Tilting-completion for gentle algebras

TL;DR

This paper proves that almost-tilting modules over gentle algebras are partial-tilting with limited complements, confirming a conjecture, and introduces surface models and induction techniques for analyzing modules.

Contribution

It establishes that almost-tilting modules over gentle algebras are partial-tilting and bounds their complements, using surface models and induction methods.

Findings

Almost-tilting modules are partial-tilting over gentle algebras.

Such modules have at most 2n possible complements.

Existence of non-partial-tilting pre-tilting modules in certain gentle algebras.

Abstract

It is demonstrated that any almost-tilting module over a gentle algebra is indeed partial-tilting, meaning it can be completed as a tilting module. Furthermore, such a module has at most $2n$ possible complements, thereby confirming a (modified) conjecture of Happel for the case of gentle algebras. Additionally, for any $n\geq 3$ and $1\leq m \leq n-2$, there always exists a (connected) gentle algebra with rank $n$ and a pre-tilting module of rank $m$ which is not partial-tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is an induction process involving surface cuts, which is hoped to be beneficial for other applications as well.