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

Abstract: This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $τ$-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.