Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3

Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3

We prove that there exist nonstandard K-linear triangulated categories with finitely many indecomposable objects and Calabi-Yau dimension one over an arbitrary algebraically closed field K of characteristic 3, using deformed preprojective algebras of generalized Dynkin type.

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Date:2007
Main Authors: Bialkowski, J., Skowronski, A.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2007
Series:Algebra and Discrete Mathematics
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/152362
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Cite this:Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 / J. Bialkowski, A. Skowronski // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 3. — С. 27–37. — Бібліогр.: 14 назв. — англ.

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spelling irk-123456789-1523622019-06-11T01:25:10Z Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 Bialkowski, J. Skowronski, A. We prove that there exist nonstandard K-linear triangulated categories with finitely many indecomposable objects and Calabi-Yau dimension one over an arbitrary algebraically closed field K of characteristic 3, using deformed preprojective algebras of generalized Dynkin type. 2007 Article Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 / J. Bialkowski, A. Skowronski // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 3. — С. 27–37. — Бібліогр.: 14 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16D50, 16G60, 18G10. http://dspace.nbuv.gov.ua/handle/123456789/152362 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 prove that there exist nonstandard K-linear triangulated categories with finitely many indecomposable objects and Calabi-Yau dimension one over an arbitrary algebraically closed field K of characteristic 3, using deformed preprojective algebras of generalized Dynkin type.
format Article
author Bialkowski, J.
Skowronski, A.
spellingShingle Bialkowski, J.
Skowronski, A.
Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3
Algebra and Discrete Mathematics
author_facet Bialkowski, J.
Skowronski, A.
author_sort Bialkowski, J.
title Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3
title_short Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3
title_full Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3
title_fullStr Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3
title_full_unstemmed Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3
title_sort nonstandard additively finite triangulated categories of calabi-yau dimension one in characteristic 3
publisher Інститут прикладної математики і механіки НАН України
publishDate 2007
url http://dspace.nbuv.gov.ua/handle/123456789/152362
citation_txt Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 / J. Bialkowski, A. Skowronski // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 3. — С. 27–37. — Бібліогр.: 14 назв. — англ.
