Auslander-Reiten components for clans (Q1977162)

It is proved that if the repetitive algebra \(\widehat\Lambda\) of a gentle algebra \(\Lambda\) is of non-polynomial growth, then the stable Auslander-Reiten quiver of \(\widehat\Lambda\) contains a finite number of components of type \(\mathbb{Z}\mathbb{A}_\infty\). Then, if the gentle algebra \(\Lambda\) has finite global dimension, \(\Lambda\) is derived equivalent to a hereditary algebra of type \(\mathbb{A}\) or \(\widetilde\mathbb{A}\), if and only if the stable Auslander-Reiten quiver of \(\widehat\Lambda\) does not contain components of type \(\mathbb{Z}\mathbb{A}_\infty\) or \(\mathbb{Z}\mathbb{A}_\infty^\infty\). These results extend to a more general situation, that is, when \(\widehat\Lambda\) is quasi-clanish. The notion of a quasi-clanish algebra generalizes the notion of clanish algebras and of special biserial algebras. Also, they study certain skew group algebras. All these results are independent of the base field \(k\).