Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type

We give a derived equivalence classification of algebras of the form Ă/〈∅〉 for some piecewise hereditary algebra A of tree type and some automorphism ∅ of Ă such that ∅(A⁽⁰⁾)=A⁽ⁿ⁾ for some positive integer n.

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Datum:	2015
Hauptverfasser:	Asashiba, H., Kimura, M.
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Schriftenreihe:	Algebra and Discrete Mathematics
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spelling	irk-123456789-1542492019-06-16T01:26:33Z Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type Asashiba, H. Kimura, M. We give a derived equivalence classification of algebras of the form Ă/〈∅〉 for some piecewise hereditary algebra A of tree type and some automorphism ∅ of Ă such that ∅(A⁽⁰⁾)=A⁽ⁿ⁾ for some positive integer n. 2015 Article Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type / H. Asashiba, M. Kimura // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 145-161 . — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC:16G30, 16E35, 16W22. http://dspace.nbuv.gov.ua/handle/123456789/154249 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We give a derived equivalence classification of algebras of the form Ă/〈∅〉 for some piecewise hereditary algebra A of tree type and some automorphism ∅ of Ă such that ∅(A⁽⁰⁾)=A⁽ⁿ⁾ for some positive integer n.
format	Article
author	Asashiba, H. Kimura, M.
spellingShingle	Asashiba, H. Kimura, M. Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type Algebra and Discrete Mathematics
author_facet	Asashiba, H. Kimura, M.
author_sort	Asashiba, H.
title	Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type
title_short	Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type
title_full	Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type
title_fullStr	Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type
title_full_unstemmed	Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type
title_sort	derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/154249
citation_txt	Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type / H. Asashiba, M. Kimura // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 145-161 . — Бібліогр.: 11 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT asashibah derivedequivalenceclassificationofgeneralizedmultifoldextensionsofpiecewisehereditaryalgebrasoftreetype AT kimuram derivedequivalenceclassificationofgeneralizedmultifoldextensionsofpiecewisehereditaryalgebrasoftreetype
first_indexed	2025-07-14T05:54:29Z
last_indexed	2025-07-14T05:54:29Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 19 (2015). Number 2, pp. 145–161 © Journal “Algebra and Discrete Mathematics” Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type Hideto Asashiba, Mayumi Kimura Communicated by D. Simson Abstract. We give a derived equivalence classification of algebras of the form Â/〈φ〉 for some piecewise hereditary algebra A of tree type and some automorphism φ of Â such that φ(A[0]) = A[n] for some positive integer n. Introduction Throughout this paper we fix an algebraically closed field k, and assume that all algebras are basic and finite-dimensional k-algebras and that all categories are k-categories. The classification of algebras under derived equivalences seems to be first explicitly investigated by Rickard in [9], which gave the derived equiva- lence classification of Brauer tree algebras (implicitly there exists an earlier work [4] by Assem–Happel giving the classification of gentle tree algebras). After that the first named author gave the classification of representation- finite self-injective algebras (see also [1] and Membrillo-Hernández [7] for type An). The technique used there (a covering technique for derived equivalences developed in [1]) is applicable also for representation-infinite 2010 MSC: 16G30, 16E35, 16W22. Key words and phrases: derived equivalence, piecewise hereditary, quivers, orbit categories. 