series Algebra and Discrete Mathematics
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AT skowronskia nonstandardadditivelyfinitetriangulatedcategoriesofcalabiyaudimensiononeincharacteristic3
first_indexed 2025-07-13T02:54:55Z
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fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2007). pp. 27 – 37 c© Journal “Algebra and Discrete Mathematics” Nonstandard additively finite triangulated categories of Calabi-Yau dimension one in characteristic 3 Jerzy Bia lkowski and Andrzej Skowroński Communicated by D. Simson Abstract. We prove that there exist nonstandard K-linear triangulated categories with finitely many indecomposable objects and Calabi-Yau dimension one over an arbitrary algebraically closed field K of characteristic 3, using deformed preprojective al- gebras of generalized Dynkin type. Throughout the paper K denotes an algebraically closed field. By a triangulated category we mean a small K-linear triangulated category T with split idempotents and finite dimensional morphism spaces. Recall that a triangulated category T admits an autoequivalence T : T → T (translation of T ) and a collection of morphisms X u −→ Y v −→ Z w −→ T X (trangulation of T ) satisfying axioms (TR1) – (TR4) (see [9]). Impor- tant examples of triangulated categories of algebraic nature are provided by the derived categories Db(mod A) of bounded complexes of finite di- mensional modules over finite dimensional K-algebras A and the stable module categories mod Λ of finite dimensional selfinjective (Frobenius) K-algebras Λ. For basic background on triangulated categories we refer to [9] and [11]. Following [6], [12], a Serre functor of a triangulated category T is an autoequivalence ν : T → T together with natural isomorphisms D HomT (X, ?) ∼ −→ HomT (?, νX) for all objects X of T , where D is Supported by the Polish Scientific Grant KBN No. 1 P03A 018 27. 2000 Mathematics Subject Classification: 16D50, 16G60, 18G10. Key words and phrases: triangulated category, Calabi-Yau category, prepro- jective algebra. Jo u rn al A lg eb ra D is cr et e M at h .28 Additively finite triangulated categories the duality HomK(−, K). A triangulated category T is called Calabi- Yau if there exists an integer d ≥ 1 such that the iteration Td of the shift functor T of T is a Serre functor of T . Then T is called d-Calabi-Yau and the smallest d with this property is the Calabi-Yau dimension of T . We refer to [7] for a complete description of Calabi-Yau stable module categories of tame symmetric algebras and their Calabi-Yau dimensions. Examples of triangulated categories of Calabi-Yau dimension 2 are pro- vided by the stable module categories of the preprojective algebras of generalized Dynkin type (see [3], [4], [8]). In this paper, we are concerned with the structure of additively fi- nite triangulated categories, that is, triangulated categories with finitely many isomorpism classes of indecomposable objects. It is known that every additively finite triangulated