Aisles in derived categories

B. Keller  D. Vossieck

Bull. Soc. Math. Belg. 40 (1988), 239-253.

Summary

The aim of the present paper is to demonstrate the usefulness of aisles for studying the tilting theory of D^{b}(mod A), where A is a finitedimensional algebra. In section 1, we establish the equivalence of “aisles” with “t-structures” in the sense of [3] and give a characterization of aisles in molecular categories. Section 2 contains an application to the generalized tilting theory of hereditary algebras. Using aisles, we then give a geometrical proof of the theorem of Happel [7] which states that a finitedimensional algebra which shares its derived category with a Dynkinalgebra A can be transformed into A by a finite number of reflections. The techniques developed so far naturally lead to the classification of the tilting sets in D^{b}(mod kA⃗_{n}) presented in section 5. Finally, we consider the classification problem for aisles in D^{b}(mod A), where A is a Dynkin-algebra. We reduce it to the classification of the silting sets in D^{b}(mod A), which we carry out for Δ = A⃗_{n}.

We thank P. Gabriel for lectures on these topics and for his helpful criticisms during the preparation of the manuscript.

Notations

Let A be a finite-dimensional algebra over a field k. We denote by

• mod A the category of finitely generated right A-modules,

• proj A the full subcategory of mod A consisting of the projective A-modules, • C^{−}(proj A) the category of differential complexes over proj A which

are bounded above,

• C^{b}(proj A) the full subcategory of C^{−}(proj A) consisting of the complexes which are bounded above and below,

• C_{b}^{−}(proj A) the full subcategory of C^{−}(proj A) consisting of the complexes X such that H^{i}X = 0 for almost all i ∈ Z,

• H^{b}(proj A) the homotopy category obtained from C^{b}(proj A) by factoring out the ideal of the morphisms homotopic to zero [12],

• D^{b}(A) := D^{b}(mod A) the bounded derived category [12] of the abelian category mod A.

1 Aisles

1.1 Let T be a triangulated category with suspension functor S. A full additive subcategory U of T is called an aisle in T if

a) SU ⊂ U,

b) U is stable under extensions, i.e. for each triangle X → Y → Z → SX of T we have Y ∈ U whenever X, Z ∈ U,

c) the inclusion U → T admits a right adjoint T → U, X ↦ X_{U}.

For each full subcategory V of T we denote by V^{⊥} (resp. ^{⊥}V) the full additive subcategory consisting of the objects Y ∈ T satisfying Hom (X, Y) = 0 (resp. Hom (Y, X) = 0) for all X ∈ V.

The following proposition shows that the assignment U ↦ (U, SU^{⊥}) is a bijection between the aisles U in T and the t-structures on T (in the sense of [3]).

Proposition. A strictly (=closed under isomorphisms) full subcategory U of T is an aisle iff it satisfies a) and c’)

c’) for each object X of T there is a triangle X_{U} → X → X^{U^{⊥}} → SX_{U} with X_{U} ∈ U and X^{U^{⊥}} ∈ U^{⊥}.

Proof. Suppose U satisfies a) and c’). The long exact sequence arising from the triangle in c’) shows that Hom (U, X_{U}) →^{∼} Hom (U, X) for each U ∈ U. If U → X → V → SU is a triangle and U, V ∈ U then Hom (X,?) vanishes on U^{⊥} and X^{U^{⊥}}. In particular, the morphism X^{U^{⊥}} → SX_{U} of c’) has a retraction, hence X^{U^{⊥}} = 0 and X ≅ X_{U} lies in U. Conversely, let U satisfy a),b),c). According to b), U is strictly full. In order to prove c), we form a triangle X_{U} →^{φ} X →^{ψ} Y →^{ε} SX_{U} over the adjunction morphism φ. Let V ∈ U and f ∈ Hom (V, Y). We insert f into a morphism of triangles

X_{U} →^{h} W → V →^{εf} SX_{U}
∥ ↓ g ↓ f ∥
X_{U} →^{φ} X →^{ψ} Y →^{ε} SX_{U}

According to b), W lies in U. By assumption, g factors uniquely through φ. Therefore, h has a retraction and εf = 0. So f factors through ψ and even through ψφ = 0 since V ∈ U.

