Indecomposables with smaller cohomological length in the derived category of gentle algebras

TL;DR

This paper extends the 'no gaps' theorem from modules to the derived category of gentle algebras, showing that indecomposables with decreasing cohomological lengths are always present.

Contribution

It proves that for gentle algebras, indecomposable objects with all smaller cohomological lengths also exist, generalizing the classical 'no gaps' result.

Findings

Indecomposables with cohomological length 2 exist if length 3 exists.

Indecomposables with length 1 exist if length 2 exists.

The 'no gaps' property holds in the derived category of gentle algebras.

Abstract

Bongartz and Ringel proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly study this "no gaps" theorem for the bounded derived module category $D^b(A)$ of a gentle algebra $A$: if there is an indecomposable object in $D^b(A)$ of cohomological length $l>1$, then there exists an indecomposable with cohomological length $l-1$.