146 Derived equivalence classification algebras; it requires that the algebras in consideration have the form of orbit categories (usually of repetitive categories of some algebras hav- ing no oriented cycles in their ordinary quivers). In fact, it was applied in [3] to give the classification of twisted multifold extensions of piecewise hereditary algebras of tree type by giving a complete invariant. Here an algebra is called a twisted multifold extension of an algebra A if it has the form Tnψ (A) := Â/〈ψ̂νnA〉 (0.1) for some positive integer n and some automorphism ψ of A, where Â is the repetitive algebra of A, νA is the Nakayama automorphism of Â and ψ̂ is the automorphism of Â naturally induced from ψ (see Definition 1.1 and Lemma 1.2 for details); and an algebra A is called a piecewise hereditary algebra of tree type if A is an algebra derived equivalent to a hereditary algebra whose ordinary quiver is an oriented tree. In this paper we extend this classification to a wider class of algebras. To state this class of algebras we introduce the following terminologies. For an integer n we say that an automorphism φ of Â has a jump n if φ(A[0]) = A[n]. An algebra of the form Â/〈φ〉 for some automorphism φ of Â with jump n for some positive integer n is called a generalized multifold extension of A. Since obviously ψ̂νnA is an automorphism of Â with jump n in the formula (0.1), twisted multifold extensions are generalized multifold extensions. We are now able to state our purpose. In this paper we will give the derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type by giving a complete invariant. Note that most algebras in this class are wild and that the tame part of the class has a big intersection with the class of self-injective algebras of Euclidean type studied by Skowroński in [10] (see Remark 1.7). The paper is organized as follows. After preparations in section 1 we first reduce the problem to the case of hereditary tree algebras in section 2. Then we investigate scalar multiples in the repetitive category of a hereditary tree algebras in section 3, which is a central part of the proof of the main result. In section 4 we show that any generalized multifold extension of a piecewise hereditary algebra of tree type is derived equivalent to a twisted multifold extension of the same type, which immediately yields the desired classification result. H. Asashiba, M. Kimura 147 1. Preliminaries For a category R we denote by R0 and R1 the class of objects and morphisms of R, respectively. A category R is said to be locally bounded if it satisfies the following: • Distinct objects of R are not isomorphic; • R(x, x) is a local algebra for all x ∈ R0; • R(x, y) is finite-dimensional for all x, y ∈ R0; and • The set {y ∈ R0 \| R(x, y) 6= 0 or R(y, x) 6= 0} is finite for all x ∈ R0. A category is called finite if it has only a finite number of objects. A pair (A,E) of an algebra A and a complete set E := {e1, . . . , en} of orthogonal primitive idempotents of A can be identified with a lo- cally bounded and finite category R by the following correspondences. Such a pair (A,E) defines a category R(A,E) := R as follows: R0 := E, R(x, y) := yAx for all x, y ∈ E, and the composition of R is defined by the multiplication of A. Then the category R is locally bounded and finite. Conversely, a locally bounded and finite category R defines such a pair (AR, ER) as follows: AR := ⊕ x,y∈R0 R(x, y) with the usual matrix multiplication (regard each element of A as a matrix indexed by R0), and ER := {(1xδ(i,j),(x,x))i,j∈R0 \| x ∈ R0}. We always regard an algebra A as a locally bounded and finite category by fixing a complete set A0 of orthogonal primitive idempotents of A. For a locally bounded category A, we denote by ModA the category of all (right) A-modules (= contravariant functors from A to the category Modk of k-vector spaces); by modA the full subcategory of ModA consisting of finitely presented objects; and by prjA the full subcategory of ModA consisting of finitely generated projective objects. Kb(A) denotes the bounded homotopy category of an additive category A. 1.1. Repetitive categories Definition 1.1. Let A be a locally bounded category. (1) The repetitive category Â of A is a k-category defined as follows (Â turns out to be locally bounded again): • Â0 := A0 × Z = {x[i] := (x, i) \| x ∈ A0, i ∈ Z}. 