Relative left Bongartz completions and their compatibility with mutations

TL;DR

This paper introduces relative left Bongartz completions for $ au$-rigid pairs in finite-dimensional algebras, demonstrating their compatibility with mutations and applications to maximal green sequences and silting theory.

Contribution

It defines relative left Bongartz completions for $ au$-rigid pairs and proves their compatibility with mutations, advancing the understanding of $ au$-tilting theory and silting theory.

Findings

Compatibility of relative left Bongartz completions with mutations

Application to existence of maximal green sequences

Extension of results to silting theory

Abstract

In this paper, we introduce relative left Bongartz completions for a given basic $\tau$-rigid pair $(U,Q)$ in the module category of a finite dimensional algebra $A$. They give a family of basic $\tau$-tilting pairs containing $(U,Q)$ as a direct summand. We prove that relative left Bongartz completions have nice compatibility with mutations. Using this compatibility we are able to study the existence of maximal green sequences under $\tau$-tilting reduction. We also explain our construction and some of the results in the setting of silting theory.