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 $A^{\mathrm{CMA}}/A^{\mathrm{CMA}}(1-ε_{\star}) A^{\mathrm{CMA}}$ of the CM--Auslander algebra of $A$ to be quasi-tilted, precisely, the following statements are equivalent: (1) $A^{\mathrm{CMA}}/A^{\mathrm{CMA}}(1-ε_{\star}) A^{\mathrm{CMA}}$ 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.

Figures (6)

• Figure 1.1: Bound quiver
• Figure 2.1: The string $\mathsf{s}$ and the four forbidden paths associated with it
• Figure 2.2: The CM--Auslander algebra of the gentle algebra given in Example \ref{['examp:gentle']}
• Figure 2.3: Recollements of $\mathsf{mod}(\tilde{C})$, $\mathsf{mod}(C)$ and $\mathsf{mod}(A)$
• Figure 2.4: The Auslander--Reiten quiver of the gentle algebra $A$ given in Example \ref{['examp:gentle']} (the modules marked by "${\color{orange}\pmb{\bigcirc}}$" are non-forbidden modules)

Theorems & Definitions (42)

• Theorem 1
• Theorem 2: Theorems \ref{['thm:main']}, \ref{['thm:main2']}
• Corollary 3: Corollary \ref{['coro:main3']}
• Definition 1.1: Gentle algebras AS1987
• Example 1.2
• Theorem 1.3: Butler, Ringel, Wald, and Waschbüsch WW1985BR1987
• Remark 1.5
• Definition 2.1: AG2008
• Definition 2.2: LMXZ2025
• Theorem 2.3: LMXZ2025