1.2 For certain triangulated categories, condition 1.1c) can still be weakened. We call an additive category T a molecular category if each object of T is a finite direct sum of objects with local endomorphism rings. In particular, if A is a finite-dimensional algebra over a field k, the category D^{b}(A) is a molecular category: For all X, Y ∈ D^{b}(A), we have dim_{k}Hom (X, Y) < ∞ and for each indecomposable U of D^{b}(A), End (U) is local since End_{C^{−}(proj A)}(V) is local for each indecomposable V of C^{−}(proj A).

If T is a molecular category, we denote by ind T a full subcategory of T whose objects form a system of representatives for the isomorphism classes of indecomposables of T .

1.3 Proposition. Let T be a triangulated molecular category and U a full additive subcategory which is closed under taking direct summands. The subcategory U is an aisle in T iff it satisfies a), b) and c”).

c”) For each object X of T the functor Hom (?, X)| U is finitely generated, i.e. there is U ∈ U and an epimorphism Hom_{U}(?, U) → Hom_{T} (?, X)| U.

Proof. The claim follows from the

Lemma. Let S be a suspended [9] molecular category and F : Sop → Ab a cohomological functor, i.e. for each triangle X →^{u} Y →^{v} Z →^{w} SX of S the sequence F X ←^{F u} F Y ←^{F v} F Z is exact. The functor F is representable iff it is finitely generated.

Proof. Let F be finitely generated. Because S is a molecular category F has a projective cover Hom_{S}(?, X) →^{φ} F in the abelian category of the additive functors Sop → Ab. We shall show that φ is a monomorphism. Let Y ∈ S and f ∈ Hom (Y, X) such that φ ∘ Hom (?, f) = 0. We form a triangle Y →^{f} X →^{g} Z →^{h} SY in S. By the ”Yoneda Lemma“ we conclude from the exactness of F Y ←^{F f} F X ←^{F g} F Z that φ factors through Hom (?, g). By construction of φ, Hom (?, g) admits a retraction. Hence g admits a retraction and f = 0.

1.4 Proposition. Let U, V be aisles in a triangulated category T such that V ⊂ U^{⊥}. Then W := U ∗ V = {X ∈ T : There is a triangle U → X → V → SU with U ∈ U,V ∈ V} is also an aisle in T (cf. [3, 1.4])

Proof. We shall verify the conditions of Proposition 1.1. Only c’) is not immediate from the definitions. Let X ∈ T . We form a diagram

[figure: X^{U^{⊥}V^{⊥}}; X^{U^{⊥}}; X; (X^{U^{⊥}})_{V}; X_{U}; Y; Δ; Δ; Δ; =]

where triangles are marked by Δ and tailed arrows denote morphisms of degree 1. By definition, Y ∈ W, X^{U^{⊥}V^{⊥}} ∈ V^{⊥}. Since X^{U^{⊥}V^{⊥}} is an extension of S((X^{U^{⊥}})_{V}) by X^{U^{⊥}} it also lies in U^{⊥}, hence in W^{⊥} = U^{⊥} ∩ V^{⊥}. We obtain the required triangle Y → X → X^{U^{⊥}V^{⊥}} → SY by forming an octahedron with base X → X^{U^{⊥}} → X^{U^{⊥}V^{⊥}}.

2 Tilting sets

2.1 Let A be a finite-dimensional algebra over a field k. We consider the problem of determining all finite-dimensional k-algebras B such that D^{b}(B) is S-equivalent [9] to D^{b}(A).

A tilting set (cf. [7]) in D^{b}(A) is a finite subset {T_{1}, …, T_{s}} ⊂ ind D^{b}(A) such that Hom (S^{l}T_{i}, T_{j}) = 0 for all i, j and all integers l ≠ 0.

Each fully faithful S-functor F : D^{b}(B) → D^{b}(A) gives rise to the tilting set {T ∈ ind D^{b}(A) : T ≅ F P for some indecomposable projective B-module P }. Conversely, let {T_{1}, …, T_{s}} be a tilting set and B := End (⨁_{i=1}^{s} T_{i}). By [9, 3.2], the obvious functor from proj B to D^{b}(A) extends to a fully faithful S-functor E : H^{b}(proj B) → D^{b}(A) (put E := C_{b}^{−}(proj A) in [9, 3.2]). We make the additional assumptions

a) Hom (T_{i}, T_{j}) = 0 ∀ i > j and Hom (T_{i}, T_{i}) is a skew field ∀i.

b) gldimA < ∞.