148 Derived equivalence classification • Â(x[i], y[j]) :=    {f [i] \| f ∈ A(x, y)} if j = i, {φ[i] \| φ ∈ DA(y, x)} if j = i+ 1, 0 otherwise, for all x[i], y[j] ∈ Â0. • For each x[i], y[j], z[k] ∈Â0 the composition Â(y[j], z[k])×Â(x[i], y[j])→ Â(x[i], z[k]) is given as follows. (i) If i = j, j = k, then this is the composition of A A(y, z) × A(x, y) → A(x, z). (ii) If i = j, j + 1 = k, then this is given by the right A-module structure of DA: DA(z, y) ×A(x, y) → DA(z, x). (iii) If i + 1 = j, j = k, then this is given by the left A-module structure of DA: A(y, z) ×DA(y, x) → DA(z, x). (iv) Otherwise, the composition is zero. (2) We define an automorphism νA of Â, called the Nakayama auto- morphism of Â, by νA(x[i]) := x[i+1], νA(f [i]) := f [i+1], νA(φ[i]) := φ[i+1] for all i ∈ Z, x ∈ A0, f ∈ A1, φ ∈ ⋃ x,y∈A0 DA(y, x). (3) For each n ∈ Z, we denote by A[n] the full subcategory of Â formed by x[n] with x ∈ A, and by 1 [n] : A ∼ → A[n] →֒ Â, x 7→ x[n], the embedding functor. We cite the following from [3, Lemma 2.3]. Lemma 1.2. Let ψ : A → B be an isomorphism of locally bounded categories. Denote by ψyx : A(y, x) → B(ψy, ψx) the isomorphism defined by ψ for all x, y ∈ A. Define ψ̂ : Â → B̂ as follows. • For each x[i] ∈ Â, ψ̂(x[i]) := (ψx)[i]; • For each f [i] ∈ Â(x[i], y[i]), ψ̂(f [i]) := (ψf)[i]; and • For each φ[i] ∈ Â(x[i], y[i+1]), ψ̂(φ[i]) := (D((ψyx)−1)(φ))[i] = (φ ◦ (ψyx)−1)[i]. Then (1) ψ̂ is an isomorphism. (2) Given an isomorphism ρ : Â → B̂, the following are equivalent. (a) ρ = ψ̂; (b) ρ satisfies the following. H. Asashiba, M. Kimura 149 (i) ρνA = νBρ; (ii) ρ(A[0]) = A[0]; (iii) The diagram A ψ −−−−→ B 1 [0] y y1 [0] A[0] −−−−→ ρ B[0] is commutative; and (iv) ρ(φ[0]) = (φ ◦ (ψyx)−1)[0] for all x, y ∈ A and all φ ∈ DA(y, x). Let R be a locally bounded category with the Jacobson radical J and with the ordinary quiver Q. Then by definition of Q there is a bijection f : Q0 → R0, x 7→ fx and injections āy,x : Q1(x, y) → J(fx, fy)/J 2(fx, fy) such that āy,x(Q1(x, y)) forms a basis of J(fx, fy)/J 2(fx, fy), where Q1(x, y) is the set of arrows from x to y in Q for all x, y ∈ Q0. For each α ∈ Q1(x, y) choose ay,x(α) ∈ J(fx, fy) such that ay,x(α) + J2(fx, fy) = āy,x(α). Then the pair (f, a) of the bijection f and the family a of injec- tions ay,x : Q1(x, y) → J(fx, fy) (x, y ∈ Q0) uniquely extends to a full functor Φ: kQ → R, which is called a display functor for R. A path µ from y to x in a quiver with relations (Q, I) is called maximal if µ 6∈ I but αµ, µβ ∈ I for all arrows α, β ∈ Q1. For a k-vector space V with a basis {v1, . . . , vn} we denote by {v∗ 1, . . . , v ∗ n} the basis of DV dual to the basis {v1, . . . , vn}. In particular if dimk V = 1, v∗ ∈ DV is defined for all v ∈ V \{0}. An algebra is called a tree algebra if its ordinary quiver is an oriented tree. Lemma 1.3. Let A be a tree algebra and Φ : kQ → A a display functor with I := Ker Φ. Then (1) Φ uniquely induces the display functor Φ̂ : kQ̂ → Â for Â, where (i) Q̂ = (Q̂0, Q̂1, ŝ, t̂) is defined as follows: • Q̂0 := Q0 × Z = {x[i] := (x, i) \| x ∈ Q0, i ∈ Z}, • Q1 × Z := {α[i] := (α, i) \| α ∈ Q1, i ∈ Z}, Q̂1 :=(Q1×Z)⊔{µ∗[i] \|µ is a maximal path in (Q, I), i∈Z}, • ŝ(α[i]) := s(α)[i], t̂(α[i]) := t(α)[i] for all α[i] ∈ Q1 × Z, and if µ is a maximal path from y to x in (Q, I) then, ŝ(µ∗[i]) := x[i], t̂(µ∗[i]) := y[i+1]. 