A Characteristic free approach to skew-gentle algebras

TL;DR

This paper introduces a characteristic-free method to relate skew-gentle algebras to gentle algebras, enabling the resolution of open problems and the strengthening of known results across different characteristics.

Contribution

It establishes a homological framework connecting skew-gentle and gentle algebras that works in all characteristics, including characteristic two, and addresses several open conjectures.

Findings

Classifies all self-injective skew-gentle algebras.

Proves the finitistic dimension and other conjectures hold for all skew-gentle algebras.

Shows skew-gentle and gentle algebras share the same singularity categories.

Abstract

To each skew-gentle algebra, one can assign a gentle algebra in terms of combinatorial data. In order to relate the structures of the two algebras, we establish a homological epimorphism and a recollement of derived module categories. This approach is characteristic free and works in particular also in characteristic two, which is the difficult case for skew-gentle algebras. This allows to solve open problems and to uniformly reprove and strengthen known results, for instance, (1) a complete classification of selfinjective skew-gentle algebras; (2) the finitistic dimension conjecture, Auslander and Reiten's conjecture, and Keller's conjecture hold for all skew-gentle algebras; (3) a precise connection of K-theory between skew-gentle algebras and gentle algebras; (4) all skew-gentle algebras are Gorenstein, and skew-gentle algebras and their gentle algebras share the same singularity categories.