Abstract

We prove that all gentle 2-Calabi-Yau tilted algebras (over an algebraically closed field $k$ , $\mathrm{char}\hskip 1.0ptk\neq 3$ ) are Jacobian, moreover their bound quiver can be obtained via block decomposition. For two related families, the $m$ -cluster-tilted algebras of type $\mathbb{A}$ and $\tilde{\mathbb{A}}$ , we prove that a module $M$ is stable Cohen-Macaulay if and only if $\Omega^{m+1}\tau M\simeq M$ .