150 Derived equivalence classification (ii) Φ̂ is defined by Φ̂(x[i]) := (Φx)[i], Φ̂(α[i]) := (Φα)[i], and Φ̂(µ∗[i]) := (Φ(µ)∗)[i] for all i ∈ Z, x ∈ Q0, α ∈ Q1 and maximal paths µ in (Q, I). (2) We define an automorphism νQ of Q̂ by νQ(x[i]) := x[i+1], νQ(α[i]) := α[i+1], νQ(µ∗[i]) := µ∗[i+1] for all i ∈ Z, x ∈ Q0, α ∈ Q1, and maxi- mal paths µ in (Q, I). (3) Ker Φ̂ is equal to the ideal Î defined by the full commutativity rela- tions on Q̂ and the zero relations µ = 0 for those paths µ of Q̂ for which there is no path t̂(µ) νQ(ŝ(µ)). (Therefore note that if a path αn · · ·α1 is in I, then α [i] n · · ·α [i] 1 is in Î for all i ∈ Z.) Let R be a locally bounded category. A morphism f : x → y in R1 is called a maximal nonzero morphism if f 6= 0 and fg = 0, hf = 0 for all g ∈ radR(z, x), h ∈ radR(y, z), z ∈ R0. Lemma 1.4. Let A be an algebra and x[i], y[j] ∈ Â0. Then there exists a maximal nonzero morphism in Â(x[i], y[j]) if and only if y[j] = νA(x[i]). Proof. This follows from the fact that Â(-, x[i+1]) ∼= DÂ(x[i], -) for all i ∈ Z, x ∈ A0. Lemma 1.5. Let A be an algebra. Then the actions of φνA and νAφ coincide on the objects of Â for all φ ∈ Aut(Â). Proof. Let x[i] ∈ Â0. Then there is a maximal nonzero morphism in Â(x[i], νA(x[i])) by Lemma 1.4. Since φ is an automorphism of Â, there is a maximal nonzero morphism in Â(φ(x[i]), φ(νA(x[i]))). Hence φ(νA(x[i])) = νA(φ(x[i])) by the same lemma. The following is immediate by the lemma above. Proposition 1.6. Let A be an algebra, n an integer, and φ an automor- phism of Â. Then the following are equivalent: (1) φ is an automorphism with jump n; (2) φ(Ai) = A[i+n] for some integer i; (3) φ(Aj) = A[j+n] for all integers j; and (4) φ = σνnA for some automorphism σ of Â with jump 0. Remark 1.7. Let A be an algebra. H. Asashiba, M. Kimura 151 (1) In Skowroński [10, 11] an automorphism φ of Â is called rigid if φ(A[j]) = A[j] for all j ∈ Z. Hence φ is rigid if and only if it is an automorphism with jump 0 by the proposition above. Therefore for an integer n, φ is an automorphism with jump n if and only if φ = σνnA for some rigid automorphism σ of Â. (2) Noting this fact we see by [11, Theorem 4.7] that the class of self- injective algebras of Euclidean type contains a lot of generalized multifold extensions of piecewise hereditary algebras of tree type. In the sequel, we always assume that n is a positive integer when we consider a morphism with jump n. 1.2. Derived equivalences and tilting subcategories Let R be a locally bounded category and φ ∈ Aut(R). Then φ induces an equivalence φ(-) : modR → modR defined by φM := M ◦ φ−1 : R → modk for all M ∈ modR. In particular for R(-, x) with x ∈ R, we have φ(R(-, x)) = R(φ−1(-), x) ∼= R(-, φx), and the last isomorphism is given by φ itself. Thus the identification φ(R(-, x)) = R(-, φx) depends on φ, and the subset {R(-, x) \| x ∈ R} of prjR is not 〈φ(-)〉-stable in a strict sense. This makes it difficult to give explicitly a complete set of representatives of isoclasses of indecomposable objects of Kb(prjR) which is 〈Kb(φ(-))〉- stable. To avoid this difficulty we used in [2] the formal additive hull addR ([5, 2.1 Example 8]) of R defined below instead of prjR. Definition 1.8. Let R be a locally bounded category. The formal additive hull addR of R is a category defined as follows. • (addR)0 := { ⊕n i=1 xi := (x1, . . . , xn) \| n ∈ N, x1, . . . , xn ∈ R0}; • For each x = ⊕m i=1 xi, y = ⊕m j=1 yi ∈ (addR)0, (addR)(x, y) := {(µj,i)j,i \| µj,i ∈ R(xi, yj) for all i = 1, . . . ,m, j = 1, . . . , n}; and • The composition is given by the matrix multiplication. We regard that R is contained in addR by the embedding (f : x → y) 7→ ((f) : (x) → (y)) for all f in R. Remark 1.9. Let R and φ be as above. 152 Derived equivalence classification (1) Define a functor ηR : addR → prjR by (x1, . . . , xn) 7→ R(-, x1) ⊕ · · ·⊕R(-, xn) and (µji)j,i 7→ (R(-, µji))j,i. Then ηR is an equivalence, called the Yoneda equivalence. (2) Let F : R → S be a functor of locally bounded categories. Then F naturally induces functors addF : addR → addS and F̃ := Kb(addF ) : Kb(addR) → Kb(addS), which are isomorphisms if F is an isomorphism. Namely, addF is defined by (x1, . . . , xn) 7→ (Fx1, . . . , Fxn) and (µji) 7→ (Fµji) for all objects (x1, . . . , xn) and all morphisms (µji) in addR; and F̃ is defined by addF component- wise. Further if G : S → T is a functor of locally bounded categories, then we have (GF )̃ = G̃F̃ . (3) The automorphism φ acts on Kb(addR) as φ̃, and φKb(ηR)(X �) ∼= Kb(ηR)(φ̃(X �)) for all X � ∈ Kb(addR). We cite the following from [2, Proposition 5.1] which follows from Keller [6] (Cf. Rickard [8], [1, Proposition 