{g_{j}^{−}} & X′_{j} \ar[r] & X_{j}}

g_{j} is a minimal left approximation of Σ^{−1}X_{j} in X_{i} if and only if g_{j}^{−} is a minimal right approximation of X_{j} in X_{i} = Σ^{−1}(ΣX_{i}).

(b) and (c) The proof uses long exact Hom sequences induced from the defining triangles of the X′_{j}. We leave it to the reader. √

Remark 7.7. In the course of the proof of Proposition 7.6 (b) and (c), one notices that the collection of endomorphism algebras of the mutated simpleminded collection is the same as that of the given simple-minded collection.

If Hom(X_{i}, ΣX_{i}) = 0, then X_{i} = add(X_{i}). In this case, all six assumptions in Proposition 7.6 (b) and (c) are satisfied.

Lemma 7.8. Let Λ be a finite-dimensional algebra and let X_{1}, …, X_{r} be a simple-minded collection in D^{b}(mod Λ). Let i = 1, …, r. Then the left mutation μ_{i}^{+}(X_{1}, …, X_{r}) and the right mutation μ_{i}^{−}(X_{1}, …, X_{r}) are again simpleminded collections.

Proof. We will show that the three assumptions in Proposition 7.6 (b) are satisfied, so the left-mutated collection μ_{i}^{+}(X_{1}, …, X_{r}) is a simple-minded collection. The case for μ_{i}^{−}(X_{1}, …, X_{r}) is similar. By Proposition 5.4, X_{1}, …, X_{r} are the simple objects in the heart of a bounded t-structure on D^{b}(mod Λ). Moreover, by Corollary 6.2, the heart is equivalent to mod Γ for some finite-dimensional algebra Γ. We identify mod Γ with the heart via this equivalence. In this way we consider X_{1}, …, X_{r} as simple Γ-modules. By [8, Section 3.1], there is a triangle functor

real : D^{b}(mod Γ) → D^{b}(mod Λ)

such that – restricted to mod Γ, real is the identity;

– for M, N ∈ mod Γ, the induced map

Ext_{Γ}^{1}(M, N) = Hom_{D^{b}(mod Γ)}(M, ΣN) → Hom_{D^{b}(mod Λ)}(M, ΣN)

is bijective;

– for M, N ∈ mod Γ, the induced map

Ext_{Γ}^{2}(M, N) = Hom_{D^{b}(mod Γ)}(M, Σ^{2}N) → Hom_{D^{b}(mod Λ)}(M, Σ^{2}N)

is injective. For j = 1, …, r, there is a short exact sequence

\xymatrix{0 \ar[r] & ΩX_{j} \ar[r] & P_{j} \ar[r] & X_{j} \ar[r] & 0,}

where P_{j} is the projective cover of X_{j} and ΩX_{j} is the first syzygy of X_{j}. Let X_{i} be the extension closure of X_{i} in mod Γ (by the second property of real listed in the preceding paragraph, this is the same as the extension closure of X_{i} in D^{b}(mod Λ)) and let X_{ij} denote the maximal quotient of ΩX_{j} belonging to X_{i}. There is the following push-out diagram

[figure: 0; ΩX_{j}; P_{j}; X_{j}; 0; ξ :; 0; X_{ij}; X′_{j}; X_{j}; 0]

(a) Suppose we are given an object Y of X_{i} and a short exact sequence

\xymatrix{η : & 0 \ar[r] & Y \ar[r] & Z \ar[r] & X_{j} \ar[r] & 0.}

Then there is a commutative diagram

[figure: 0; ΩX_{j}; P_{j}; X_{j}; 0; η :; 0; Y; Z; X_{j}; 0.]

Because X_{ij} is the maximal quotient of ΩX_{j} belonging to X_{i}, this morphism of short exact sequences factors through ξ. In other words, the morphism g_{j} : X_{j} → ΣX_{ij} corresponding to ξ is a minimal left ΣX_{i}-approximation.

(b) The dimension of the space Hom_{D^{b}(mod Λ)}(ΣX_{ij}, ΣX_{i}) ≅ Hom_{Γ}(X_{ij}, X_{i}) over End(X_{i}) equals the number of indecomposable direct summands of top(X_{ij}), which clearly equals the dimension of Ext_{Γ}^{1}(X_{j}, X_{i}) ≅ Hom_{D^{b}(mod Λ)}(X_{j}, ΣX_{i}) over End(X_{i}). Therefore the induced map

Hom(g_{j}, ΣX_{i}) : Hom_{D^{b}(mod Λ)}(ΣX_{ij}, ΣX_{i}) ⟶ Hom_{D^{b}(mod Λ)}(X_{j}, ΣX_{i})

is injective since by (a) it is surjective.

