Stable equivalence of quadratic monomial algebras

Abstract

It is well known that gentle algebras are closed under derived equivalence. However, it is easy to verify that they are not closed under stable equivalence. We prove that the classes of quadratic monomial, almost gentle and string almost gentle algebras are closed under stable equivalence. In fact, in each of these cases, stable equivalence is characterized by isomorphism after deleting all nodes and removing semisimple direct summands. The underlying isomorphism theorem gives a positive answer to Martínez-Villa's question on selfinjective-free algebras for these classes. We also prove that the nonprojective simple modules form the unique weakly simple-minded system of a quadratic monomial algebra. Consequently, every stable equivalence of Morita type involving such an algebra lifts to a Morita equivalence when neither algebra has semisimple direct summands. Finally, we describe the effect of node deletion on stable categories of Gorenstein-projective modules and singularity categories, and give combinatorial descriptions of the equivalences for almost gentle algebras using marked configurations.