category has a Serre functor (see [1, Theorem 1.1.1]). Fundamental examples of additively finite triangulated categories are provided by the stable module categories modΛ of all self- injective algebras of finite representation type (we refer to [13] for descrip- tion of these categories). Moreover, the authors gave in [5] necessary and sufficent conditions for these categories to be Calabi-Yau. In general, it follows from [14] that if T is an additively finite triangulated K-category then the Auslander-Reiten quiver ΓT of T is of the form Z∆/G, where ∆ is a Dynkin quiver of type An, Dn, E6, E7 or E8, and G is a weakly admissible group of automorphisms of the translation quiver Z∆. More- over, such a triangulated category T is called standard if T is K-linearly equivalent to an orbit category Db(mod K∆)/H, where K∆ is the path algebra of ∆ and H is an automorphism group of Db(modK∆). In [1], C. Amiot proved that, for most groups G, the triangulated categories T with ΓT ∼= Z∆/G are standard. Moreover, C. Amiot gave in [1] sufficient conditions for the category proj Λ of finite dimensional projective mod- ules over a selfinjective algebra Λ to be a triangulated category. Then, invoking the main result of [4], C. Amiot proved in [1] that the class of 1- Calabi-Yau additively finite triangulated categories T coincides with the class of the categories projP f (∆) over deformed preprojective algebras P f (∆) of generalized Dynkin types An, Dn, Ln, E6, E7, E8, described bellow. This allowed to construct nonstandard 1-Calabi-Yau additively finite triangulated categories over an arbitrary algebraically closed field K of characteristic 2. The main aim of this paper is to show existence of nonstandard 1- Calabi-Yau additively finite triangulated categories over an arbitrary al- gebraically closed field K of characteristic 3. We also note that the nonstandard stable module categories of self- injective algebras of finite type there exist only in characteristic 2 (see [13]). Jo u rn al A lg eb ra D is cr et e M at h .J. Bia lkowski, A. Skowroński 29 We recall now the deformed preprojective algebras of generalized Dynkin type introduced in [4]. Let ∆ be a generalized Dynkin graph of type An(n ≥ 1), Dn(n ≥ 4), En(n = 6, 7, 8) or Ln(n ≥ 1). Let Q∆ be the following associated quiver: ∆ = An : (n ≥ 1) 0 a0 // 1 ā0 oo a1 // 2 ā1 oo ... n − 2 an−2 // n − 1 ān−2 oo ∆ = Dn : (n ≥ 4) 0 a0 ��= == == == 2 ā0 ^^======= ā1���� �� �� � a2 // 3 ā2 oo ... n − 2 an−2 // n − 1 ān−2 oo 1 a1 @@������� ∆ = En : (n = 6, 7, 8) 0 a0 �� 1 a1 // 2 ā1 oo a2 // 3 ā2 oo a3 // ā0 OO 4 ā3 oo ... n − 2 an−2 // n − 1 ān−2 oo ∆ = Ln : (n ≥ 1) 0@ABGFEε=ε̄ �� a0 // 1 ā0 oo a1 // 2 ā1 oo ... n − 2 an−2 // n − 1 ān−2 oo . The preprojective algebra P (∆) associated to the graph ∆ is the bound quiver algebra KQ∆/I∆, where KQ∆ is the path algebra of Q∆ and I∆ is the ideal in KQ∆ generated by the relations of the form ∑ a,sa=i aā, i vertices of Q∆, where sa denotes the source of an arrow a of Q∆. The preprojective al- gebra P (∆) is a