(c) First observe that the following diagram is commutative

\xymatrix{Hom_{D^{b}(mod Λ)}(ΣX_{ij}, Σ^{2}X_{i}) \ar[r]^{Hom_{Λ}(g_{j},Σ^{2}X_{i})} & Hom_{D^{b}(mod Λ)}(X_{j}, Σ^{2}X_{i}) \\ Hom_{D^{b}(mod Γ)}(ΣX_{ij}, Σ^{2}X_{i}) \ar[u]^{real} \ar[r]^{Hom_{Γ}(g_{j},Σ^{2}X_{i})} & Hom_{D^{b}(mod Γ)}(X_{j}, Σ^{2}X_{i}) \ar[u]^{real}}

The left vertical map is a bijection and the right vertical map is injective, so to prove the injectivity of Hom_{Λ}(g_{j}, Σ^{2}X_{i}) it suffices to prove the injectivity of Hom_{Γ}(g_{j}, Σ^{2}X_{i}). Writing

Hom_{D^{b}(mod Γ)}(ΣX_{ij}, Σ^{2}X_{i}) = Ext_{Γ}^{1}(X_{ij}, X_{i}) = Hom_{Γ}(ΩX_{ij}, X_{i})

and

Hom_{D^{b}(mod Γ)}(X_{j}, Σ^{2}X_{i}) = Ext_{Γ}^{2}(X_{j}, X_{i}) = Ext_{Γ}^{1}(ΩX_{j}, X_{i}) = Hom_{Γ}(Ω^{2}X_{j}, X_{i}),

we see that Hom_{Γ}(g_{j}, Σ^{2}X_{i}) is Hom_{Γ}(α, X_{i}), where α is defined by the following commutative diagram

\xymatrix{0 \ar[r] & Ω^{2}X_{j} \ar[r] \ar[d]^{α} & P^{0} \ar[r] \ar[d]^{β} & ΩX_{j} \ar[r] \ar[d]^{γ} & 0 \\ 0 \ar[r] & ΩX_{ij} \ar[r] & Q^{0} \ar[r] & X_{ij} \ar[r] & 0,}

Here, P^{0} and Q^{0} are projective covers of ΩX_{j} and X_{ij}, respectively, and γ is the canonical quotient map. As the map γ is surjective, the map β is a split epimorphism. By the snake lemma, there is an exact sequence

\xymatrix{ker(γ) \ar[r] & cok(α) \ar[r] & 0.}

Since X_{ij} is the maximal quotient of ΩX_{j} in X_{i}, it follows that Hom_{Γ}(ker(γ), X_{i}) = 0, and hence Hom_{Γ}(cok(α), X_{i}) = 0. Therefore Hom_{Γ}(α, X_{i}) is injective. √

For two simple-minded collections {X_{1}, …, X_{r}} and {X′_{1}, …, X′_{r}} of C, define

{X_{1}, …, X_{r}} ≥ {X′_{1}, …, X′_{r}}

if Hom(X′_{i}, Σ^{m}X_{j}) = 0 for any m < 0 and any i, j = 1, …, r.

Proposition 7.9. The relation ≥ defined above is a partial order on the set of equivalence classes of simple-minded collections of C.

Proof. The reflexivity is clear by the definition of a simple-minded collection. Next we show the antisymmetry and transitivity. Let {X_{1}, …, X_{r}} and {X′_{1}, …, X′_{r}} be two simple-minded collections of C and let (C^{≤0}, C^{≥0}) and (C^{′≤0}, C^{′≥0}) be the corresponding t-structures given in Proposition 5.4 (the general case). Then

{X_{1}, …, X_{r}} ≥ {X′_{1}, …, X′_{r}}
⇔ Hom(X′_{i}, Σ^{m}X_{j}) = 0 for any m < 0 and i, j = 1, …, r
⇔ Hom(Σ^{m′}X′_{i}, Σ^{m}X_{j}) = 0 for any m < 0, m′ ≥ 0
and i, j = 1, …, r
⇔ C^{′≤0} ⊥ Σ^{−1}C^{≤0}
⇔ C^{′≤0} ⊆ C^{≤0}.

(a) If {X_{1}, …, X_{r}} ≥ {X′_{1}, …, X′_{r}} and {X′_{1}, …, X′_{r}} ≥ {X_{1}, …, X_{r}}, then (C^{≤0}, C^{≥0}) = (C^{′≤0}, C^{′≥0}). In particular, the two t-structures have the same heart. Therefore, both {X_{1}, …, X_{r}} and {X′_{1}, …, X′_{r}} are complete sets of pairwise non-isomorphic simple objects of the same abelian category, and hence they are equivalent.

