Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets
Osamu Iyama, Junyang Liu
TL;DR
The paper establishes a precise triangle-equivalence between the stable category of $ abla$-graded Cohen–Macaulay modules over an $ abla$-graded Gorenstein tiled order $A$ and the perfect derived category of the incidence algebra of a finite poset $\mathbb{V}_A^{op}$, via a standard tilting object $V_A$ with $\mathrm{End}(V_A) \cong k\mathbb{V}_A^{op}$. It proves that a finite poset $P$ is realizable as an incidence algebra $kP$ as the endomorphism algebra of a standard tilting object if and only if $P$ is empty or has a maximum, and analyzes how $\mathbb{V}_A$ behaves under graded Morita equivalences and coverings. The work also develops a detailed classification program for Gorenstein tiled orders with small posets, notably giving complete descriptions for $|\mathbb{V}_A^{op}|\leq 3$, including explicit exponent-matrix types that realize various endomorphism algebras $\Gamma_A$. Overall, the results connect noncommutative singularity categories to poset-incidence algebras, enabling combinatorial control of tilting objects and stable CM categories in tiled orders.
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 $|\mathbb{V}_A^{op}|\leq 3$.