finite dimensional selfinjective algebra and its Nakayama permutation is identity for ∆ = A1, D2n, E7, E8 and Ln, and of order 2 in all other cases. Further, consider the associated algebra R(∆) as follows R(An) = K; R(Dn) = K〈x, y〉/(x2, y2, (x + y)n−2); R(En) = K〈x, y〉/(x2, y3, (x + y)n−3); R(Ln) = K[x]/(x2n). Moreover, choose the exceptional vertex of the quiver Q∆ of P (∆) as follows 0, for ∆ = An or Ln; 2, for ∆ = Dn; 3, for ∆ = En. Jo u rn al A lg eb ra D is cr et e M at h .30 Additively finite triangulated categories We note that if e is the idempotent of P (∆) corresponding to the ex- ceptional vertex then R(∆) is isomorphic to eP (∆)e, and hence is local, finite dimensional and selfinjective. Let f be an element of the square rad2 R(∆) of the Jacobson radical of R(∆). Then the deformed prepro- jective algebra P f (∆) of generalized Dynkin type ∆ is the bound quiver algebra KQ∆/If ∆, where If ∆ is the ideal in the path algebra KQ∆ of Q∆ generated by the relations of the form ∑ a,sa=i aā, for each nonexceptional vertex i of Q∆, and a0ā0, for ∆ = An; ā0a0 + ā1a1 + a2ā2 + f(ā0a0, ā1a1), (ā0a0 + ā1a1) n−2, for ∆ = Dn; ā0a0 + ā2a2 + a3ā3 + f(ā0a0, ā2a2), (ā0a0 + ā2a2) n−3, for ∆ = En; ε2 + a0ā0 + εf(ε), ε2n, for ∆ = Ln. Therefore, P f (∆) is obtained from P (∆) by deforming the relation at the exceptional vertex of Q∆, and P f (∆) = P (∆) if f = 0. Moreover, P f (∆) is a selfinjective algebra with dimK P f (∆) = dimK P (∆) and the Cartan matrices of P f (∆) and P (∆) coincide. It is shown in [4, Theorem 1.3] that, for an algebraically closed field K of characteristic 2 and a generalized Dynkin graph ∆ other than An and L1, there exists a deformed preprojective algebra P f (∆) over K which is not isomorphic to the preprojective algebra P (∆). In such a case, projP f (∆) is a nonstandard 1-Calabi-Yau additively finite triangu- lar category (see [1, Theorem 9.3.3]). The following theorem is the main result of the paper. Theorem. Let K be an algebraically closed field of characteristic 3, ∆ a Dynkin graph En(n = 6, 7, 8) and f = y2x + (x2, y3, (x + y)n−3) ∈ R(∆). Then (i) P f (∆) is not isomorphic to P (∆). (ii) projP f (∆) is a nonstandard 1-Calabi-Yau additively finite triangu- lated category. Proof. (i) Let K be an algebraically closed field of characteristic 3, ∆ a Dynkin graph En(n = 6, 7, 8) and f = y2x + (x2, y3, (x + y)n−3) ∈ R(∆). We will show that P f (∆) is not isomorphic to P (∆). Jo u rn al A lg eb ra D is cr et e M at h .J. Bia lkowski, A. Skowroński 31 Suppose that ϕ : P f (∆) → P (∆) is an algebra isomorphism. Then ϕ is determined by the elements of the form ϕ(a0) = α (0) 0 a0 + α (0) 0 α (0) 1 a0ā2a2 + α (0) 0 α (0) 2 a0ā2a2ā0a0+ +α (0) 0 α (0) 3 a0ā2a2ā2a2 + . . . ϕ(ā0) = ᾱ (0) 0 ā0 + ᾱ (0) 0 ᾱ (0) 1 ā2a2ā0 + ᾱ (0) 0 ᾱ (0) 2 ā0a0ā2a2ā0+ +ᾱ (0) 0 ᾱ (0) 3 ā2a2ā2a2ā0 + . . . ϕ(a1) = α (1) 0 a1 + α (1) 0 α (1) 1 a1a2ā0a0ā2 + . . . ϕ(ā1) = ᾱ (1) 0 ā1 + ᾱ (1) 0 ᾱ (1) 1 a2ā0a0ā2ā1 + . . . ϕ(a2) = α (2) 0 a2 + α (2) 0 α (2) 1 a2ā0a0 + α (2) 0 α (2) 2 a2ā2a2+ +α (2) 0 α (2) 