(b) Let {X′′_{1}, …, X′′_{r}} be a third simple-minded collection of C, with corresponding t-structure (C^{′′≤0}, C^{′′≥0}). Suppose {X_{1}, …, X_{r}} ≥ {X′_{1}, …, X′_{r}} and {X′_{1}, …, X′_{r}} ≥ {X′′_{1}, …, X′′_{r}}. Then C^{′′≤0} ⊆ C^{′≤0} ⊆ C^{≤0}. Consequently, {X_{1}, …, X_{r}} ≥ {X′′_{1}, …, X′′_{r}}. √

7.3. t-structures. Let (C^{≤0}, C^{≥0}) be a bounded t-structure of C such that the heart A is a length category which has only finitely many simple objects S_{1}, …, S_{r} up to isomorphism. Then {S_{1}, …, S_{r}} is a simple-minded collection. Let F = S_{i} be the extension closure of S_{i} in A and let T = ^{⊥}S_{i} be the left perpendicular category of S_{i} in A. It is easy to show that (T , F) is a torsion pair of A. Define the left mutation μ_{i}^{+}(C^{≤0}, C^{≥0}) = (C^{′≤0}, C^{′≥0}) by

C^{′≤0} = {M ∈ C | H^{m}(M) = 0 for m > 0 and H^{0}(M) ∈ T },
C^{′≥0} = {M ∈ C | H^{m}(M) = 0 for m < −1 and H^{−1}(M) ∈ F}.

Similarly one defines the right mutation μ_{i}^{−}(C^{≤0}, C^{≥0}). These mutations provide an effective method to compute the space of Bridgeland’s stability conditions on C by gluing different charts, see [14, 48].

Proposition 7.10. The pairs μ_{i}^{+}(C^{≤0}, C^{≥0}) and μ_{i}^{−}(C^{≤0}, C^{≥0}) are bounded tstructures of C. The heart of μ_{i}^{+}(C^{≤0}, C^{≥0}) has a torsion pair (ΣF, T ) and the heart of μ_{i}^{−}(C^{≤0}, C^{≥0}) has a torsion pair (S_{i}^{⊥}, Σ^{−1}S_{i}). Moreover, μ_{i}^{+} ∘ μ_{i}^{−}(C^{≤0}, C^{≥0}) = (C^{≤0}, C^{≥0}) = μ_{i}^{−} ∘ μ_{i}^{+}(C^{≤0}, C^{≥0}).

Proof. This follows from [24, Proposition 2.1, Corollary 2.2] and [14, Proposition 2.5]. √ In general the heart of the mutation of a bounded t-structure with length heart is not necessarily a length category. For an example, let Q be the quiver

[figure: 1; 2]

and consider the bounded derived category C = D^{b}(nil. rep Q) of finitedimensional nilpotent representations of Q. Let S_{1} and S_{2} be the onedimensional nilpotent representations associated to the two vertices. Let F = S_{1} be the extension closure of S_{1} and T = ^{⊥}F = {M ∈ nil. rep Q | top(M) ∈ add(S_{2})}. Then the heart A′ of the left mutation at 1 of the standard t-structure has a torsion pair (ΣF, T ). Due to nil. rep Q being hereditary, there are no extensions of ΣF by T , and hence any indecomposable object of A′ belongs to either T or ΣF. Suppose that A′ is a length category. Then A′ has two isomorphism classes of simple modules, which respectively belong to T and ΣF, say S′_{2} ∈ T and S′_{1} ∈ ΣF. For n ∈ ℕ define an indecomposable object M_{n} in T as

[figure: k; J_{n}(0); k,; (0,...,0,1)^{tr}]

where J_{n}(0) is the (upper triangular) Jordan block of size n and with eigenvalue 0. There are no morphisms from S′_{1} to M_{n} for any n. Suppose that the Loewy length of S′_{2} in A is l. Then for n > l, any morphism from S′_{2} to M_{n} factors through rad^{n−l}M_{n} which lies in F, and hence the morphism has to be zero. Therefore M_{n} (n > l), considered as an object in A′, does not have finite length, a contradiction.

For two bounded t-structures (C^{≤0}, C^{≥0}) and (C^{′≤0}, C^{′≥0}) on C, define

(C^{≤0}, C^{≥0}) ≥ (C^{′≤0}, C^{′≥0})

if C^{≤0} ⊇ C^{′≤0}. This defines a partial order on the set of bounded t-structures on C.

7.4. Co-t-structures. Let (C_{≥0}, C_{≤0}) be a bounded co-t-structure of C. Assume that the co-heart admits a basic additive generator M = M_{1} ⊕ … ⊕ M_{r} with M_{i} indecomposable. Then M is a silting object of C. Let i = 1, …, r. Define C′_{≤0} as the additive closure of the extension closure of Σ^{m}M_{j}, j ≠ i, and Σ^{m+1}M_{i} for m ≥ 0 and define C′_{≥0} as the left perpendicular category of ΣC′_{≤0}. The left mutation μ_{i}^{+}(C_{≥0}, C_{≤0}) is defined as the pair (C′_{≥0}, C′_{≤0}). Similarly one defines the right mutation μ_{i}^{−}(C_{≥0}, C_{≤0}).

Proposition 7.11. The pairs μ_{i}^{+}(C_{≥0}, C_{≤0}) and μ_{i}^{−}(C_{≥0}, C_{≤0}) are bounded co-t-structures on C. Moreover, μ_{i}^{+} ∘ μ_{i}^{−}(C_{≥0}, C_{≤0}) = (C_{≥0}, C_{≤0}) = μ_{i}^{−} ∘ μ_{i}^{+}(C_{≥0}, C_{≤0}).