1.1]). Proposition 1.10. Let R and S be locally bounded categories. Then the following are equivalent: (1) There is a triangle equivalence D(ModS) → D(ModR); and (2) There is a full subcategory E of Kb(addR) such that (a) Kb(addR)(T,U [n]) = 0 for all T,U ∈ E and all n 6= 0; (b) R is contained in the smallest full triangulated subcategory of Kb(addR) containing E that is closed under direct summands and isomorphisms; and (c) E is isomorphic to S. Definition 1.11. We say that locally bounded categories R and S are derived equivalent if one of the equivalent conditions above holds. In (2) the triple (R,E, S) is called a tilting triple and E ⊆ Kb(addR) is called a tilting subcategory for R. Theorem 1.5 in [1] is interpreted as follows. Theorem 1.12. If (A,E,B) is a tilting triple of locally bounded categories with an isomorphism ψ : E → B, then (Â, Ê, B̂) is also a tilting triple with the isomorphism ψ̂ : Ê → B̂, where Ê is isomorphic to (and identified with) the full subcategory of Kb(add Â) consisting of the (1[n])̃ (T ) with T ∈ E, n ∈ Z. H. Asashiba, M. Kimura 153 For a group G acting on a category S we say that a subclass E of the objects of S is G-stable (resp. G-stable up to isomorphisms) if gx ∈ E (resp. if gx is isomorphic to some object in E) for all g ∈ G and x ∈ E. Proposition 1.13. Let (A,E,B) be a tilting triple of locally bounded categories with an isomorphism ψ : E → B, g an automorphism of Â and h an automorphism of B̂. Then Â/〈g〉 is derived equivalent to B̂/〈h〉 if (1) g is of infinite order and 〈g〉 acts freely on Â; (2) Ê is 〈g̃〉-stable; and (3) The following diagram commutes: Ê ψ̂ −−−−→ B̂ g̃ y yh Ê −−−−→ ψ̂ B̂. Remark 1.14. Let E be a tilting subcategory for a locally bounded categoryR andG a group acting on R. IfE is G-stable up to isomorphisms, then there exists a tilting subcategory E′ for R such that E ∼= E′ and E′ is G-stable (see [1, Remark 3.2] and [2, Lemma 5.3.3 and Remark 5.3(2)]). 2. Reduction to hereditary tree algebras Let Q be a quiver. We denote by Q̄ the underlying graph of Q, and call Q finite if both Q0 and Q1 are finite sets. Each automorphism of Q is regarded as an automorphism of Q̄ preserving the orientation of Q, thus Aut(Q) can be regarded as a subgroup of Aut(Q̄). Suppose now that Q is a finite oriented tree. Then it is also known that Aut(Q) 6 Aut0(Q̄) := {f ∈ Aut(Q̄) \| f(x) = x for some x ∈ Q0}. We say that Q is an admissibly oriented tree if Aut(Q) = Aut0(Q̄). We quote the following from [3, Lemma 4.1]: Lemma 2.1. For any finite tree T there exists an admissibly oriented tree Q with a unique source such that Q̄ = T . We cite the following from [3, Lemma 5.4]. Lemma 2.2. Let A be a piecewise hereditary algebra of type Q for an admissibly oriented tree Q. Then there is a tilting triple (A,E, kQ) such that E is 〈φ̃〉-stable up to isomorphisms for all φ ∈ Aut(A). 154 Derived equivalence classification By the following proposition we can reduce the derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type to the corresponding problem of generalized multifold extensions of hereditary tree algebras. The second statement also enables us to compare the generalized multifold extension and a twisted version corresponding to it using the repetitive category of the common hereditary algebra. Proposition 2.3. Let A be a piecewise hereditary algebra of tree type Q̄ for an admissibly oriented tree Q, and n a positive integer. Then we have the following: (1) For any φ ∈ Aut(Â) with jump n, there exists some ψ ∈ Aut(k̂Q) with jump n such that Â/〈φ〉 is derived equivalent to k̂Q/〈ψ〉; and (2) If we set φ′ := νnAφ̂0 ∈ Aut(Â), where φ0 := (1[0])−1ν−n A φ1[0], then there exists some ψ′ ∈ Aut(k̂Q) with jump n such that Â/〈φ′〉 is derived equivalent to k̂Q/〈ψ′〉, and that the actions of ψ and ψ′ coincide on the objects of k̂Q. Proof. (1) We set ν := νA and φi := (1[i])−1ν−n A φ1[i] ∈ Aut(A) for all i ∈ Z. By Lemma 2.2, there exists a tilting triple (A,E,kQ) with an isomorphism ζ : E → kQ such that E is 〈η̃〉-stable up to isomorphisms for all η ∈ Aut(A). In particular, E is 〈φ̃i〉-stable up to isomorphisms for