3 a2ā0a0ā2a2 + α (2) 0 α (2) 4 a2ā2a2ā0a0 + . . . ϕ(ā2) = ᾱ (2) 0 ā2 + ᾱ (2) 0 ᾱ (2) 1 ā0a0ā2 + ᾱ (2) 0 ᾱ (2) 2 ā2a2ā2+ +ᾱ (2) 0 ᾱ (2) 3 ā0a0ā2a2ā2 + ᾱ (2) 0 ᾱ (2) 4 ā2a2ā0a0ā2 + . . . ϕ(a3) = α (3) 0 a3 + α (3) 0 α (3) 1 ā0a0a3 + α (3) 0 α (3) 2 ā2a2a3+ +α (3) 0 α (3) 3 ā0a0ā2a2a3 + α (3) 0 α (3) 4 ā2a2ā0a0a3+ +α (3) 0 α (3) 5 ā2a2ā2a2a3 + . . . ϕ(ā3) = ᾱ (3) 0 ā3 + ᾱ (3) 0 ᾱ (3) 1 ā3ā0a0 + ᾱ (3) 0 ᾱ (3) 2 ā3ā2a2+ +ᾱ (3) 0 ᾱ (3) 3 ā3ā0a0ā2a2 + ᾱ (3) 0 ᾱ (3) 4 ā3ā2a2ā0a0+ +ᾱ (3) 0 ᾱ (3) 5 ā3ā2a2ā2a2 + . . . ϕ(a4) = α (4) 0 a4 + α (4) 0 α (4) 1 ā3a3a4 + α (4) 0 α (4) 2 ā3ā0a0a3a4+ +α (4) 0 α (4) 3 ā3ā2a2a3a4 + . . . ϕ(ā4) = ᾱ (4) 0 ā4 + ᾱ (4) 0 ᾱ (4) 1 ā4ā3a3 + ᾱ (4) 0 ᾱ (4) 2 ā4ā3ā0a0a3+ +ᾱ (4) 0 ᾱ (4) 3 ā4ā3ā2a2a3 + . . . , for n = 6, 7, 8, ϕ(a5) = α (5) 0 a5 + α (5) 0 α (5) 1 ā4a4a5 + . . . ϕ(ā5) = ᾱ (5) 0 ā5 + ᾱ (5) 0 ᾱ (5) 1 ā5ā4a4 + . . . , for n = 7, 8, and ϕ(a6) = α (6) 0 a6 + . . . ϕ(ā6) = ᾱ (6) 0 ā6 + . . . , for n = 8, for some parameters α (l) i , ᾱ (l) i ∈ K, with α (l) 0 , ᾱ (l) 0 non-zero, for all l ∈ {0, 1, . . . , n}, α (3) 5 = ᾱ (3) 5 = α (4) 1 = ᾱ (4) 1 = α (4) 3 = ᾱ (4) 3 = 0 for n = 6, and α (4) 3 = ᾱ (4) 3 = α (5) 1 = ᾱ (5) 1 = 0 for n = 7. Denote α = α (0) 0 ᾱ (0) 0 . Note that α (i) i ᾱ (i) i = α 6= 0, for all i ∈ {0, 1, . . . , n}. Jo u rn al A lg eb ra D is cr et e M at h .32 Additively finite triangulated categories Invoking the relation at the vertex 0, we obtain 0 = α−2ϕ (a0ā0) = a0ā0 + ( α (0) 1 + ᾱ (0) 1 ) a0ā2a2ā0+ + ( α (0) 1 ᾱ (0) 1 + α (0) 3 + ᾱ (0) 3 ) a0ā2a2ā2a2ā0 + . . . , and thus α (0) 1 + ᾱ (0) 1 = 0 = α (0) 1 ᾱ (0) 1 + α (0) 3 + ᾱ (0) 3 . Similarly, using the relation at the vertex 1, we obtain 0 = α−2ϕ (a1ā1) = a1ā1 + ( α (1) 1 + ᾱ (1) 1 ) a1a2ā0a0ā2ā1 + . . . , and thus α (1) 1 + ᾱ (1) 1 = 0. Further, invoking the relation at the vertex 2, we obtain 0 = α−2ϕ (ā1a1 + a2ā2) = ā1a1 − ᾱ (1) 1 a2ā0a0ā2a2ā2 − α (1) 1 a2ā2a2ā0a0ā2+ +a2ā2 + ( α (2) 1 + ᾱ (2) 1 ) a2ā0a0ā2+ + ( α (2) 3 + ᾱ (2) 3 + α (2) 1 ᾱ (2) 2 ) a2ā0a0ā2a2ā2+ + ( α (2) 4 + ᾱ (2) 4 + α (2) 2 ᾱ (2) 1 ) a2ā2a2ā0a0ā2 + . . . , hence α (2) 1 + ᾱ (2) 1 = α (2) 3 + ᾱ (2) 3 +α (2) 1 ᾱ (2) 2 −α (1) 1 = α (2) 4 + ᾱ (2) 4 +α (2) 2 ᾱ (2) 1 − ᾱ (1) 1 = 0 (note that a2ā2a2ā2 = ā1a1ā1a1 = 0). Applying now α (1) 1 + ᾱ (1) 1 = 0 we obtain the equality α (2) 3 + ᾱ (2) 3 + α (2) 1 ᾱ (2) 2 + α (2) 4 + ᾱ (2) 4 + α (2) 2 ᾱ (2) 1 = 0. Assume that n = 6. Using the relation at the vertex 5, we obtain 0 = α−2ϕ (ā4a4) = ā4a4 + ( α (4) 2 + ᾱ (4) 2 ) ā4ā3ā0a0a3a4 + . . . , and hence α (4) 2 + ᾱ (4) 2 = 0. Similarly, using the relation at the vertex 4, we obtain 0 = α−2ϕ (ā3a3 + a4ā4) = ā3a3 + ( α (3) 1 + ᾱ (3) 1 − α (3) 2 − ᾱ (3) 2 ) ā3ā0a0a3 + ( α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 − α (3) 2 ᾱ (3) 2 ) ā3ā0a0ā2a2a3 + ( α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 − α (3) 2 ᾱ (3) 2 ) ā3ā2a2ā0a0a3 +a4ā4 + α (4) 2 ā3ā0a0ā2a2a3 + ᾱ (4) 2 ā3ā2a2ā0a0a3 + . . . Jo u rn al A lg eb ra D is cr et e M at h .J. Bia lkowski, A. Skowroński 33 Note that we have ā3a3ā3a3 = ā4a4 = 0. Hence α (3) 1 + ᾱ (3) 1 − α (3) 2 − ᾱ (3) 2 = α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 − α (3) 2 ᾱ (3) 2 + α (4) 2 = α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 − α (3) 2 ᾱ (3) 2 + ᾱ (4) 2 = 0. Applying α (4) 2 + ᾱ (4) 2 = 0 to the above equations we obtain α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 + α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 + α (3) 2 ᾱ (3) 2 = 0 (note that the calculations are in a field K of characteristic 3). Now assume that n ∈ {7, 6}. Using the relation at the vertex 5, we obtain 0 = α−2ϕ (ā4a4 + a5ā5) = ā4a4 + ( α (4) 1 + ᾱ (4) 1 ) ā4ā3a3a4 + ( α (4) 2 + ᾱ (4) 2 − α (4) 3 − ᾱ (4) 3 ) ā4ā3ā0a0a3a4 +a5ā5 + ( α (5) 1 + ᾱ (5) 1 ) ā4ā3a3a4 + . . . = ā4a4 + a5ā5 + ( α (4) 1 + ᾱ (4) 1 + α (5) 1 + ᾱ (5) 1 ) ā4ā3a3a4 + ( α (4) 2 + ᾱ (4) 2 − α (4) 3 − ᾱ (4) 3 ) ā4ā3ā0a0a3a4 + . . . , and hence α (4) 1 + ᾱ (4) 1 + α (5) 1 + ᾱ (5) 1 = 0 = α (4) 2 + ᾱ (4) 2 − α (4) 3 − ᾱ (4) 3 (for n = 7, we have α (4) 1 + ᾱ (4) 1 = α (4) 2 + ᾱ (4) 2 = 0). Assume that n = 7. Using the relation at the vertex 4, we obtain 0 = α−2ϕ (ā3a3 + a4ā4) = ā3a3 + ( α (3) 1 + ᾱ (3) 1 ) ā3ā0a0a3 + ( α (3) 2 + ᾱ (3) 2 ) ā3ā2a2a3 + ( α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 − α (3) 2 ᾱ (3) 2 − α (3) 5 − ᾱ (3) 5 ) ā3ā0a0ā2a2a3 + ( α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 − α (3) 2 ᾱ (3) 2 − α (3) 5 − ᾱ (3) 5 ) ā3ā2a2ā0a0a3 +a4ā4 + ( α (4) 1 + ᾱ (4) 1 ) ā3ā0a0a3 + ( α (4) 1 + ᾱ (4) 1 ) ā3ā2a2a3 +α (4) 2 ā3ā0a0ā2a2a3 + ᾱ (4) 2 ā3ā2a2ā0a0a3 + . . . Note that we have ā3a3ā3a3ā3a3 = ā4a4ā4a4 = 0. Hence α (3) 1 + ᾱ (3) 1 + α (4) 1 + ᾱ (4) 1 = 0, α (3) 2 + ᾱ (3) 2 + α (4) 1 + ᾱ (4) 1 = 0, α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 − α (3) 2 ᾱ (3) 2 − α (3) 5 − ᾱ (3) 5 + α (4) 2 = 0, α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 − α (3) 2 ᾱ (3) 2 − α (3) 5 − ᾱ (3) 5 + ᾱ (4) 2 = 0. Jo u rn al A lg eb ra D is cr et e M at h .34 Additively finite triangulated categories Summing up the last two equalities and applying the equality α (4) 2 + ᾱ (4) 2 = 0 we obtain α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 + α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 + α (3) 2 ᾱ (3) 2 + α (3) 5 + ᾱ (3) 5 = 0 (again note that the calculations are in a field K of characteristic 3). Assume that n = 8. Applying the relation at the vertex 4, we obtain 0 = α−2ϕ (ā3a3 + a4ā4) = ā3a3 + ( α (3) 1 + ᾱ (3) 1 ) ā3ā0a0a3 + ( α (3) 2 + ᾱ (3) 2 ) ā3ā2a2a3 + ( α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 ) ā3ā0a0ā2a2a3 + ( α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 ) ā3ā2a2ā0a0a3 + ( α (3) 2 ᾱ (3) 2 + α (3) 5 + ᾱ (3) 5 ) ā3ā2a2ā2a2a3 +a4ā4 + ( α (4) 1 + ᾱ (4) 1 ) ā3ā0a0a3 + ( α (4) 1 + ᾱ (4) 1 ) ā3ā2a2a3 + ( α (4) 2 + α (4) 3 − α (4) 1 ᾱ (4) 1 ) ā3ā0a0ā2a2a3 + ( ᾱ (4) 2 + ᾱ (4) 3 − α (4) 1 ᾱ (4) 1 ) ā3ā2a2ā0a0a3 + ( α (4) 3 + ᾱ (4) 3 − α (4) 1 ᾱ (4) 1 ) ā3ā2a2ā2a2a3 + . . . Note that we have ā3a3ā3a3ā3a3 = ā4a4ā4a4 = 0. Hence α (3) 1 + ᾱ (3) 1 + α (4) 1 + ᾱ (4) 1 = 0, α (3) 2 + ᾱ (3) 2 + α (4) 1 + ᾱ (4) 1 = 