Recollements of derived categories of graded gentle algebras and surface cuts

Wen Chang, Haibo Jin, Sibylle Schroll

Published 2022-06-22Version 1

In this paper we study recollements of the derived categories of graded quadratic monomial algebras induced by idempotents. In particular, we show that in such recollements, all three terms are given by graded quadratic monomial algebras which we explicitly determine. We then apply these results to graded gentle algebras and their associated surface model, in which case, all three terms are derived categories of graded gentle algebras. We show that these recollements correspond to cutting the corresponding surfaces along simple curves connecting boundary components. For homologically smooth and proper graded gentle algebras, these recollements restrict to the bounded derived categories and induce recollements of the corresponding partially wrapped Fukaya categories. We then apply our results to show that silting reductions and reductions of simple-minded collections of graded gentle algebras correspond to cutting the associated surfaces. Furthermore, we determine for which homologically smooth and proper graded gentle algebras the perfect derived category admits full exceptional sequences. We also show that almost all homologically smooth and proper graded gentle algebras admit silting objects or, equivalently, simple-minded collections. We further conjecture that our classification of graded gentle algebras admitting silting objects is complete and that, in particular, there is an explicitly determined infinite family of graded gentle algebras which do not admit silting objects.