Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line
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Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line Chao Sun1 Received: 12 March 2019 / Accepted: 9 October 2019 / © Springer Nature B.V. 2019
Abstract We use recollement and HRS-tilt to describe bounded t-structures on the bounded derived category Db (X) of coherent sheaves over a weighted projective line X of domestic or tubular type. We will see from our description that the combinatorics in the classification of bounded t-structures on Db (X) can be reduced to that in the classification of bounded t-structures on the bounded derived categories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras. Keywords Weighted projective line · Derived category · T-structure · Derived equivalence Mathematics Subject Classification (2010) Primary 14F05; Secondary 18E30
1 Introduction 1.1 Background and Aim In an attempt to give a geometric treatment of Ringel’s canonical algebras [43], Geigle and Lenzing introduced in [17] a class of noncommutative curves, called weighted projective lines, and each canonical algebra is realized as the endomorphism algebra of a tilting bundle in the category of coherent sheaves over some weighted projective line. A stacky point of view to weighted projective lines is that for a weighted projective line X defined over a field k, there is a smooth algebraic k-stack X with the projective line over k as its coarse moduli space such that cohX cohX and QcohX QcohX, where coh (resp. Qcoh) denotes the category of coherent (resp. quasi-coherent) sheaves. As an indication of the importance of the notion of weighted projective lines, a famous theorem of Happel [20] states that if A is a connected hereditary category linear over an algebraically closed field k with finite Presented by: Henning Krause. Chao Sun
[email protected] 1
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
C. Sun
dimensional morphism and extension spaces such that its bounded derived category Db (A) admits a tilting object then Db (A) is triangle equivalent to the bounded derived category of finite dimensional modules over a finite dimensional hereditary algebra over k or to the bounded derived category of coherent sheaves on a weighted projective line defined over k. The notion of t-structures is introduced by Beilinson, Bernstein and Deligne in [7] to serve as a categorical framework for defining perverse sheaves in the derived category of constructible sheaves over a stratified space. Recently, there has been a growing interest in t-structures ever since Bridgeland [12] introduced the notion of stability conditions. To give a stability condition on a triangulated category requires specifying a bounded t-structure. On the other hand, there are many works on bounded t-structures on the bounded derived category Db () of finite dimensional modules over a finite dimensional algebra in recent years. Remarkably, K¨onig and Yang proved th
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