Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tilting Objects via Recollements and $p$-Cycles on Weighted Projective Lines

Qiang Dong, Hongxia Zhang

TL;DR

This work develops a recollement-based approach to construct tilting objects in the stable category of vector bundles over weighted projective lines, using the explicit $p$-cycle description of line and extension bundles. Central to the method is a main tilting-gluing theorem that glues tilting data from two side categories across a recollement, provided specific Hom-vanishing conditions hold, and is complemented by reduction/insertion functors organized into ladders. The authors provide explicit $p$-cycle interpretations for line and extension bundles, enabling closed-form actions of the reduction and insertion functors, and they apply these tools to reprove the tilting cuboid and Auslander-bundle tiltings while producing new tilting objects in $\underline{\text{vect}}-\mathbb{X}(2,p_2,p_3)$. Overall, the paper deepens connections between recollement theory, ladder structures, and tilting theory in the geometry of weighted projective lines, with potential links to related areas such as singularity theory and Nakayama algebras.

Abstract

In this paper, we provide a new method for constructing tilting objects in a triangulated category via recollements. The $p$-cycle approach to exceptional curve processes significant advantages in constructing recollements and ladders, due to the existence of reduction/insertion functors. In order to construct tilting objects in the stable category of vector bundles over a weighted projective line, we give explicit expressions for line bundles and extension bundles due to the $p$-cycles constuctions. Furthermore, we provide an essential proof for tilting cuboic object and tilting objects consisting of Auslander bundles. Moreover, we construct certain new tilting objects in the stable category of vector bundles over a weighted projective line.

Tilting Objects via Recollements and $p$-Cycles on Weighted Projective Lines

TL;DR

This work develops a recollement-based approach to construct tilting objects in the stable category of vector bundles over weighted projective lines, using the explicit -cycle description of line and extension bundles. Central to the method is a main tilting-gluing theorem that glues tilting data from two side categories across a recollement, provided specific Hom-vanishing conditions hold, and is complemented by reduction/insertion functors organized into ladders. The authors provide explicit -cycle interpretations for line and extension bundles, enabling closed-form actions of the reduction and insertion functors, and they apply these tools to reprove the tilting cuboid and Auslander-bundle tiltings while producing new tilting objects in . Overall, the paper deepens connections between recollement theory, ladder structures, and tilting theory in the geometry of weighted projective lines, with potential links to related areas such as singularity theory and Nakayama algebras.

Abstract

In this paper, we provide a new method for constructing tilting objects in a triangulated category via recollements. The -cycle approach to exceptional curve processes significant advantages in constructing recollements and ladders, due to the existence of reduction/insertion functors. In order to construct tilting objects in the stable category of vector bundles over a weighted projective line, we give explicit expressions for line bundles and extension bundles due to the -cycles constuctions. Furthermore, we provide an essential proof for tilting cuboic object and tilting objects consisting of Auslander bundles. Moreover, we construct certain new tilting objects in the stable category of vector bundles over a weighted projective line.

Paper Structure

This paper contains 21 sections, 14 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Exceptional curve
  4. $p$-cycle construction
  5. Simple sheaves in $\mathcal{H}(p;x)$
  6. $p$-cycle construction for weighted projective lines
  7. $p$-cycle interpretation for line bundles
  8. $p$-cycle interpretation for extension bundles
  9. Reduction functor $\psi^j$ and insertion functor $\psi_j$
  10. The functors $\psi^j,\psi_j$ act on line bundles
  11. The reduction functor acts on extension bundles
  12. The reduction functor acts on extension bundles
  13. The insertion functor acts on extension bundles
  14. Tilting objects in a triangulated category via recollement
  15. The adjoint pair
  16. ...and 6 more sections

Key Result

Proposition 3.1

Assume that $\vec{y}$ has the normal form $\vec{y}=\sum_{i=1}^{t}k_i\vec{x}_i+k\vec{c}$. Denote by $\vec{y}^{\prime}=\sum_{i=1}^{t-1}k_i\vec{x}_i+k\vec{c}$. Then ${\mathcal{O}}(\vec{y})\in {\rm coh}\hbox{-}\mathbb{X}(\mathbf{p}, {\boldsymbol\lambda})$ corresponds to the follwing $p_t$-cycle \xymatri

Figures (5)

  • Figure 1: Ladder for $\underline{{\rm vect}\hbox{-}}{\mathbb X}(2,2,p_3)$
  • Figure 2: Ladder for $\underline{{\rm vect}\hbox{-}}{\mathbb X}(2,p_2,p_3)$
  • Figure 3: Ladder for $\underline{{\rm vect}\hbox{-}}{\mathbb X}(p_1,p_2,p_3)$
  • Figure :
  • Figure :

Theorems & Definitions (27)

  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • ...and 17 more