Assumption a) implies gldimB < ∞. By composing E with a quasiinverse of the equivalence H^{b}(proj B) → D^{b}(B) we obtain a fully faithful S-functor F : D^{b}(B) → D^{b}(A).

Proposition. The essential image of F is an aisle in D^{b}(A). In particular, F has a right adjoint.

Proof. Let U_{i} be the strictly full triangulated subcategory of D^{b}(A) generated by T_{i}. Since D^{b}(End (T_{i})) →^{∼} U_{i}, assumption b) and proposition 1.3 imply that U_{i} is an aisle in D^{b}(A). The essential image of F equals U_{s} ∗ U_{s−1} ∗ ⋯ ∗ U_{2} ∗ U_{1} (by [3, 1.3.10], ∗ is associative). The claim now follows from proposition 1.4.

2.2 Theorem. Let A be a hereditary finite-dimensional k-algebra and {T_{1}, …, T_{s}} a tilting set in D^{b}(A). The functor F is an S-equivalence iff s equals the number of isomorphism classes of simple A-modules.

Proof. By [7, 7.3] assumption a) is satisfied for an appropriate numbering of the T_{i} and, since A is hereditary, so is assumption b). An S-equivalence D^{b}(B) →^{∼} D^{b}(A) induces an isomorphism of the Grothendieck-groups K_{0}(B) →^{∼} K_{0}(A). Therefore, since s equals the number of isomorphism classes of simple B-modules, the condition is necessary. For the converse, it is enough to show that {F X : X ∈ D^{b}(B)}^{⊥} = 0, by proposition 2.1. Let Y ∈ D^{b}(A) be indecomposable and such that Hom (F X, Y) = 0, ∀X ∈ D^{b}(B). This implies < [F X], [Y] >= 0, where [.] denotes the canonical map D^{b}(A) → K_{0}(A) and < . , . > denotes the canonical bilinear form on K_{0}(A) [7]. F induces a section K_{0}(B) → K_{0}(A) (the right adjoint of F yields a retraction). Since rankK_{0}(B) = s = rank K_{0}(A), we have K_{0}(A) = {[F X] : X ∈ D^{b}(B)}, hence [Y] = 0 and, since A is hereditary and Y is indecomposable, Y = 0 [7].

3 Dynkin-algebras

3.1 Let B be a basic, connected, finite-dimensional algebra over an algebraically closed field k. Assume that there exists a simple, projective, non-injective right B-module P. Define Q by B ≅ Q ⊕ P. Then T = τ^{−}P ⊕ Q is a tilting module in mod B (cf. [5], [8]), where τ^{−}P denotes a preimage of P under the Auslander-Reiten-translation. The derived functors F = \underline{R}Hom_{B}(T,?) : D^{b}(B) → D^{b}(B_{P}) and G = \underline{L}(?⊗_{B_{P}} T) are quasi-inverse S-equivalences, where B_{P} = End (T_{B}) is obtained from B by reflection in P [4]. Thus, the derived category D^{b}(B) is “invariant under reflections”. Conversely, we have the

Theorem. (Happel) Let D^{b}(B) be S-equivalent to D^{b}(A) where A is a Dynkin-algebra (=path algebra [6, 4.1] of a Dynkin quiver). Then A is obtained from B by a finite number of reflections.

Proof. Let U = {X ∈ D^{b}(B) : H^{i}X = 0 ∀i > 0} be the natural aisle in D^{b}(B) [3, 1.3.1], V the natural aisle in D^{b}(A) and E : D^{b}(B) → D^{b}(A) an S-equivalence with [V] ⊂ [EU], where [?] denotes the set of isomorphism classes of indecomposables in ?. If [V] = [EU], E induces an equivalence of the hearts (=hearts of the corresponding t-structures) of U and V, i.e. of mod B and mod A. In general, [V] is obtained from [EU] by the omission of finitely many isomorphism classes, as it is apparent from Happel’s description of ind D^{b}(A) [7]. By induction, we are reduced to the case [V] = [EU]\{EM}, where M is a source of ind U, i.e. Hom (V, M) = 0 ∀V ∈ ind U, V ≠ M and S^{−}M ∉ U. The sources of ind U are isomorphic to the simple projectives of mod B. It is therefore enough to prove the

3.2 Lemma. In the setting of 3.1 let U be the natural aisle in D^{b}(B), W the natural aisle in D^{b}(B_{P}) and W′ its essential image under G. Then W′ = {X ∈ U : P is not a direct summand of X}.