Proof. This can be proved directly. Here we alternatively make use of the results in Sections 3.1 and 7.1. Recall from Theorem 7.1 that there is a mutated silting object μ_{i}^{+}(M). It is straightforward to check, using the defining triangle for μ_{i}^{+}(M), that μ_{i}^{+}(C_{≥0}, C_{≤0}) is the bounded co-t-structure associated to μ_{i}^{+}(M) as defined in Proposition 3.5, and similarly for μ_{i}^{−}. The second statement follows from Theorem 7.1. √

For two bounded co-t-structures (C_{≥0}, C_{≤0}) and (C′_{≥0}, C′_{≤0}) on C, define

(C_{≥0}, C_{≤0}) ≥ (C′_{≥0}, C′_{≤0})

if C_{≤0} ⊇ C′_{≤0}. This defines a partial order on the set of bounded co-t-structures on C.

7.5. The bijections commute with mutations. Let Λ a finite-dimensional algebra over K.

Theorem 7.12. The ϕ_{ij}’s defined in Section 5 commute with the left and right mutations defined in previous subsections.

A priori it it not known that the heart of the mutation of a bounded t-structure with length heart is again a length category. So the theorem becomes wellstated only when the proof has been finished.

Proof. In view of Lemma 6.6, Theorem 7.1, and Propositions 7.6, 7.10 and 7.11, it suffices to prove that ϕ_{41}, ϕ_{31} and ϕ_{23} commute with the corresponding left mutations.

(a) ϕ_{41} commutes with μ_{i}^{+}: this was already shown in the proof of Proposition 7.11.

(b) ϕ_{31} commutes with μ_{i}^{+}: Let M = M_{1} ⊕…⊕M_{r} be a silting object with M_{i} indecomposable and (C^{≤0}, C^{≥0}) = ϕ_{31}(M). We want to show μ_{i}^{+}(C^{≤0}, C^{≥0}) = ϕ_{31}(μ_{i}^{+}(M)). Let Γ̃ be the truncated dg endomorphism algebra of M as in Section 5.1. Then

there is a triangle equivalence F =? ⊗^{L}Γ̃ M : D_{fd}(Γ̃) → D^{b}(mod Λ), which takes Γ̃ to M and takes the standard t-structure (D^{≤0}, D^{≥0}) on D_{fd}(Γ̃) to (C^{≤0}, C^{≥0}). There is a decomposition 1 = e_{1} +…+e_{r}, where e_{1}, …, e_{r} are (not necessarily primitive) idempotents of Γ̃ such that F takes e_{j}Γ̃ to M_{j} for 1 ≤ j ≤ r. Let Γ = H^{0}(Γ̃) and π : Γ̃ → Γ be the canonical projection. By abuse of notation, write e_{1} = π(e_{1}), …, e_{r} = π(e_{r}). Then e_{1}Γ, …, e_{r}Γ are indecomposable projective Γ-modules. Let S_{1}, …, S_{r} be the corresponding simple modules. Recall that the heart of the t-structure (D^{≤0}, D^{≥0}) is mod Γ. Let F = add(S_{i}) ⊆ mod Γ and T = ^{⊥}S_{i}. Define D^{′≤0} (respectively, D^{′≥0}) to be the extension closure of ΣD^{≤0} and T (respectively, of ΣF and D^{≥0}). Then F (D^{′≤0}, D^{′≥0}) = μ_{i}^{+}(C^{≤0}, C^{≥0}). The left mutation of Γ̃ at e_{i}Γ̃ is μ_{i}^{+}(Γ̃) = Q_{i} ⊕ ⨁_{j≠i} e_{j}Γ̃, where Q_{i} is defined by the triangle

\xymatrix{e_{i}Γ̃ \ar[r]^{f} & E \ar[r] & P′_{i} \ar[r] & Σe_{i}Γ̃} , (7.1)

where f is a minimal left add(⨁_{j≠i} e_{j}Γ̃)-approximation. Then F (μ_{i}^{+}(Γ̃)) = μ_{i}^{+}(M). Define

D^{′′≤0} = {N ∈ D_{fd}(Γ̃) | Hom(μ_{i}^{+}(Γ̃), Σ^{m}N) = 0, ∀m > 0},
D^{′′≥0} = {N ∈ D_{fd}(Γ̃) | Hom(μ_{i}^{+}(Γ̃), Σ^{m}N) = 0, ∀m < 0}.

