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Bijections between $τ$-rigid modules

Gabriella D'Este, H. Melis Tekin Akcin

TL;DR

The paper investigates extending classical tilting bijections to the $\tau$-tilting setting, asking whether a correspondence between indecomposable summands persists when moving from tilting to $\tau$-tilting. It develops a framework of permutations on indecomposable $\tau$-rigid modules and demonstrates both positive correspondences and negative results via explicit constructions, including a permutation $t$ that transforms one $\tau$-tilting module into another by swapping non-shared summands. Through detailed examples and algebras $A$ and $B\simeq A^{op}$, it shows how a bijection $s$ between $\tau$-rigid modules can be linked by $\beta = s\alpha s^{-1}$ and highlights failures of a universal Ext-based summand bijection. Overall, the work clarifies the nuanced combinatorics of $\tau$-tilting theory and provides a concrete framework to compare $\tau$-tilting structures across algebras.

Abstract

We describe a special bijection between the indecomposable summands of two basic $τ$-tilting modules.

Bijections between $τ$-rigid modules

TL;DR

The paper investigates extending classical tilting bijections to the -tilting setting, asking whether a correspondence between indecomposable summands persists when moving from tilting to -tilting. It develops a framework of permutations on indecomposable -rigid modules and demonstrates both positive correspondences and negative results via explicit constructions, including a permutation that transforms one -tilting module into another by swapping non-shared summands. Through detailed examples and algebras and , it shows how a bijection between -rigid modules can be linked by and highlights failures of a universal Ext-based summand bijection. Overall, the work clarifies the nuanced combinatorics of -tilting theory and provides a concrete framework to compare -tilting structures across algebras.

Abstract

We describe a special bijection between the indecomposable summands of two basic -tilting modules.

Paper Structure

This paper contains 2 sections, 1 theorem, 15 equations.

Table of Contents

  1. Introduction
  2. Examples and Results

Key Result

Theorem 2.5

Let $X= \oplus_{i=1}^{n} X_{i}$, $Y= \oplus_{i=1}^{n} Y_{i}$ be $\tau$-tilting modules over an algebra $A$ and let $(*)$ denote the following property: $(*)$ There is a permutation $s$ of $\{1,\ldots, n\}$ such that, for any $i$, we have either $X_{i}\simeq Y_{s(i)}$ or $Ext^1_{A}(X_{i}, Y_{s(i)}) \

Theorems & Definitions (12)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 2 more