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...ts of shifts of P1(n), n ≥ 1. The Auslander–Reiten translation τ takes P1(n) to Σ −1P1(n). It is straightforward to check that P1 is a 0-spherical object of Db(modΛ) in the sense of Seidel and Thomas =-=[45]-=-. The additive closure of this component is the triangulated subcategory generated by P1. This component will be referred to as the 0-spherical component. 28 STEFFEN KOENIG AND DONG YANG The ZA∞∞ comp...

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... a graded H∗(A)-module structure, and hence a graded Ā = H0(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 di−1→ M i di→ M i+1 → . . . be a dg A-module. Consider the standard truncation functors τ≤0 and τ>0: τ≤0M = τ>0M = ....

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... if Λ/Λ(1− ei)Λ is projective as a Λ-module. When Λ/Λ(1− ei)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 ...

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...) denote the category of (right) dg modules over A and K(A) the homotopy category. Let D(A) denote the derived category of dg Amodules, i.e. the triangle quotient of K(A) by acyclic dg A-modules, cf. =-=[29, 30]-=-, and let Dfd(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 ...

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... a graded H∗(A)-module structure, and hence a graded Ā = H0(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 di−1→ M i di→ M i+1 → . . . be a dg A-module. Consider the standard truncation functors τ≤0 and τ>0: τ≤0M = τ>0M = ....

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...etting) 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 bijection...

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...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 t-structures 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 pai...

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...he 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 [...

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...ation 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...

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...odules is a simple-minded collection in Db(modΛ). 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−1C≥0 ⊆ C≥0; · Hom(M,Σ−1N) = 0 for M ∈ C≤0 and N ∈ C≥0, · for each M ∈ C t...

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...extension closure of Σm{Si | i ∈ I} for m ≥ 0 respectively for m ≤ 0. (e) C = thick(Si, i ∈ I). (f) If I is finite, then {Si | 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−...

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...j, · X1, . . . ,Xr generate C (i.e. C = thick(X1, . . . ,Xr)). 6 STEFFEN KOENIG AND DONG YANG 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 finitedimensional algebra Λ, a complete collection of pairwise non-isomorphic simple modules is a simple-minded collection in Db(mo...

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...) denote the category of (right) dg modules over A and K(A) the homotopy category. Let D(A) denote the derived category of dg Amodules, i.e. the triangle quotient of K(A) by acyclic dg A-modules, cf. =-=[29, 30]-=-, and let Dfd(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 ...

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...proximation Σ−1Xj gj // Xij . Similarly one defines the right mutation µ−i (X1, . . . ,Xr). 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 (X1, . . . ,Xr) ∼= (X1, . . . ,Xr) ∼= µ−i ◦ µ+i (X1, . . . ,Xr). (b) Assume that · for any j 6= i the object Σ−1Xj...

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...e as a Λ-module. When Λ/Λ(1− ei)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....

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...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 t-structures 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 pai...

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...bounded t-structure on Db(modΛ) with length heart. Proof. Because (C≤0, C≥0) = φ31 ◦ φ14(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]-=-. Define φ34(C≥0, C≤0) = (C≤0, C≥0). If Λ has finite global dimension, then Kb(projΛ) is identified with Db(modΛ). As a consequence, C≤0 = C≤0 and C≥0 = νC≥0. Thus the t-structure (C≤0, C≥0) is right ...

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...the heart and for any m < 0. The t-structure (C≤0, C≥0) is said to be bounded if ⋃ n∈Z ΣnC≤0 = C = ⋃ n∈Z ΣnC≥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(ModΛ) of an (ordinary) algebra Λ, where D≤0 consists of complexes with vanishing cohomologies in positive degrees,...

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... 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...

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...al 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-struct...

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.... 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...

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...> 0. The co-t-structure (C≤0, C≥0) is said to be bounded [12] if ⋃ n∈Z ΣnC≤0 = C = ⋃ n∈Z ΣnC≥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 Kb(projΛ) of a finite-dimensional algebra Λ, where K≥0 consists of complexes which are homotopy equivalent to a ...

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...of C. This notion was introduced by Keller and Vossieck in [33] to study t-structures 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,ΣmM) = 0 for m < 0. For an algebra Λ, a tilt...

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...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 b...

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...ished by Keller and Nicolás [32] for homologically smooth non-positive dg algebras, by Assem, Souto SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES 3 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 collecti...

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...els 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-dim...

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...stablished 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 betw...

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...ppears 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 Ni...

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... 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...

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...ightmost components have been put in degree 0. 8.2. The Auslander–Reiten quiver. The Auslander–Reiten quiver of Db(modΛ) consists of three components: two ZA∞ components and one ZA ∞ ∞ component (see =-=[10, 28]-=-) ◦ ◦ Σ−1P1 ◦ ◦ ◦ P1 ◦ ◦ ΣP1 ?? ?? ?? ❄ ❄ ?? ❄ ?? ❄ ❄ ?? ❄ ❄ ❄ ◦ ◦ I2 ◦ S1 ΣS1 ◦ P2 ΣP2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ❄ ❄ ?? ❄ ❄ ?? ?? ❄ ❄ ?? ?? ?? ?? ❄ ?? ❄ ?? ❄ ❄ ...

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... projective Λ-modules corresponding to the vertices 1 and 2. Then up to isomorphism and up to shift an indecomposable object in Db(modΛ) belongs to one of the following four families (see for example =-=[19, 9]-=-) · P1(n) = P1 → P1 → . . .→ P1 → P1, n ≥ 1, · R(n) = P1 → P1 → . . .→ P1 → P1 → P2, n ≥ 0, · L(n) = P2 → P1 → P1 → . . .→ P1 → P1, n ≥ 0, · B(n) = P2 → P1 → P1 → . . .→ P1 → P1 → P2, n ≥ 1, where the...

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...ightmost components have been put in degree 0. 8.2. The Auslander–Reiten quiver. The Auslander–Reiten quiver of Db(modΛ) consists of three components: two ZA∞ components and one ZA ∞ ∞ component (see =-=[10, 28]-=-) ◦ ◦ Σ−1P1 ◦ ◦ ◦ P1 ◦ ◦ ΣP1 ?? ?? ?? ❄ ❄ ?? ❄ ?? ❄ ❄ ?? ❄ ❄ ❄ ◦ ◦ I2 ◦ S1 ΣS1 ◦ P2 ΣP2 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ❄ ❄ ?? ❄ ❄ ?? ?? ❄ ❄ ?? ?? ?? ?? ❄ ?? ❄ ?? ❄ ❄ ...

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...nded 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 t...

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... a graded H∗(A)-module structure, and hence a graded Ā = H0(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 di−1→ M i di→ M i+1 → . . . be a dg A-module. Consider the standard truncation functors τ≤0 and τ>0: τ≤0M = τ>0M = ....

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...gebra and Hom(Xi,Xj) vanishes for i 6= j, · X1, . . . ,Xr generate C (i.e. C = thick(X1, . . . ,Xr)). 6 STEFFEN KOENIG AND DONG YANG 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 finitedimensional algebra Λ, a complete collection of pairwise non-isomorphic simple modules...

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...Nicolás [32] for homologically smooth non-positive dg algebras, by Assem, Souto SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES 3 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-str...