Semiorthogonal decompositions for bounded derived categories of gentle algebras

TL;DR

This paper explores the relationship between semiorthogonal decompositions of derived categories of gentle algebras and geometric cuts of associated surfaces, providing a classification framework.

Contribution

It establishes a one-to-one correspondence between semiorthogonal decompositions and surface cuts, linking algebraic and geometric perspectives.

Findings

Correspondence between decompositions and surface cuts

Characterization of basis morphisms between indecomposables

Framework for understanding derived categories geometrically

Abstract

We study semiorthogonal decompositions of bounded derived categories of gentle algebras and how they are manifested in the geometric model of these categories as constructed by Opper, Plamondon and Schroll. We prove that there is a one-to-one correspondence between such semiorthogonal decompositions and suitable cuts of the marked surface underlying the geometric model. Our main tool is the characterization of basis morphisms between indecomposable objects due to Arnesen, Laking and Pauksztello.