Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets

TL;DR

This paper establishes a correspondence between Gorenstein tiled orders and incidence algebras of posets, revealing new triangle equivalences and classifying certain Gorenstein tiled orders based on their associated posets.

Contribution

It proves a triangle equivalence between the stable category of Gorenstein tiled orders and derived categories of incidence algebras, and characterizes when incidence algebras are endomorphism algebras of tilting objects.

Findings

Triangle equivalence between stable categories and derived categories of incidence algebras.

Characterization of incidence algebras as endomorphism algebras of tilting objects.

Classification of Gorenstein tiled orders with small associated posets.

Abstract

We prove that for any $\mathbb{N}$-graded Gorenstein tiled order $A$, the stable category $\underline{\mathrm{CM}}^{\mathbb{Z}}A$ is triangle equivalent to the perfect derived category of the incidence algebra of a finite poset $\mathbb{V}_A^{op}$. Moreover, for a finite poset $P$, we prove that the incidence algebra of $P$ can be realized as the endomorphism algebra of a standard tilting object if and only if $P$ is either empty or has the maximum. We also study the behaviors of the corresponding poset under graded Morita equivalences and coverings of a Gorenstein tiled order. Finally, we classify Gorenstein tiled orders $A$ satisfying $|\mathbb{V}_A^{op}|\leq 3$.