Thick subcategories of derived categories of gentle algebras
Callum Page
TL;DR
We classify thick subcategories of the bounded derived category $\mathsf{D}^b(\mathrm{mod}\ \Lambda)$ for gentle algebras, showing they are generated either by string objects or by band objects. When a thick subcategory is Ext-connected and contains a string object, it is generated by string objects, and finitely generated cases arise from finitely many such generators; conversely, Ext-connected subcategories containing bands can behave differently. For the string-generated case, thicks correspond to connected finite arc-collections on the geometric model, yielding an isomorphism $\ Arc(\Sigma)/\sim_{\text{gen}} \cong \text{thick-st}(\mathsf{D})$ that translates algebraic containment into a generation relation of arcs. The framework leverages homotopy strings/bands and a geometric ribbon surface $\Sigma_{\Lambda}$ to convert morphisms and cones into intersection data of arcs, providing a concrete combinatorial classification and revealing when thick subcategories are generated by exceptional/spherelike objects. The results extend thick-subcategory classifications beyond discrete derived categories by giving a precise geometric/combinatorial description for gentle algebras.
Abstract
We study thick subcategories of derived categories of gentle algebras. Any thick subcategory of a derived category of a gentle algebra is generated by a set of string objects or a set of band objects. We show the thick subcategories generated by string objects are in bijection with sets of non-crossing paths on the geometric model of the derived category.