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A theorem on support $τ$--tilting pairs

Gabriella D'Este

TL;DR

The paper addresses aligning basic support $ au$-tilting pairs by establishing a structured bijection between their indecomposable summands. It introduces a precise framework with $ au$-rigidity concepts and proves Theorem 2.1, which guarantees a permutation $s rom S_{n}$ mapping $X_i$ to $Y_{s(i)}$ with explicit $ au$-rigidity relations. The results extend known bijections for basic tilting and basic $ au$-tilting modules to the broader setting of support pairs and are supported by examples illustrating sharpness and necessity of the hypotheses. These insights deepen the structural understanding of the landscape of support $ au$-tilting pairs and may inform mutation-like processes in representation theory.

Abstract

We show that there is a special bijection between the indecomposable summands of the two modules which form a basic support $τ$--tilting pair and the indecomposable summands of the two modules which form another basic support $τ$--tilting pair.

A theorem on support $τ$--tilting pairs

TL;DR

The paper addresses aligning basic support -tilting pairs by establishing a structured bijection between their indecomposable summands. It introduces a precise framework with -rigidity concepts and proves Theorem 2.1, which guarantees a permutation mapping to with explicit -rigidity relations. The results extend known bijections for basic tilting and basic -tilting modules to the broader setting of support pairs and are supported by examples illustrating sharpness and necessity of the hypotheses. These insights deepen the structural understanding of the landscape of support -tilting pairs and may inform mutation-like processes in representation theory.

Abstract

We show that there is a special bijection between the indecomposable summands of the two modules which form a basic support --tilting pair and the indecomposable summands of the two modules which form another basic support --tilting pair.

Paper Structure

This paper contains 2 sections, 5 theorems.

Table of Contents

  1. Introduction and definitions
  2. A theorem and some examples

Key Result

Lemma 1.1

7 Let $T$ be a tilting module, and let $X$ be a module such that $T \oplus X$ is a partial tilting module. Then $X \in \operatorname{add} T$.

Theorems & Definitions (17)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 2.1
  • proof
  • Example 2.2
  • proof : Construction
  • Corollary 2.3
  • proof
  • Example 2.4
  • ...and 7 more