Proof. Let U_{P} = {X ∈ U : P is not a direct summand of X}. It follows from pdim_{B_{P}} T ≤ 1 that GW^{⊥} ⊂ SU^{⊥} hence SU ⊂ W′. We also have SU ⊂ U_{P} and U_{P} = (SU)∗(U_{P} ∩mod B), W′ = (SU)∗(W′∩mod B). By [2], U_{P} ∩ mod B is just the torsion theory generated by T, which by [5] coincides with the essential image of mod B_{P} under H^{0}G ≅? ⊗_{B_{P}} T.

4 Complete tilting sets in D^{b}(kA⃗_{n})

Let A be a finite-dimensional algebra over a field k. We define the spectrum of a tilting set M in D^{b}(A) as the full subcategory of D^{b}(A) whose objects are the elements of M. We call a tilting set in D^{b}(A) complete if its cardinality equals the number of isomorphism classes of simple Amodules.

Let A be the path algebra of the quiver A⃗_{n} : 1 → 2 → … → n [6]. We identify [7] the category ind D^{b}(A) with the mesh-category k(ZA⃗_{n}) [11, 2.1] of the translation quiver ZA⃗_{n}:

[figure: (0,1); (0,2); (0,n−1); (0,n); (1,1); (1,2); (1,3); (1,n−1); (1,n); …; …; ^{. .}; ^{. .}; ^{. .}]

By an A⃗_{n}-quiver we mean an oriented tree K having n vertices and whose set of arrows is decomposed into a class of α-arrows and a class of β-arrows such that in each vertex of K there terminate at most one αarrow and one β-arrow and there originate at most one α-arrow and one β-arrow. Now let K be an A⃗_{n}-quiver. For each vertex x of K let x^{α} (resp. x_{α}) be the number of vertices y of K such that the shortest walk from y to x ends (resp. begins) with an α-arrow terminating (resp. originating) in x. Analogously, we define x^{β} and x_{β}. Then there is exactly one map of the underlying sets of vertices K_{0} → (ZA⃗_{n})_{0}, x ↦ (gx, hx) such that

a) min_{x∈K} gx = 0,

b) (gy, hy) = (gx, hx + x^{β} + y_{β} + 1) for each α-arrow x →^{α} y and

c) (gy, hy) = (gx + x^{α} + y_{α} + 1, hx − x^{α} − y_{α} − 1) for each β-arrow

x →^{β} y.

Because hx = 1 + x^{α} + x_{β}, this map is indeed well defined. Let M_{K} denote its image.

Theorem. The assignment K ↦ M_{K} induces a bijection from the isomorphism classes of A⃗_{n}-quivers to the complete tilting sets M in ind D^{b}(kA⃗_{n}) = k(ZA⃗_{n}) with min_{(g,h)∈M} g = 0. Moreover, the spectrum of M_{K} is described by the quiver K bound by all possible relations αβ = 0 and βα = 0 (cf. [1]).

Proof. Let K be an A⃗_{n}-tree. We call x ∈ K_{0} a knot of K if x is contained in a full subtree of one of the forms

• →^{α} x →^{α} •, • →^{α} x ←^{β} •, • ←^{β} x →^{α} •, • ←^{β} x ←^{β} •.

The other vertices of K are called peaks. We term (g, h) ∈ (ZA⃗_{n})_{0} a marginal vertex if h = 1 or h = n and we call the other vertices of ZA⃗_{n} inner vertices. We use induction on the number m of knots of K.

If m = 0 , K has one of the forms

K_{α} = • →^{α} • →^{β} • →^{α} …

or

K_{β} = • →^{β} • →^{α} • →^{β} …

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; SP; P; S^{−}P; Y_{1}; Y_{2}; Y_{s}; X_{1}; X_{2}; X_{r}]

Figure 1: ZA⃗_{n} with given vertex P

It is easy to see that the corresponding sets M_{K_{α}}, M_{K_{β}} are exactly the complete tilting sets M of ind D^{b}(kA⃗_{n}) which only consist of marginal vertices and satisfy min_{(g,h)∈M} g = 0.