Thus showing μ_{i}^{+}(C^{≤0}, C^{≥0}) = ϕ_{31}(μ_{i}^{+}(M)) is equivalent to showing the equality (D^{′≤0}, D^{′≥0}) = (D^{′′≤0}, D^{′′≥0}), equivalently, the inclusions D^{′≤0} ⊆ D^{′′≤0} and D^{′≥0} ⊆ D^{′′≥0}. It suffices to prove T ⊆ D^{′′≤0}, ΣD^{≤0} ⊆ D^{′′≤0}, ΣF ⊆ D^{′′≥0} and D^{≥0} ⊆ D^{′′≥0}. We only show the first inclusion, the other three are easy. Let T ∈ T . To show T ∈ D^{′′≤0}, it suffices to show Hom(Q_{i}, ΣT) = 0. Applying Hom(?, T) to the triangle (7.1), we obtain a long exact sequence

\xymatrix{Hom(E, T) \ar[r]^{f^{∗}} & Hom(e_{i}Γ̃, ΣT) \ar[r] & Hom(Q_{i}, ΣT) \ar[r] & Hom(E, ΣT) = 0} .

We claim that f^{∗} is surjective. Then the desired result follows. Consider the commutative diagram

\xymatrix{Hom(e_{i}Γ, T) \ar[r]^{π_{i}^{∗}} & Hom(e_{i}Γ̃, T) \\ Hom(H^{0}(E), T) \ar[r]^{π_{E}^{∗}} \ar[u]_{H^{0}(f)^{∗}} & Hom(E, T), \ar[u]_{f^{∗}}}

(7.2)

where π_{i} : e_{i}Γ̃ → e_{i}Γ and π_{E} : E → H^{0}(E) are the canonical projections. Let C = ker(π_{i}). Then there is a triangle

\xymatrix{C \ar[r] & e_{i}Γ̃ \ar[r]^{π_{i}} & e_{i}Γ \ar[r] & ΣC} .

Note that C belongs to ΣD^{≤0}, which implies that Hom(C, T) = 0 = Hom(ΣC, T). It follows that the map π_{i}^{∗} is bijective. Similarly, the map π_{E}^{∗} is also bijective. Thus it suffices to show the surjectivity of H^{0}(f)^{∗}. Now let P_{T} be a projective cover of T in mod Γ. Then P_{T} belongs to add(⨁_{j≠i} e_{j}Γ) because T ∈ T = ^{⊥}S_{i}. It follows that any morphism e_{i}Γ → T factors through P_{T}, and hence factors through H^{0}(f) : e_{i}Γ → H^{0}(E), since H^{0}(f) is a minimal left add(⨁_{j≠i} e_{j}Γ)-approximation (for H^{0}|_{add(Γ̃)} : add(Γ̃) → add(Γ) is an equivalence). This shows that H^{0}(f)^{∗} is surjective, completing the proof of the claim.

(c) ϕ_{23} commutes with μ_{i}^{+}: Let (C^{≤0}, C^{≥0}) be a bounded t-structure on D^{b}(mod Λ) with length heart. Let {X_{1}, …, X_{r}} = ϕ_{23}(C^{≤0}, C^{≥0}). The mutated simple-minded collection μ_{i}^{+}(X_{1}, …, X_{r}) is contained in the heart of the mutated t-structure μ_{i}^{+}(C^{≤0}, C^{≥0}). Consequently, the aisle and coaisle of ϕ_{32} ∘ μ_{i}^{+}(X_{1}, …, X_{r}) are respectively contained in the aisle and coaisle of μ_{i}^{+}(C^{≤0}, C^{≥0}), and hence ϕ_{32} ∘ μ_{i}^{+}(X_{1}, …, X_{r}) = μ_{i}^{+}(C^{≤0}, C^{≥0}), i.e. ϕ_{23} ∘ μ_{i}^{+}(C^{≤0}, C^{≥0}) = μ_{i}^{+} ∘ ϕ_{23}(C^{≤0}, C^{≥0}). √ 7.6. The bijections are isomorphisms of partially ordered sets. Let Λ be a finite-dimensional algebra over K.

Theorem 7.13. The ϕ_{ij}’s defined in Section 5 are isomorphisms of partially ordered sets with respect to the partial orders defined in previous subsections.

Proof. In view of Theorem 6.1 and Lemma 6.6, it suffices to show that f(x) ≥ f(y) if and only if x ≥ y for f = ϕ_{41}, ϕ_{32} and ϕ_{34}.

(a) For ϕ_{41} the desired result follows from [1, Proposition 2.14].

(b) For ϕ_{32} the desired result is included in the proof of Proposition 7.9.

(c) Let (C_{≥0}, C_{≤0}) and (C′_{≥0}, C′_{≤0}) be two bounded co-t-structures on C and let (C^{≤0}, C^{≥0}) and (C^{′≤0}, C^{′≥0}) be their respective images under ϕ_{34}. Then by definition

C^{≤0} = {M ∈ D^{b}(mod Λ) | Hom(N, M) = 0 ∀N ∈ Σ^{−1}C_{≥0}},
C^{′≤0} = {M ∈ D^{b}(mod Λ) | Hom(N, M) = 0 ∀N ∈ Σ^{−1}C′_{≥0}}.

