${\tau }^2$-stable tilting complexes over weighted projective lines

Abstract

Let $\mathbb{X}$ be a weighted projective line and $\mathrm{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\mathrm{coh}\mathbb{X})$ such that ${\tau }^{2}T\cong T$, where $\tau$ is the Auslander-Reiten translation in $D^b(\mathrm{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we obtain a classification of the selfinjective cluster-tilted algebras of canonical-type. This complements Ringel's classification of the selfinjective cluster-tilted algebras.

28 pages, corrected typos, minor edits

Cited by in corpus (7)

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• Higher Nakayama algebras I: Construction
• d-Representation-finite self-injective algebras
• Tilting theory for Gorenstein rings in dimension one
• From groups to clusters
• Weighted projective lines of tubular type and equivariantization
• The dual actions, equivariant autoequivalences and stable tilting objects