Torsion pairs and simple-minded systems in triangulated categories

TL;DR

This paper explores the structure and mutation of simple-minded systems in triangulated categories, establishing connections with torsion pairs, Serre functors, and stable module categories of self-injective algebras.

Contribution

It introduces a mutation framework for simple-minded systems in triangulated categories, generalizing existing concepts and linking to stable module categories of self-injective algebras.

Findings

Constructs torsion pairs associated to subsets of simple-minded systems.

Defines mutation procedures that preserve simple-minded systems under certain conditions.

Links mutations of simple modules to stable equivalences between algebras.

Abstract

Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings is a simple-minded system if its closure under extensions is all of T. We construct torsion pairs in T associated to any subset X of a simple-minded system S, and use these to define left and right mutations of S relative to X. When T has a Serre functor \nu, and S and X are invariant under \nu[1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where T is the stable module category of a self-injective algebra \Lambda. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in the derived category of \Lambda. It follows that the mutation of the set of simple \Lambda-modules relative to X yields the images of the simple \Gamma-modules under a stable equivalence between \Gamma\ and \Lambda, where \Gamma\ is the tilting mutation of \Lambda\ relative to X.