On the $\tau$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras

TL;DR

This paper characterizes $ au$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras, linking algebraic properties to surface models and extending previous results.

Contribution

It proves that skew-gentle algebras are $ au$-tilting finite iff they are representation-finite and provides a geometric criterion for silting-discreteness using surface models.

Findings

Skew-gentle algebra is $ au$-tilting finite iff it is representation-finite.

Silting-discreteness corresponds to genus zero surface with non-zero winding numbers.

Extended geometric characterization to skew-gentle algebras via orbifold models.

Abstract

This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $\tau$-tilting finite if and only if it is representation-finite, which extends the result for gentle algebras by Plamondon (2019). Second, using surface models, we characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models.