Now let m > 0. We first describe the complete tilting sets in k(ZA⃗_{n}) which contain a given inner vertex P. The tilting sets {X_{1}, …, X_{r}, P } and {Y_{1}, …, Y_{s}, P } (cf. Fig. 1) give rise to fully faithful embeddings j_{X} : D^{b}(kA⃗_{r+1}) → D^{b}(kA⃗_{n}) and j_{Y} : D^{b}(kA⃗_{s+1}) → D^{b}(kA⃗_{n}). We may assume that j_{X}(ZA⃗_{r+1})_{0} ⊂ (ZA⃗_{n})_{0} and j_{Y} (ZA⃗_{s+1})_{0} ⊂ (ZA⃗_{n})_{0}, in particular j_{X}P_{X} = P and j_{Y} P_{Y} = P where P_{X} = (0, r + 1) ∈ (ZA⃗_{r+1})_{0} and P_{Y} = (0, s + 1) ∈ (ZA⃗_{s+1})_{0}. With these notations, the complete tilting sets M in k(ZA⃗_{n}) containing P are exactly the sets j_{X}(L) ∪ j_{Y} (N) where L ⊂ (ZA⃗_{r+1})_{0} and N ⊂ (ZA⃗_{s+1})_{0} are complete tilting sets containing P_{X} and P_{Y}, respectively. Here, the set of marginal points of M equals {j_{X}(R) : R is a marginal point of L \ P_{X} }∪{j_{Y} (R) : R is a marginal point of N \ P_{Y} }. For the corresponding spectra we have the pushout diagram

k →^{i_{X}} L
i_{Y} ↓  ↓ j_{X}
N →^{j_{Y}} M

where k is considered as a category with one object and i_{X} and i_{Y} are fully faithful with i_{X}(k) = P_{X} and i_{Y} (k) = P_{Y}. Combined with the induction hypothesis, this description shows that the spectra of the complete tilting sets in k(ZA⃗_{n}) are described by the A⃗_{n}-quivers with all relations αβ = 0 = βα and that peaks are mapped to marginal points by the corresponding isomorphisms of categories. So let M be a complete tilting set in k(ZA⃗_{n}) whose spectrum is described by K. We claim that the corresponding bijection e : K_{0} → M ⊂ (ZA⃗_{n})_{0} satisfies conditions b) and c). This is obvious if x, y are peaks of K since then ex, ey are marginal points. If for example x is a knot we apply the above construction with P = ex and the claim follows from the induction hypothesis.

5 Aisles in D^{b}(kΔ)

5.1 Let U be an aisle in a triangulated category T . The heart of U is the full subcategory U^{0} = U ∩ SU^{⊥} of T ; the associated cohomology functor H_{U}^{0} : T → U^{0} is given by X ↦ (X_{U})^{SU^{⊥}} [3, 1.3.1-6]. The aisle U is faithful if the inclusion U^{0} → T extends to an S-equivalence D^{b}(U^{0}) → ⋃_{n∈N} S^{−n}U; it is separated if ⋂_{n∈N} S^{n}U = 0.

Let kΔ be the path algebra [6] of a Dynkin-quiver Δ. For each subset M ⊂ ind D^{b}(kΔ) let F(M) be the smallest strictly full subcategory of D^{b}(kΔ) which contains M and is stable under S and closed under extensions and direct summands. By proposition 1.3, F(M) is an aisle. The assignment {T_{1}, …, T_{s}} ↦ F(T_{1}, …, T_{s}) is a bijection between the tilting sets in D^{b}(kΔ) and the faithful aisles. We shall generalize the concept of a tilting set in order to obtain an analogous description of all separated aisles in D^{b}(kΔ).

A silting set in D^{b}(kΔ) is a finite subset {R_{1}, …, R_{s}} ⊂ ind D^{b}(kΔ) such that Hom (R_{i}, S^{l}R_{j}) = 0 ∀i, j and ∀l > 0.

Theorem.

a) The assignment {R_{1}, …, R_{s}} ↦ F(R_{1}, …, R_{s}) is a bijection between the silting sets in D^{b}(kΔ) and the separated aisles in D^{b}(kΔ).

If {R_{1}, …, R_{s}} is a silting set and W = F(R_{1}, …, R_{s}) we have

b) W ∩^{⊥}(SW) ∩ ind D^{b}(kΔ) = {R_{1}, …, R_{s}}

c) s ≤ |Δ_{0}|, and s = |Δ_{0}| ⇔ ⋃_{n∈N} S^{−n}W = D^{b}(kΔ)

d) W = {X ∈ ⋃_{n∈N} S^{−n}W : Hom (R_{i}, S^{l}X) = 0 ∀i, ∀l > 0}

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; •; S^{2}Q; •; SQ; •; Q]

Figure 2: Example of V (•) and U (∘) in ZA⃗_{n}

e) H_{W}^{0}R_{1}, …, H_{W}^{0}R_{s} form a system of representatives of the isomorphism classes of indecomposable projectives of W^{0}, and H_{W}^{0} induces an equivalence between the full subcategories {R_{1}, …, R_{s}} and {H_{W}^{0}R_{1}, …, H_{W}^{0}R_{s}} of D^{b}(kΔ).

