Recollements of Cohen-Macaulay Auslander algebras for gentle algebras

We construct two recollements of module categories for the Cohen–Macaulay Auslander algebra $A^{\mathrm{C}\mathrm{M}\mathrm{A}}$ of a gentle algebra $A$. In this paper, we establish three equivalent characterizations for the quotient algebra $A^{\mathrm{C}\mathrm{M}\mathrm{A}}\mathrm{/}{A}^{\mathrm{C}\mathrm{M}\mathrm{A}}(1-{ϵ}_{\star })A^{\mathrm{C}\mathrm{M}\mathrm{A}}$ of the CM–Auslander algebra of $A$ to be quasi-tilted, precisely, the following statements are equivalent: (1) $A^{\mathrm{C}\mathrm{M}\mathrm{A}}\mathrm{/}{A}^{\mathrm{C}\mathrm{M}\mathrm{A}}(1-{ϵ}_{\star })A^{\mathrm{C}\mathrm{M}\mathrm{A}}$ is quasi-tilted; (2) $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{m}A⩽2$, and for each forbidden $A$-module $M$, $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{.}\mathrm{d}\mathrm{i}\mathrm{m}M+\mathrm{i}\mathrm{n}\mathrm{j}\mathrm{.}\mathrm{d}\mathrm{i}\mathrm{m}M⩽2$; (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 $A^{\mathrm{C}\mathrm{M}\mathrm{A}}$ is bounded by 2 in the case where $A$ is gentle one-cycle.