Automorphisms and homological properties of locally gentle algebras

Abstract

In this work, we consider the infinite dimensional generalizations of string algebras, referred to as locally string algebras, giving special attention the generalizations of gentle algebras, known as locally gentle algebras. We describe the prime spectrum and Jacobson radical of a locally string algebra. We show that, up to an inner automorphism and a unique graded automorphism, an automorphism of a string algebra acts as permutations on stationary paths and decomposes into a composition of exponential automorphisms. For the locally gentle algebras, we give an explicit injective resolution and combinatorial descriptions of their homological dimensions. We classify the Artin-Schelter Gorenstein, Artin-Schelter regular, and Cohen-Macaulay locally gentle algebras, and provide analogues of Stanley's theorem for locally gentle algebras.