all i ∈ Z. Then (Â, Ê, k̂Q) is a tilting triple with the isomorphism ζ̂ by Theorem 1.12 and the following holds. Claim 1. Ê is 〈φ̃〉-stable up to isomorphisms. Indeed, for each T ∈ E0 and i ∈ Z, we have φ̃(1[i])̃ (T ) = (νnν−nφ1[i])̃ (T ) = (νn1[i]φi)̃ (T ) = (1[i+n])̃ φ̃i(T ). (2.1) Since E is 〈φ̃i〉-stable up to isomorphisms, we have φ̃i(T ) ∼= T ′ for some T ′ ∈ E, and hence φ̃((1[i])̃ (T )) ∼= (1[i+n])̃ (T ′) ∈ Ê, as desired. By Remark 1.14, we have a 〈φ̃〉-stable tilting subcategory Ê′ and an isomorphism θ : Ê′ ∼ → Ê. Therefore by Proposition 1.13 Â/〈φ〉 and Ê′/〈φ̃〉 are derived equivalent. If we set ψ := (ζ̂θ)φ̃(ζ̂θ)−1, then (2.1) shows that ψ is an automorphism with jump n, and that Ê′/〈φ̃〉 ∼= k̂Q/〈ψ〉. Hence Â/〈φ〉 and k̂Q/〈ψ〉 are derived equivalent. H. Asashiba, M. Kimura 155 (2) Note that φ′ is also an automorphism with jump n. By the same argument we see that Ê is also 〈φ̃′〉-stable up to isomorphisms; there exists a 〈φ̃′〉-stable tilting subcategory Ê′′ and an isomorphism θ′ : Ê′′ ∼ → Ê; and Â/〈φ′〉 and Ê′′/〈φ̃′〉 are derived equivalent. Set ψ′ := (ζ̂θ′)φ̃′(ζ̂θ′)−1, then ψ′ is an automorphism with jump n, Ê′′/〈φ̃′〉 ∼= k̂Q/〈ψ′〉, and Â/〈φ′〉 and k̂Q/〈ψ′〉 are derived equivalent. Now for i = 0 the equality (2.1) shows that φ̃(1[0])̃ (T ) = (1[n])̃ φ̃0(T ) for all T ∈ E0. Since φ′ 0 = φ0, the same calculation shows that φ̃′(1[0])̃ (T ) = (1[n])̃ φ̃0(T ) for all T ∈ E0. Thus the actions of φ̃ and φ̃′ coincide on the objects of E[0], which shows that the actions of ψ and ψ′ coincide on the objects of kQ[0]. Hence by Lemma 1.5 their actions coincide on the objects of k̂Q. Indeed, ψ(x[i]) = ψνi(x[0]) = νiψ(x[0]) = νiψ′(x[0]) = ψ′νi(x[0]) = ψ′(x[i]) for all x ∈ Q0 and i ∈ Z. 3. Hereditary tree algebras Remark 3.1. Let Q be an oriented tree. (1) We may identify k̂Q = kQ̂/Î as stated in Lemma 1.3, and we denote by µ the morphism µ+ Î in k̂Q for each morphism µ in kQ̂. (2) Let x, y ∈ Q̂0. Since Î contains full commutativity relations, we have dimk k̂Q(x, y)6 1, and in particular Q̂ has no double arrows. (3) Let α : x → y be in Q̂1 and φ ∈ Aut(k̂Q). Then there exists a unique arrow φx → φy in Q̂, which we denote by (π̂φ)(α), and we have φ(α) = φα(π̂φ)(α) ∈ k̂Q(φx, φy) for a unique φα ∈ k × := k \ {0}. This defines an automorphism π̂φ of Q̂, and thus a group homomorphism π̂ : Aut(k̂Q) → Aut(Q̂). (4) Similarly, let α : x → y be in Q1 and ψ ∈ Aut(kQ). Then there exists a unique arrow ψx → ψy in Q, which we denote by (πψ)(α). This defines an automorphism πψ of Q, and thus a group homomorphism π : Aut(kQ) → Aut(Q). We cite the following from [3, Proposition 7.4]. Proposition 3.2. Let R be a locally bounded category, and g, h auto- morphisms of R acting freely on R. If there exists a map ρ : R0 → k × such that ρ(y)g(f) = h(f)ρ(x) for all morphisms f : x → y in R, then R/〈g〉 ∼= R/〈h〉. 156 Derived equivalence classification Definition 3.3. (1) For a quiver Q = (Q0, Q1, s, t) we set Q[Q−1 1 ] to be the quiver Q[Q−1 1 ] := (Q0, Q1 ⊔ {α−1 \| α ∈ Q1}, s′, t′), where s′\|Q1 := s, t′\|Q1 := t, s′(α−1) := t(α) and t′(α−1) := s(α) for all α ∈ Q1. A walk in Q is a path in Q[Q−1 1 ]. (2) Suppose that Q is a finite oriented tree. Then for each x, y ∈ Q0 there exists a unique shortest walk from x to y in Q, which we denote by w(x, y). If w(x, y) = αεn n · · ·αε1 1 for some α1, · · · , αn ∈ Q1 and ε1, . . . , εn ∈ {1,−1}, then we define a subquiver W (x, y) of Q by W (x, y) := (W (x, y)0,W (x, y)1, s ′, t′), where W (x, y)0 := {s(αi), t(αi) \| i = 1, . . . , n}, W (x, y)1 := {α1, . . . , αn}, and s′, t′ are restrictions of s, t to W (x, y)1, respectively. Since Q is an oriented tree, w(x, y) is uniquely recovered by W (x, y). Therefore we can identify w(x, y) with W (x, y), and define a sink and a source of w(x, y) as those in W (x, y). The following is a central part of the proof of the main result. Proposition 3.4. Let Q be a finite oriented tree and φ, ψ automorphisms of k̂Q acting freely on k̂Q. If the actions of φ and ψ coincide on the objects of k̂Q, then there exists a map ρ : (Q̂0 =) k̂Q0 → k × such that ρ(y)ψ(f) = φ(f)ρ(x) for all morphisms f : x → y in