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) $\mathrm{findim}A⩽2$, and for each forbidden $A$-module $M$, $proj.dimM+inj.dimM⩽2$; (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 $⩽2$. 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.

29 pages, 6 figures. The following modifications have been made to the submitted manuscript: 1. Added some examples to the main results; 2. Updated the new funding informations; 3. Corrected some citation errors in the LaTeX source code