On derived equivalence classification of gentle two-cycle algebras

Abstract

We complete the derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobiński-Malicki.

Derived categories of quasi-hereditary algebras and their derived composition series

“…We finish with a complete description of derived equivalence classes of gentle algebras of finite global dimension with three vertices and four arrows, this also follows from Bobiński [9]. In particular, the algebras A 1 and A 2 are derived unique.…”

“…Recently, Amiot confirmed other cases for gentle algebras arising from a torus with one boundary component, see [2]. Upon receiving a preliminary version of this article Grzegorz Bobiński kindly informed us that using an extension of Amiot's techniques, he is able to establish the conjecture in all cases, see [9]. Our alternative approach might nevertheless be useful to understand derived categories of gentle algebras, see also Remark 4.30.…”

“…(c) The algebra A 1 appears in several different places in the literature. We already mentioned [11]. Moreover, Burban & Drozd show that A 1 is derived equivalent to coherent sheaves over a certain non-commutative irreducible nodal cubic curve [19] -Burban also conjectured Corollary 4.7 for A 1 .…”

“…Indeed this follows from the classification of Bessenrodt & Holm [8,Example 3.7] in combination with Ladkani [49] and the definition of the AG-invariant, which detects oriented cycles with full zero relations. In particular, Proposition 1.4 provides a minimal 4 example showing that the AG-invariant is not sufficient to distinguish derived categories of gentle algebras, see [11] & [2] for further examples. (2) In algebraic geometry, Bondal & Orlov [14] showed that the derived category D b (coh X) of a smooth projective variety X with ample canonical or anticanonical bundle determines X.…”

“…(a) Proposition 1.4 is part of a conjecture by Bobiński & Malicki [11] (A 1 = Λ ′ 0 (1, 0) and A 2 = Λ 0 (2, 1) in their notation, see also the paragraph after Lemma 4.8). We also show that Proposition 1.4 allows to distinguish the derived categories of a whole family of finite dimensional algebras confirming further parts of this conjecture, see Corollaries 4.29 & 4.31.…”