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Pré-Publication, Document De Travail Année : 2007

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot

Résumé

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category is of the form $\mathbb{Z}\Delta/G$ where $\Delta$ is a disjoint union of simply laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}\Delta$. Then we prove that for `most' groups $G$, the category $\T$ is standard, \emph{i.e.} $k$-linearly equivalent to an orbit category $\mathcal{D}^b(\modd k\Delta)/\Phi$. This happens in particular when $\T$ is maximal $d$-Calabi-Yau with $d\geq2$. Moreover, if $\T$ is standard and algebraic, we can even construct a triangle equivalence between $\T$ and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard $1$-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type.
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Dates et versions

hal-00118624 , version 1 (05-12-2006)
hal-00118624 , version 2 (12-01-2007)

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Claire Amiot. On the structure of triangulated categories with finitely many indecomposables. 2007. ⟨hal-00118624v2⟩
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