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An open question for $d$-tilting bundles on Geigle-Lenzing projective spaces

Jianmin Chen, Weikang Weng

TL;DR

The paper investigates when a $d$-tilting bundle on a Geigle–Lenzing projective space $\,\mathbb{X}$ lifts to a $d$-tilting object in the corresponding Cohen–Macaulay category $\,\underline{\mathsf{CM}}^{\mathbb{L}} R$. Focusing on $d=2$ and GL type $(2,2,p,q)$, it constructs a 2-tilting bundle $M$ on $\,\mathbb{X}$ via a prescribed combination of CM-tilting summands and line bundles, and proves that $\,\pi(M)$ remains a 2-tilting bundle; the paper also derives exact non-tilting criteria in the CM category for specific $(p,q)$ values and uses $2$-APR mutations to relate different tilting bundles. Two non-examples are provided: one showing the open-question negative in weight type $(2,2,2,4)$, and another demonstrating that the converse of a partial tilting result can fail even when a 2-tilting bundle/object exists. These results illuminate the limitations of lifting tilting structures from the geometric setting to the CM category and highlight the role of mutations in navigating the landscape of $2$-tilting objects on GL spaces.

Abstract

We construct a family of $2$-tilting bundles on a Geigle-Lenzing projective space of type $(2,2,p,q)$ via the action of iterated $2$-APR mutations. As an application, we give some non-examples for an open question raised by Herschend, Iyama, Minamoto and Oppermann in the paper "Representation theory of Geigle-Lenzing complete intersections".

An open question for $d$-tilting bundles on Geigle-Lenzing projective spaces

TL;DR

The paper investigates when a -tilting bundle on a Geigle–Lenzing projective space lifts to a -tilting object in the corresponding Cohen–Macaulay category . Focusing on and GL type , it constructs a 2-tilting bundle on via a prescribed combination of CM-tilting summands and line bundles, and proves that remains a 2-tilting bundle; the paper also derives exact non-tilting criteria in the CM category for specific values and uses -APR mutations to relate different tilting bundles. Two non-examples are provided: one showing the open-question negative in weight type , and another demonstrating that the converse of a partial tilting result can fail even when a 2-tilting bundle/object exists. These results illuminate the limitations of lifting tilting structures from the geometric setting to the CM category and highlight the role of mutations in navigating the landscape of -tilting objects on GL spaces.

Abstract

We construct a family of -tilting bundles on a Geigle-Lenzing projective space of type via the action of iterated -APR mutations. As an application, we give some non-examples for an open question raised by Herschend, Iyama, Minamoto and Oppermann in the paper "Representation theory of Geigle-Lenzing complete intersections".

Paper Structure

This paper contains 8 sections, 14 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $d$-cluster tilting subcategories
  4. Geigle-Lenzing complete intersections and projective spaces
  5. $d$-tilting objects and slices
  6. $d$-APR mutations of $d$-tilting bundles
  7. Proof of Theorem \ref{['main theorem']}
  8. examples

Key Result

Theorem 1.2

(See Theorem main theorem_1, Propositions pro1, pro2 for details) Let $d=2$, $n=4$ and $(R,\mathbb{L})$ be a GL complete intersection of type $(2,2,p,q)$ with integers $p,q \ge 2$, and $\mathbb{X}$ the corresponding GL projective space. Let $J$ be an upset restricted to $[\vec{s},\vec{s}+\vec{\delta Then $M$ gives a $2$-tilting bundle on $\mathbb{X}$. Moreover, we have the following.

Theorems & Definitions (26)

  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: HIMO
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 16 more