Proof. 1st step: The following variant of a construction by Parthasarathy [10] allows us to use induction on |Δ_{0}|.

Let W be a separated aisle in D^{b}(kΔ) and Q a source (3.1) of ind W. We may assume that Δ admits a unique source q and that Q = (0, q). Let Q be the full subcategory of D^{b}(kΔ) whose objects are the direct sums of objects S^{n}Q, n ∈ Z. The tilting set {(0, r) : r ∈ Δ_{0}, r ≠ q} ⊂ k(ZΔ) yields a fully faithful S-functor D^{b}(kΔ′) → D^{b}(kΔ) (2.1), where Δ′ is obtained from Δ by omitting the vertex q and all arrows originating in q. The essential image of this S-functor equals ^{⊥}Q. We claim that W = U ∗ V, where U = W ∩^{⊥} Q and V = W ∩ Q. Obviously, we have W ⊃ U ∗ V. Conversely, let X be an indecomposable in W which is not isomorphic to Q. We have the triangle X_{^{⊥}Q} → X → X^{Q} → SX_{^{⊥}Q} (1.3). The assumptions on X imply that X^{Q} lies in SV. Thus, as an extension of X by S^{−}X^{Q} ∈ V ⊂ W, X_{^{⊥}Q} lies in W ∩^{⊥}Q = U.

2nd step: b) Let the numbering be chosen in such a way that Hom (R_{j}, R_{i}) = 0 ∀i < j. We apply the construction of the first step to the source Q = R_{1} of ind W. Let X ∈ W ∩^{⊥}(SW) be indecomposable. We have the triangle X_{^{⊥}Q} → X → X^{Q} → SX_{^{⊥}Q}. As in the first step, either X ≅ R_{1} or X^{Q} ∈ SV and in this case we infer X^{Q} = 0 and X ∈ U ∩^{⊥} (SU). The claim now follows from the induction hypothesis.

a) Because of b) we only have to show surjectivity. In the setting of the first step let R_{1} = Q. We complete R_{1} to a system of representatives {R_{1}, …, R_{s}} of the indecomposables of W ∩^{⊥}(SW). Then {R_{2}, …, R_{s}} is a system of representatives of the indecomposables of U ∩^{⊥}(SU). According to the induction hypothesis, we have U = F(R_{2}, …, R_{s}), and therefore W = U ∗ V = F(R_{1}, …, R_{s}).

The proof of c) is left to the reader. d) Obviously, W is contained in the aisle given in the assertion. Conversely, let X = S^{−n}Y (Y ∈ W, n ∈ N) and Hom (R_{i}, S^{l}X) = 0 ∀i, ∀l > 0. By induction we conclude S^{−n}Y_{U} ∈ U. Using Hom (R_{1}, SU) = 0 and the triangle Y_{U} ≅ Y_{^{⊥}Q} → Y → Y^{Q} → SY_{^{⊥}Q} we infer Hom (S^{−l}R_{1}, S^{−n}Y^{Q}) = 0 for all l > 0 and S^{−n}Y^{Q} ∈ V.

e) From Hom (R_{1}, SW) = 0 it follows that R_{1} ≅ H_{W}^{0}R_{1} is projective in W^{0}. The rest of the assertion follows from the

Lemma. Let U, V be aisles in a triangulated category T such that U ⊂^{⊥} V and let W = U ∗ V. (cf. proposition 1.4)

a) V^{0} ⊂ W^{0} and H_{V}^{0}| W^{0} is right adjoint to this inclusion.

b) H_{U}^{0}| W^{0} is exact and H_{W}^{0}| U^{0} is left adjoint to H_{U}^{0}| W^{0} and fully faithful.

c) For each A ∈ W^{0} we have an exact sequence

H_{W}^{0}H_{U}^{0}A → A → H_{W}^{0}A^{U^{⊥}} → 0

d) H_{W}^{0}| W ∩^{⊥}(SW) is fully faithful and for X ∈ U we have H_{W}^{0}X ≅ H_{W}^{0}H_{U}^{0}X.