0, α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 + α (4) 2 + ᾱ (4) 3 − α (4) 1 ᾱ (4) 1 = 0, α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 + ᾱ (4) 2 + α (4) 3 − α (4) 1 ᾱ (4) 1 = 0, α (3) 2 ᾱ (3) 2 + α (3) 5 + ᾱ (3) 5 + α (4) 3 + ᾱ (4) 3 − α (4) 1 ᾱ (4) 1 = 0. Summing up the last three equalities and applying the equality α (4) 2 + ᾱ (4) 2 − α (4) 3 − ᾱ (4) 3 = 0, we obtain α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 + α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 + α (3) 2 ᾱ (3) 2 + α (3) 5 + ᾱ (3) 5 = 0. Assume that n = {6, 7, 8}. Applying the relation at the vertex 3, we obtain 0 = α−2ϕ (ā0a0 + ā2a2 + a3ā3 + ā2a2ā2a2ā0a0) = ā0a0 + α (0) 1 ā0a0ā2a2 + ᾱ (0) 1 ā2a2ā0a0 + α (0) 1 ᾱ (0) 1 ā2a2ā0a0ā2a2 + ( α (0) 2 + ᾱ (0) 2 ) ā0a0ā2a2ā0a0 +ᾱ (0) 3 ā2a2ā2a2ā0a0 + α (0) 3 ā0a0ā2a2ā2a2 +ā2a2 + ᾱ (2) 1 ā0a0ā2a2 + α (2) 1 ā2a2ā0a0 + ( α (2) 2 + ᾱ (2) 2 ) ā2a2ā2a2 + ( α (2) 1 ᾱ (2) 2 + α (2) 4 ) ā2a2ā2a2ā0a0 + ( α (2) 2 ᾱ (2) 1 + ᾱ (2) 3 ) ā0a0ā2a2ā2a2 Jo u rn al A lg eb ra D is cr et e M at h .J. Bia lkowski, A. Skowroński 35 + ( α (2) 3 + ᾱ (2) 4 ) ā2a2ā0a0ā2a2 +α (2) 1 ᾱ (2) 1 ā0a0ā2a2ā0a0 +a3ā3 − ( α (3) 1 + ᾱ (3) 2 ) ā0a0ā2a2 − ( ᾱ (3) 1 + α (3) 2 ) ā2a2ā0a0 − ( α (3) 2 + ᾱ (3) 2 ) ā2a2ā2a2 + ( −ᾱ (3) 4 − α (3) 5 − α (3) 2 ᾱ (3) 1 ) ā2a2ā2a2ā0a0 + ( −α (3) 3 − ᾱ (3) 5 − α (3) 1 ᾱ (3) 2 ) ā0a0ā2a2ā2a2 + ( −α (3) 4 − ᾱ (3) 3 − α (3) 2 ᾱ (3) 2 ) ā2a2ā0a0ā2a2 + ( −α (3) 1 ᾱ (3) 1 − α (3) 3 − ᾱ (3) 4 ) ā0a0ā2a2ā0a0 +α4ā2a2ā2a2ā0a0 + . . . = ā0a0 + ā2a2 + a3ā3 + ( α (0) 1 + ᾱ (2) 1 − α (3) 1 − ᾱ (3) 2 ) ā0a0ā2a2 + ( ᾱ (0) 1 + α (2) 1 − ᾱ (3) 1 − α (3) 2 ) ā0a0ā2a2 + ( α (2) 2 + ᾱ (2) 2 − α (3) 2 − ᾱ (3) 2 ) ā2a2ā2a2 + ( ᾱ (0) 3 + α (2) 1 ᾱ (2) 2 + α (2) 4 − ᾱ (3) 4 − α (3) 5 − α (3) 2 ᾱ (3) 1 + α4 ) ā2a2ā2a2ā0a0 + ( α (0) 3 + α (2) 2 ᾱ (2) 1 + ᾱ (2) 3 − α (3) 3 − ᾱ (3) 5 − α (3) 1 ᾱ (3) 2 ) ā0a0ā2a2ā2a2 + ( α (0) 1 ᾱ (0) 1 + α (2) 3 + ᾱ (2) 4 − α (3) 4 − ᾱ (3) 3 − α (3) 2 ᾱ (3) 2 ) ā2a2ā0a0ā2a2 + ( α (0) 2 + ᾱ (0) 2 + α (2) 1 ᾱ (2) 1 − α (3) 1 ᾱ (3) 1 − α (3) 3 − ᾱ (3) 4 ) ā0a0ā2a2ā0a0 + . . . Note that ā2a2ā2a2ā0a0, ā0a0ā2a2ā2a2, ā2a2ā0a0ā2a2, ā0a0ā2a2ā0a0 are linearly independent, for n = 7, 8, and ā2a2ā2a2ā0a0 + ā0a0ā2a2ā2a2 + ā2a2ā0a0ā2a2 + ā0a0ā2a2ā0a0 = 0, for n = 6. Denote β = α (0) 2 + ᾱ (0) 2 + α (2) 1 ᾱ (2) 1 − α (3) 1 ᾱ (3) 1 − α (3) 3 − ᾱ (3) 4 , for n = 6, and β = 0, for n = 7, 8. Then we have ( ᾱ (0) 3 + α (2) 1 ᾱ (2) 2 + α (2) 4 − ᾱ (3) 4 − α (3) 5 − α (3) 2 ᾱ (3) 1 + α4 ) − β = 0, ( α (0) 3 + α (2) 2 ᾱ (2) 1 + ᾱ (2) 3 − α (3) 3 − ᾱ (3) 5 − α (3) 1 ᾱ (3) 2 ) − β = 0, ( α (0) 1 ᾱ (0) 1 + α (2) 3 + ᾱ (2) 4 − α (3) 4 − ᾱ (3) 3 − α (3) 2 ᾱ (3) 2 ) − β = 0. Jo u rn al A lg eb ra D is cr et e M at h .36 Additively finite triangulated categories Summing up the above three equations we obtain 0 = α4 + 3β + ( α (0) 1 ᾱ (0) 1 + α (0) 3 + ᾱ (0) 3 ) + ( α (2) 3 + ᾱ (2) 3 + α (2) 1 ᾱ (2) 2 + α (2) 4 + ᾱ (2) 4 + α (2) 2 ᾱ (2) 1 ) − ( α (3) 3 + ᾱ (3) 3 + α (3) 2 ᾱ (3) 1 + α (3) 4 + ᾱ (3) 4 + α (3) 1 ᾱ (3) 2 + α (3) 2 ᾱ (3) 2 +α (3) 5 + ᾱ (3) 5 ) = α4. This is a