k̂Q. Hence in particular, k̂Q/〈φ〉 is isomorphic to k̂Q/〈ψ〉. Proof. Assume that the actions of φ, ψ ∈ Aut(k̂Q) coincides on the objects of k̂Q. Then φ and ψ induce the same quiver automorphism q = π̂φ = π̂ψ of Q̂, and there exist (φα) α∈Q̂1 , (ψα) α∈Q̂1 ∈ (k×)Q̂1 such that for each α ∈ Q̂1 we have φ(α) = φαq(α), ψ(α) = ψαq(α). For each path λ = αn · · ·α1 in Q̂ with α1, . . . , αn ∈ Q̂1 we set φλ := φαn · · ·φα1 . Then we have φ(λ) = φλq(λ), where q(λ) := q(αn) · · · q(α1) because φ(αn) · · ·φ(α1) = φαn · · ·φα1q(αn) · · · q(α1). To show the statement we may assume that ψα = 1 for all α ∈ Q̂1. Since for each x, y ∈ Q̂0 the morphism space k̂Q(x, y) is at most 1- dimensional and has a basis of the form µ for some path µ, it is enough H. Asashiba, M. Kimura 157 to show that there exists a map ρ : Q̂0 → k × satisfying the following condition: ρ(v[j]) = φβρ(u[i]) for all β : u[i] → v[j] in Q̂1. (3.1) We define a map ρ as follows: Fix a maximal path µ : y x in Q. Then x is a sink and y is a source in Q. We can write µ as µ = αl · · ·α1 for some α1, . . . , αl ∈ Q1. First we set ρ(x[0]) := 1. By induction on 0 6 i ∈ Z we define ρ(x[i]) and ρ(x[−i]) by the following formulas: ρ(x[i+1]) := φµ[i+1]φµ∗[i]ρ(x[i]), (3.2) ρ(x[i−1]) := φ−1 µ∗[i−1]φ −1 µ[i]ρ(x[i]). (3.3) Now for each i ∈ Z and u ∈ Q0 if w(u, x) = βεm m · · ·βε1 1 for some β1, . . . , βm ∈ Q1 and ε1, . . . , εm ∈ {1,−1}, then we set ρ(u[i]) := φ−ε1 β [i] 1 · · ·φ−εm β [i] m ρ(x[i]). (3.4) We have to verify the condition (3.1). Case 1. β = α[i] : u[i] → v[i] for some i ∈ Z, and α : u → v in Q1. Since Q is an oriented tree, we have either w(u, x) = w(v, x)α or w(v, x) = w(u, x)α−1. In either case we have ρ(v[i])=φα[i]ρ(u[i]) by the formula (3.4). Case 2. Otherwise, we have β = λ∗[i] : u[i] → v[i+1] for some maximal path λ : v u in Q and i ∈ Z. In this case the condition (3.1) has the following form: ρ(v[i+1]) = φλ∗[i]ρ(u[i]). (3.5) Two paths are said to be parallel if they have the same source and the same target. We prepare the following for the proof. Claim 2. If ζ and η are parallel paths in Q̂, then we have φζ = φη. Indeed, since ζ − η ∈ Î, we have φ(ζ) = φ(η), which shows φζq(ζ) = φηq(η). Here we have q(ζ) = ψ(ζ) = ψ(η) = q(η), and ψ(ζ) 6= 0 because ζ 6= 0. Hence φζ = φη, as required. We now set d(a, b) to be the number of sinks in w(a, b) for all a, b ∈ Q0. By induction on d(y, v) we show that the condition (3.5) holds. Note that both v and y (resp. u and x) are sources (resp. sinks) in Q. 158 Derived equivalence classification y x y x Q[i] ... Q[i+1] ... v1 z2 v1 z2 z1 u1 z1 u1 v u v u µ // νm )) ν′ 0 // ν2 "" ν1 55 ν3 ;; λ // ν0 // µ∗[i] // (ν0ν)∗[i] 77 (ν0ν1)∗[i] '' λ∗[i] // µ // νm ** ν′ 0 // ν2 ## ν1 55 ν3 :: λ // ν0 // Figure 1. Assume d(y, v) = 0. Then y = v because these are sources in Q. By formulas (3.4) and (3.2) we have ρ(v[i+1]) = ρ(y[i+1]) = φ−1 α [i+1] 1 · · ·φ−1 α [i+1] l ρ(x[i+1]) = φµ∗[i]ρ(x[i]). If u = x, then λ = µ and hence φµ∗[i]ρ(x[i]) = φλ∗[i]ρ(u[i]). Thus (3.5) holds. If u 6= x, then φµ∗[i]φµ[i] = φλ∗[i]φλ[i] by Claim 2. Since Q is an oriented tree, we have w(u, x) = µλ−1, and ρ(u[i]) = φλ[i]φ−1 µ[i]ρ(x[i]). Therefore ρ(v[i+1]) = φµ∗[i]ρ(x[i]) = φλ∗[i]φλ[i]φ−1 µ[i]ρ(x[i]) = φλ∗[i]ρ(u[i]), and (3.5) holds. Assume d(y, v) > 1. Then we can write w(y, v) = ν−1 1 ν2 · · · ν−1 m−1νm for some paths ν1, . . . , νm of length at least 1 and m > 2. Set z1 := t(ν2), z2 := s(ν2). Then z1 is a sink and z2 is a source in w(y, v). Since Q is a tree, there exists a unique maximal path of the form ν0ν2ν ′ 0 : v1 u1 in Q for some paths ν0, ν ′ 0. We set ν := ν2ν ′ 0. (See Figure 1, where we omitted the notations [i], [i + 1] for paths in Q[i], Q[i+1], respectively.) Since d(v1, y) = d(v, y) − 1, we have ρ(v [i+1] 1 ) = φ(ν0ν)∗[i]ρ(u [i] 1 ) (3.6) H. Asashiba, M. Kimura 159 by induction hypothesis. Since the paths ν[i+1](ν0ν)∗[i] and ν [i+1] 1 (ν0ν1)∗[i] are parallel, we have φν[i+1]φ(ν0ν)∗[i] = φ ν [i+1] 1 φ(ν0ν1)∗[i] (3.7) by Claim 1. Further by the result of Case 1 we have ρ(v[i+1]) = φ−1 ν [i+1] 1 φν[i+1]ρ(v [i+1] 1 ). (3.8) It follows from (3.6), (3.7) and (3.8) that ρ(v[i+1]) = φ(ν0ν1)∗[i]ρ(u [i] 1 ). (If u1 = u, then ν0ν1 = λ and this already gives (3.5).) Again by the result of Case 1 we have ρ(u [i] 1 ) = φ(ν0ν1)[i]φ−1 λ[i]ρ(u[i]). Since the paths λ∗[i]λ[i] and (ν0ν1)∗[i](ν0ν1)[i] are parallel, we have φλ∗[i]φλ[i] = φ(ν0ν1)∗[i]φ(ν0ν1)[i] by Claim 1. The last three equalities give (3.5). 