We leave the proof of the lemma to the reader (compare with [3]).

5.2 In the setting of 5.1 let {T_{1}, …, T_{s}} be a tilting set in D^{b}(kΔ). Suppose that the numbering has been chosen in such a way that Hom (T_{j}, T_{i}) = 0 ∀j > i. Let p : {1, …, s} → N be a non-decreasing function with p(1) = 0.

Proposition.

a) R_{1} := S^{p (1)}T_{1}, …, R_{s} := S^{p (s)}T_{s} form a silting set in D^{b}(kΔ).

b) With U = F(T_{1}, …, T_{s}), W = F(R_{1}, …, R_{s}) we have for each i ∈ Z

U^{0} ∩ S^{−i}W = {X ∈ U^{0} : Hom (T_{j}, X) = 0 for each j with i < p(j)}

W = {X ∈ U : H_{U}^{i}X ∈ U^{0} ∩ S^{i}W, ∀i ∈ Z}

c) The full subcategory {R_{1}, …, R_{s}} of D^{b}(kΔ) is isomorphic to the disjoint sum of the full subcategories C_{j} = {T_{i} : p(i) = j} , j ∈ N, of {T_{1}, …, T_{s}}.

We leave the proof to the reader.

5.3 Theorem. Each silting set in D^{b}(kA⃗_{n}) (cf. section 4) is of the form given in 5.2

Proof. Let {R_{1}, …, R_{s}} be a silting set in D^{b}(kA⃗_{n}).

1st step: {R_{1}, …, R_{s}} is contained in a complete silting set (= silting set of maximal cardinality).

If s < n we have T^{⊥} ≠ 0 (5.1 c), where T = ⋃_{m∈N} S^{−m}F(R_{1}, …, R_{s}). Because gldimkΔ < ∞ we can find an indecomposable R_{0} ∈ T^{⊥} such that Hom (R_{0}, S^{l}R_{i}) = 0 for all l > 0, i = 1, …, s. Then {R_{0}, … R_{s}} is a silting set. The claim now follows by induction on n − s.

2nd step: By the first step we may assume n = s. Let the numbering of the R_{i} be non-decreasing with respect to the order on ind D^{b}(kA⃗_{n}) generated by the arrows of ZA⃗_{n}. With the notations of the first step of the proof of theorem 5.1 we set Q = R_{1}. The induction hypothesis applied to {R_{2}, …, R_{n}} ⊂ ^{⊥}Q yields a tilting set {T_{2}, …, T_{n}} which corresponds to a complete tilting set in D^{b}(kΔ′). Therefore the connected components of the spectrum of {T_{2}, …, T_{n}} are precisely the intersections of {T_{2}, …, T_{n}} with the connected components of ind^{⊥}Q.

[figure: …; …; ∘; ∘; ∘; ∘]

Figure 3: A silting set in ZD⃗_{4}

Let C be a connected component of ind^{⊥}Q. Because Hom (Q,?)| C ≠ 0 and {T_{2}, …, T_{n}} ∩ C is a complete tilting set in C, Hom (Q,?) does not vanish on S^{l}{T_{2}, …, T_{n}} ∩ C for some l ∈ Z. Hence we may assume that {Q, T_{2}, …, T_{n}} is connected. Because {X ∈ C : Hom (Q, X) ≠ 0} is linearly ordered by X ≤ X′ :⇔ Hom (X, X′) ≠ 0 (cf. figure 2), {Q, T_{2}, …, T_{n}} must be a tilting set in D^{b}(kA⃗_{n}). Setting T_{1} = Q we have R_{i} = S^{p (i)}T_{i} for some function p : {1, …, n} → Z with p(1) = 0. It is easy to see that Hom (T_{i}, T_{j}) ≠ 0 implies p(i) ≤ p(j).

5.4 Remarks: a) Silting sets can always be completed (cf. the first step of the above proof) but in general, tilting sets cannot : {(0, 1), (3, 3)} ⊂ ZA⃗_{3}.

b) The silting set of figure 3 is not of the form given in 5.2 since 5.2 c) cannot be satisfied.

References

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B.K. : Mathematik, G 28.2, ETH-Zentrum, 8092 Zürich, Switzerland; D.V. : Mathematisches Institut, Universität Zürich, Rämistrasse 74, 8051 Zürich, Switzerland