Recollements of Cohen-Macaulay Auslander algebras for gentle algebras

Abstract

We construct two recollements of module categories for the Cohen--Macaulay Auslander algebra $A^{\mathrm{CMA}}$ of a gentle algebra $A$. In this paper, we establish three equivalent characterizations for the quotient algebra $A^{\mathrm{CMA}}/A^{\mathrm{CMA}}(1-\epsilon_{\star}) A^{\mathrm{CMA}}$ of the CM--Auslander algebra of $A$ to be quasi-tilted, precisely, the following statements are equivalent: (1) $A^{\mathrm{CMA}}/A^{\mathrm{CMA}}(1-\epsilon_{\star}) A^{\mathrm{CMA}}$ is quasi-tilted; (2) $\mathrm{findim} A\leqslant 2$, and for each forbidden $A$-module $M$, $\mathrm{proj.dim}M+\mathrm{inj.dim}M\leqslant 2$; (3) for any homotopy string/band $\mathsf{h}$ none of whose arrows lie on any forbidden cycle, the cohomological width of the indecomposable object in $\mathsf{D}^b(A)$ corresponding to $\mathsf{h}$ is $\leqslant 2$. Moreover, we prove that the Krull--Gabriel dimension of $A$ is bounded by 2 if and only if the Krull--Gabriel dimension of $A^{\mathrm{CMA}}$ is bounded by 2 in the case where $A$ is gentle one-cycle.