4. Main result Theorem 4.1. Let A be a piecewise hereditary algebra of tree type and φ an automorphism of Â with jump n. Then Â/〈φ〉 and Tnφ0 (A) are derived equivalent, where we set φ0 := (1[0])−1ν−nφ1[0]. Proof. Let T be the tree type of A. Then by Lemma 2.1 there exists an admissibly oriented tree Q with Q̄ = T . We set φ′ := νnAφ̂0 (= φ̂0ν n A). Then Tnφ0 (A) = Â/〈φ′〉. By Proposition 2.3(2) there exist some ψ,ψ′ ∈ Aut(k̂Q) both with jump n such that Â/〈φ〉 (resp. Â/〈φ′〉) is derived equivalent to k̂Q/〈ψ〉 (resp. k̂Q/〈ψ′〉), and the actions of ψ and ψ′ coincide on the objects of k̂Q. Then by Proposition 3.4 we have k̂Q/〈ψ〉 ∼= k̂Q/〈ψ′〉. Hence Â/〈φ〉 and Tnφ0 (A) are derived equivalent. Definition 4.2. Let Λ be a generalized n-fold extension of a piecewise hereditary algebra A of tree type T , say Λ = Â/〈φ〉 for some φ ∈ Aut(A) with jump n. Further let Q be an admissibly oriented tree with Q̄ = T . 160 Derived equivalence classification Then by Proposition 2.3 there exists ψ ∈ Aut(k̂Q) with jump n such that Â/〈φ〉 is derived equivalent to k̂Q/〈ψ〉. We define the (derived equivalence) type type(Λ) of Λ to be the triple (T, n, π(ψ0)), where ψ0 := (1[0])−1ν−n kQψ1 [0] and π(ψ0) is the conjugacy class of π(ψ0) in Aut(T ). type(Λ) is uniquely determined by Λ. By Theorem 4.1, we can extend the main theorem in [3] as follows. Theorem 4.3. Let Λ, Λ′ be generalized multifold extensions of piecewise hereditary algebras of tree type. Then the following are equivalent: (i) Λ and Λ′ are derived equivalent. (ii) Λ and Λ′ are stably equivalent. (iii) type(Λ) = type(Λ′). Finally we pose a question concerning a refinement of Theorem 4.1. Question. Under the setting of Theorem 4.1, when are the algebras Â/〈φ〉 and Tnφ0 (A) isomorphic? By Proposition 3.4 this is affirmative if A is hereditary. Acknowledgements This work is partially supported by Grant-in-Aid for Scientific Research (C) 21540036 from JSPS. References [1] H. Asashiba, A covering technique for derived equivalence, J. Algebra 191, (1997) 382–415. [2] H. Asashiba, The derived equivalence classification of representation-finite selfin- jective algebras, J. of Algebra 214, (1999) 182–221. [3] H. Asashiba, Derived and stable equivalence classification of twisted multifold extensions of piecewise hereditary algebras of tree type, J. Algebra 249, (2002) 345–376. [4] I.Assem, D. Happel, Generalized tilted algebras of type An, Comm. Algebra 9 (1981), no. 20, 2101–2125. [5] P. Gabriel, A. V. Roiter, Representations of finite-dimensional algebras, Ency- clopaedia of Mathematical sciences Vol. 73, Springer, 1992. [6] B. Keller, Deriving DG categories, Ann. scient. Éc. Norm. Sup., (4) 27 (1994), 63–102. [7] F. H. Membrillo-Hernández, Brauer tree algebras and derived equivalence, J. Pure Appl. Algebra 114 (1997), no. 3, 231–258. H. Asashiba, M. Kimura 161 [8] J. Rickard, Morita theory for derived categories, J. London Math. Soc., 39 (1989), 436–456. [9] J. Rickard, Derived categories and stable equivalence, J. Pure and Appl. Alg. 61 (1989), 303–317. [10] A. Skowroński, Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177–199. [11] A. Skowroński, Selfinjective algebras: finite and tame type, In Trends in Represen- tation Theory of Algebras and Related Topics, Contemporary Mathematics 406, Amer. Math. Soc., Providence, RI, 2006, 169–238. Contact information Hideto Asashiba, Mayumi Kimura Faculty of Science, Shizuoka University, 836 Ohya, Suruga-ku, Shizuoka, 422-8529, Japan E-Mail(s): asashiba.hideto@shizuoka.ac.jp, f5144005@ipc.shizuoka.ac.jp Received by the editors: 25.04.2013 and in final form 25.05.2013.

Derived equivalence classification of generalized multifold extensions of piecewise hereditary algebras of tree type

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