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Semibrick-cosilting correspondence

Ramin Ebrahimi, Alireza Nasr-Isfahani

Abstract

Let $Λ$ be a finite dimensional algebra. In this paper we show that there is a natural bijection between cosilting modules in Mod$Λ$ and semibricks in Mod$Λ$ satisfying some condition. Also this bijection restricts to a bijection between all semibricks in mod$Λ$ and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence [7] between support $τ^-$-tilting modules and right finite semibricks.

Semibrick-cosilting correspondence

Abstract

Let be a finite dimensional algebra. In this paper we show that there is a natural bijection between cosilting modules in Mod and semibricks in Mod satisfying some condition. Also this bijection restricts to a bijection between all semibricks in mod and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence [7] between support -tilting modules and right finite semibricks.
Paper Structure (5 sections, 24 theorems, 13 equations)

This paper contains 5 sections, 24 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. preliminaries
  4. Torsion free, almost torsion modules in a cotilting torsion pair
  5. The main theorems

Key Result

Theorem 1

Let $\Lambda$ be a finite dimensional algebra. Then there exists a bijection between support $\tau^-$-tilting modules and semibricks in $\mathop{\mathrm{mod}}\nolimits \Lambda$ that generate a functorially finite torsion free class in $\mathop{\mathrm{mod}}\nolimits\Lambda$.

Theorems & Definitions (49)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • ...and 39 more