Homological conditions on locally gentle algebras
S. Ford, A. Oswald, J. J. Zhang
TL;DR
This work extends the homological framework from gentle to locally gentle algebras, providing explicit combinatorial descriptions of centers, prime spectra, and Hilbert series, and establishing graded projective/injective resolutions. It proves that global and injective dimensions are governed by maximal paths via the Koszul dual, and it classifies AS properties: the only AS regular case is $A=\Bbbk\widetilde{A}_n$, with AS Gorenstein occurring in two broad quiver-structure families. The Cohen-Macaulay theory is developed for these algebras, revealing a Stanley-type Hilbert-series criterion and linking Cohen-Macaulayness to the absence of finite maximal paths. These results yield a coherent semi-commutative homological picture for locally gentle algebras and identify exactly when Stanley-type symmetry holds.
Abstract
Gentle algebras are a class of special biserial algebra whose representation theory has been thoroughly described. In this paper, we consider the infinite dimensional generalizations of gentle algebras, referred to as locally gentle algebras. We give combinatorial descriptions of the center, prime spectrum, and homological dimensions of a locally gentle algebra, including an explicit injective resolution. We classify when these algebras are Artin-Schelter Gorenstein, Artin-Schelter regular, and Cohen-Macaulay, and provide an analogue of Stanley's theorem for locally gentle algebras.