contradiction, because α 6= 0. This shows that P f (∆) is not isomorphic to P (∆). (ii) It follows from (i) and [1, Corollary 9.3.2] that the categories projP f (∆) and projP (∆) are not equivalent 1-Calabi-Yau categories with the same Auslander-Reiten quiver Q∆. In particular, projP (∆) is standard and projP f (∆) is nonstandard. Remark. We would like to mention that is not clear if there exist non- standard additively finite K-linear triangulated categories of Calabi-Yau dimension one over algebraically closed fields K of characteristic different from 2 and 3. References [1] C. Amiot, On the structure of triangulated categories with finitely many indecom- posables, Preprint (2006), arXiv: math. CT/0612141. [2] I. Assem, D. Simson and A. Skowroński, “Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory”, London Math. Soc. Student Texts, Vol. 65, Cambridge Univ. Press, 2006. [3] M. Auslander and I. Reiten, D Tr-periodic modules and functors, In: Represen- tation Theory of Algebras, CMS Conf. Proc. Vol. 18, Amer. Math. Soc., 1996, pp. 39-50. [4] J. Bia lkowski, K. Erdmann and A. Skowroński, Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc., 359, 2007, pp. 2625-2650. [5] J. Bia lkowski and A. Skowroński, Calabi-Yau stable module categories of finite type, Colloq. Math., 109 (2007), 257-269. [6] A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat., 53, 1989, pp. 1183-1205. [7] K. Erdmann and A. Skowroński, The stable Calabi-Yau dimension of tame sym- metric algebras, J. Math. Soc. Japan, 58, 2006, pp. 97-128. [8] K. Erdmann and N. Snashall, Preprojective algebras of Dynkin type, periodic- ity and the second Hochschild cohomology, In: Algebras and Modules II, CMS Conference Proceedings, Vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 183-193. [9] D. Happel, “Triangulated Categories in the Representation Theory of Finite Di- mensional Algebras”, London Math. Soc. Lecture Note Series, Vol. 119, Cam- bridge Univ. Press, 1988. Jo u rn al A lg eb ra D is cr et e M at h .J. Bia lkowski, A. Skowroński 37 [10] B. Keller, On triangulated orbit categories, Documenta Math., 10, 2005, pp. 551- 581. [11] A. Neeman, “Triangulated categories”, Annals of Mathematics Studies, Vol. 148, Princeton University Press, Princeton, 2001. [12] I. Reiten and M. van den Bergh, Noetherian hereditary abelian categories satisfy- ing Serre duality , J. Amer. Math. Soc., 15, 2002, pp. 295-366. [13] A. Skowroński, Selfinjective algebras: finite and tame type, In: Trends in Repre- sentation Theory of Algebras and Related Topics, Contemp. Math., 406, 2006, pp. 169-238. [14] J. Xiao and B. Zhu, Locally finite triangulated categories, J. Algebra, 290, 2005, pp. 473-490. Contact information J. Bia lkowski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland E-Mail: jb@mat.uni.torun.pl A. Skowroński Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland E-Mail: skowron@mat.uni.torun.pl Received by the editors: 31.05.2007 and in final form 01.02.2008.

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