Here, C^{≤0} ⊇ C^{′≤0} if and only if C_{≥0} ⊇ C′_{≥0}, and hence by definition (C^{≤0}, C^{≥0}) ≥ (C^{′≤0}, C^{′≥0}) if and only if (C_{≥0}, C_{≤0}) ≥ (C′_{≥0}, C′_{≤0}). √

8. A concrete example

Let Λ be the finite-dimensional K-algebra given by the quiver

\xymatrix{1 \ar[r]^{α} & 2 \ar[l]^{β}}

with relation αβ = 0. This algebra has many manifestations: It is, possibly up to Morita equivalence, the Auslander algebra of k[x]/x^{2}, the Schur algebra S(2, 2) (charK = 2) and the principal block of the category O for 𝔰𝔩_{2}(ℂ) (K = ℂ). In this section we will compute the derived Picard group for Λ and classify all silting objects/simple-minded collections in D^{b}(mod Λ). As a consequence of this classification and a result of Woolf [48, Theorem 3.1], the space of stability conditions on D^{b}(mod Λ) is exactly ℂ^{2}.

8.1. Indecomposable objects. Let P_{1} and P_{2} be the indecomposable projective Λ-modules corresponding to the vertices 1 and 2. Then up to isomorphism and up to shift an indecomposable object in D^{b}(mod Λ) belongs to one of the following four families (see for example [19, 9])

· P_{1}(n) = P_{1} → P_{1} → … → P_{1} → P_{1}, n ≥ 1,
· R(n) = P_{1} → P_{1} → … → P_{1} → P_{1} → P_{2}, n ≥ 0,
· L(n) = P_{2} → P_{1} → P_{1} → … → P_{1} → P_{1}, n ≥ 0,
· B(n) = P_{2} → P_{1} → P_{1} → … → P_{1} → P_{1} → P_{2}, n ≥ 1,

where the homomorphisms are the unique non-isomorphisms, n is the number of occurrences of P_{1} and the rightmost components have been put in degree 0.

8.2. The Auslander–Reiten quiver. The Auslander–Reiten quiver of D^{b}(mod Λ) consists of three components: two ℤA_{∞} components and one ℤA_{∞}^{∞} component (see [10, 28])

[figure: ∘; ∘; Σ^{−1}P_{1}; ∘; ∘; ∘; P_{1}; ∘; ∘; ΣP_{1}; ∘; ∘; I_{2}; ∘; S_{1}; ΣS_{1}; ∘; P_{2}; ΣP_{2}; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ΣS_{2}; ∘; ∘; ∘; S_{2}; ∘; ∘; Σ^{−1}S_{2}]

The abelian category mod Λ has five indecomposable objects up to isomorphism: the two simple modules S_{1} and S_{2}, their projective covers P_{1} and P_{2} and their injective envelopes I_{1} = P_{1} and I_{2}. They are marked on the above Auslander–Reiten quiver. The left ℤA_{∞} component consists of shifts of P_{1}(n), n ≥ 1. The Auslander– Reiten translation τ takes P_{1}(n) to Σ^{−1}P_{1}(n). It is straightforward to check that P_{1} is a 0-spherical object of D^{b}(mod Λ) in the sense of Seidel and Thomas [45]. The additive closure of this component is the triangulated subcategory generated by P_{1}. This component will be referred to as the 0-spherical component. The ℤA_{∞}^{∞} component consists of shifts of R(n) and L(n), n ≥ 0. Note that S_{1} = L(1), P_{2} = R(0) = L(0) and I_{2} = L(2). The Auslander–Reiten translation τ takes R(n) (n ≥ 2) to ΣR(n − 2), takes R(1) to L(1) and takes L(n) to Σ^{−1}L(n + 2). The right ℤA_{∞} component consists of shifts of B(n), n ≥ 1. The Auslander– Reiten translation takes B(n) to ΣB(n). The simple module S_{2} = B(1) is a 2-spherical object of D^{b}(mod Λ) and the additive closure of this component is the triangulated subcategory generated by S_{2}. This component will be referred to as the 2-spherical component.

8.3. The derived Picard group. Let E be a spherical object of a triangulated category C in the sense of Seidel and Thomas [45]. Then the twist functor Φ_{E} defined by

Φ_{E}(M) = Cone(⨁_{m∈ℤ} Hom(Σ^{m}E, M) ⊗ Σ^{m}E ⟶^{ev} M),

where ev is the evaluation map, is an auto-equivalence of C by [45, Proposition 2.10].

Recall from the preceding subsection that P_{1} is a 0-spherical object and S_{2} is a 2-spherical object of D^{b}(mod Λ). Thus the associated twist functors Φ_{P_{1}} and Φ_{S_{2}} are two auto-equivalences of D^{b}(mod Λ).

Lemma 8.1. For M in D^{b}(mod Λ) there are isomorphisms Φ_{S_{2}}(M) ≅ Φ_{P_{1}} ∘ Σ^{−1}(M) and Φ_{P_{1}}^{2}(M) ≅ ν^{−1} ∘ Σ^{2}(M). Moreover, if M is indecomposable and belongs to the ℤA_{∞}^{∞} component, there exists a unique pair of integers (n, n′) such that M ≅ Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}(P_{2}).

Proof. Observe that Φ_{P_{1}}(S_{1}) ≅ ΣP_{2}, Φ_{P_{1}}(P_{1}) ≅ ΣP_{1}, Φ_{P_{1}}(S_{2}) ≅ S_{2} and Φ_{S_{2}}(S_{1}) ≅ P_{2}, Φ_{S_{2}}(P_{1}) ≅ P_{1}, Φ_{S_{2}}(S_{2}) ≅ Σ^{−1}S_{2}. Since auto-equivalences preserve the shape of the Auslander–Reiten quiver, the statements follow. √

Remark 8.2. Inspecting the action of Φ_{P_{1}} and Φ_{S_{2}} on maps shows that the isomorphism Φ_{P_{1}}^{2}(M) ≅ ν^{−1} ∘ Σ^{2}(M) is functorial, while Φ_{S_{2}}(M) ≅ Φ_{P_{1}} ∘ Σ^{−1}(M) is not.

Let Aut D^{b}(mod Λ) denote the group of algebraic auto-equivalences of D^{b}(mod Λ), i.e. those which admits a dg lift. By [29, Lemma 6.4], such an auto-equivalence is naturally isomorphic to the derived tensor functor of a complex of bimodules.

Lemma 8.3. Aut D^{b}(mod Λ) is isomorphic to ℤ^{2} × K^{×}.

Proof. Let F ∈ Aut D^{b}(mod Λ). Since F preserves the Auslander–Reiten quiver, the object F (P_{2}) is in the ℤA_{∞}^{∞} component. Thus there is a pair of integers (n_{F} , n′_{F}) such that F (P_{2}) ≅ Φ_{P_{2}}^{n_{F}} ∘Φ_{S_{2}}^{n′_{F}} (P_{2}). This allows us to define a map

\xymatrix{f : & Aut D^{b}(mod Λ) \ar[r] & ℤ^{2} \\  & F \ar@{|->}[r] & (n_{F} , n′_{F}).}

This map is clearly a surjective group homomorphism. Moreover, the group homomorphism

\xymatrix{ℤ^{2} \ar[r] & Aut D^{b}(mod Λ) \\ (n, n′) \ar@{|->}[r] & Φ_{P_{2}}^{n} ∘ Φ_{S_{2}}^{n′}}

is a retraction of f. Therefore Aut D^{b}(mod Λ) ≅ ℤ^{2} × ker(f). Let F ∈ ker(f). Then F (P_{2}) ≅ P_{2}. This forces F (P_{1}) ≅ P_{1}, and hence F is induced from an outer automorphism of Λ which fixes the two primitive idempotents e_{1} and e_{2}. Thus ker(f) ≅ K^{×}, finishing the proof. √

8.4. Morphism spaces. We first compute the morphism spaces between the two ℤA_{∞} components.

Lemma 8.4. (a) For n ≥ 2, Hom(P_{1}(n), Σ^{m}P_{1}(n)) does not vanish for some m > 0 and for some m < 0. For n = 1, Hom(P_{1}, Σ^{m}P_{1}) is isomorphic to K[x]/x^{2} for m = 0 and vanishes for m ≠ 0.

(b) For n ≥ 2, Hom(B(n), Σ^{m}B(n)) does not vanish for some m > 0 and for some m < 0. For n = 1, Hom(S_{2}, Σ^{m}S_{2}) is K for m = 0, 2 and vanishes for m ≠ 0, 2.

Proof. Direct computation, or apply some general result (e.g. [25, Section 2]) to the triangulated categories generated by P_{1} and S_{2}. √

Next we compute the morphism spaces between P_{2} and the objects on the ℤA_{∞}^{∞} component.

Lemma 8.5. Let n ≥ 0.

(a) Hom(P_{2}, Σ^{m}R(n)) is K if −n ≤ m ≤ 0 and is 0 otherwise.

(a’) Hom(R(n), Σ^{m}P_{2}) is K if 2 ≤ m ≤ n or if n = 0, m = 0 and is 0

otherwise.

(b) Hom(P_{2}, Σ^{m}L(n)) is K if 2 − n ≤ m ≤ 0 or if n = 0, m = 0 and is 0 otherwise.

(b’) Hom(L(n), Σ^{m}P_{2}) is K if 0 ≤ m ≤ n and is 0 otherwise.

Proof. (a) and (b) Because Hom(P_{2}, M) = H^{0}(M)e_{2}. (a’) and (b’) are obtained from (a) and (b) by applying the Auslander–Reiten formula D Hom(M, N) ≅ Hom(N, τΣM). √

8.5. Silting objects and simple-minded collections. Now we are ready to classify the silting objects and simple-minded collections in D^{b}(mod Λ).

Proposition 8.6. Up to isomorphism, any basic silting object of D^{b}(mod Λ) belongs to one of the following two families

· Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}(P_{1} ⊕ P_{2}), n, n′ ∈ ℤ, the corresponding simple-minded collec-

tion is Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}{S_{1}, S_{2}},

· Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}(Σ^{m}S_{1} ⊕ P_{2}), n, n′ ∈ ℤ and m ≤ −1, the corresponding

simple-minded collection is Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}{Σ^{m}S_{1}, I_{2}}.

Proof. Let N be an indecomposable direct summand of a silting object. By Lemma 8.4, N does not belong to the 2-spherical component, and N belongs to the 0-spherical component if and only if N is a shift of P_{1}. Moreover, a basic silting object can have at most one shift of P_{1} as a direct summand. It follows that a silting object has at least one indecomposable direct summand from the ℤA_{∞}^{∞} component. Let M = M_{1}⊕M_{2} be a silting object with M_{1} and M_{2} indecomposable. Assume that M_{1} belongs to the ℤA_{∞}^{∞} component. Up to an auto-equivalence of the form Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}, we may assume that M_{1} = P_{2}. Then, if M_{2} belongs to the 0-spherical component it has to be P_{1}. Thus we assume that M_{2} also belongs to the ℤA_{∞}^{∞} component. Then it follows from Lemma 8.5 that M_{2} is isomorphic to Σ^{m}S_{1} for some m ≤ −1 or to Σ^{m}R(1) for some m ≥ 0. Observing P_{2} ⊕ Σ^{m}R(1) = Φ_{P_{1}}^{−m−1} ∘ Φ_{S_{2}}^{m}(P_{2} ⊕ Σ^{−m−1}S_{1}) for m ≥ 0 finishes the proof for the silting-object part. That the simple-minded collection corresponding to a silting object is the desired one follows from the Hom-duality they satisfy. √ 8.6. The silting quiver. Recall from [1] that the silting quiver has as vertices the isomorphism classes of basic silting objects and there is an arrow from M to M′ if M′ can be obtained from M by a left mutation. The vertex set of the silting quiver of D^{b}(mod Λ) is {(n, n′, m) | n, n′ ∈ ℤ, m ∈ ℤ_{≤0}}, where (n, n′, 0) represents the silting object Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}(P_{1} ⊕ P_{2}) and (n, n′, m) (m ≤ −1) represents the silting object Φ_{P_{1}}^{n} ∘ Φ_{S_{2}}^{n′}(Σ^{m}S_{1} ⊕ P_{2}). It is straightforward to show that from each vertex (n, n′, m) there are precisely two outgoing arrows whose targets are respectively

· (n, n′ − 1, m) and (n + 1, n′, m − 1) if m = 0, · (n + 1, n′ − 1, m − 1) and (n, n′, m + 1) if m ≤ −1.

8.7. Hearts and the space of stability conditions.

Lemma 8.7. The heart of any t-structure on D^{b}(mod Λ) is a length category.

Proof. Let A be the heart of a t-structure on D^{b}(mod Λ). We will show that A has only finitely many isomorphism classes of indecomposable objects. Such an abelian category must be a length category. Due to vanishing of negative extensions, it follows from Lemma 8.4 that A contains at most one indecomposable object from the 0-spherical component respectively the 2-spherical component. Suppose that A contains an indecomposable object from the ℤA_{∞}^{∞} component. Without loss of generality we may assume that it is P_{2}. It follows from Lemma 8.5 that for n ≥ 3 and m ∈ ℤ either Hom(P_{2}, Σ^{m′}Σ^{m}R(n)) ≠ 0 for some m′ < 0 or Hom(Σ^{m}R(n), Σ^{m′}P_{2}) ≠ 0 for some m′ < 0. Similarly for L(n). Therefore an indecomposable object M belongs to the heart only if it is isomorphic to one of Σ^{m}P_{2}, Σ^{m}R(1), Σ^{m}R(2), Σ^{m}L(1) and Σ^{m}L(2), m ∈ ℤ. But at most one shift of a nonzero object can belong to a heart. So A contains at most 7 indecomposable objects up to isomorphism. √

In view of Lemma 8.7, the result in the preceding subsection shows that all bounded t-structures on D^{b}(mod Λ) are related to each other by a sequence of left or/and right mutations. In particular, this implies that the t-structures Woolf considered in [48, Section 3.1] are already all bounded t-structures on D^{b}(mod Λ). Therefore we have

Corollary 8.8. (a) The Bridgeland space of stability conditions on

D^{b}(mod Λ) is ℂ^{2}.

(b) An abelian category is the heart of some bounded t-structure on D^{b}(mod Λ) if and only if it is equivalent to mod Γ for Γ = Λ or

Γ = K( \xymatrix{· \ar[r] & ·}) or Γ = K ⊕ K.

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Steffen Koenig

Universität Stuttgart

Institut für Algebra

und Zahlentheorie Pfaffenwaldring 57 D-70569 Stuttgart Germany skoenig@mathematik.unistuttgart.de Dong Yang Department of Mathematics Nanjing University Nanjing 210093 P. R. China dongyang2002@gmail.com