On the structure of triangulated categories with finitely many indecomposables

Claire Amiot

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 3, page 435-474
  • ISSN: 0037-9484

Abstract

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field k . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category 𝒯 is of the form ℤ Δ / G where Δ is a disjoint union of simply-laced Dynkin diagrams and G a weakly admissible group of automorphisms of ℤ Δ . Then we prove that for ‘most’ groups G , the category 𝒯 is standard,i.e. k -linearly equivalent to an orbit category 𝒟 b ( m o d k Δ ) / Φ . This happens in particular when 𝒯 is maximal d -Calabi-Yau with d ≥ 2 . 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.

How to cite

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Amiot, Claire. "On the structure of triangulated categories with finitely many indecomposables." Bulletin de la Société Mathématique de France 135.3 (2007): 435-474. <http://eudml.org/doc/272357>.

@article{Amiot2007,
abstract = {We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $\mathcal \{T\}$ 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 $\mathcal \{T\}$ is standard,i.e.$k$-linearly equivalent to an orbit category $\mathcal \{D\}^b(~mod \;k\Delta )/\Phi $. This happens in particular when $\mathcal \{T\}$ is maximal $d$-Calabi-Yau with $d\ge 2$. Moreover, if $\mathcal \{T\}$ is standard and algebraic, we can even construct a triangle equivalence between $\mathcal \{T\}$ 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.},
author = {Amiot, Claire},
journal = {Bulletin de la Société Mathématique de France},
keywords = {locally finite triangulated category; Calabi-Yau category; Dynkin diagram; Auslander-Reiten quiver; orbit category},
language = {eng},
number = {3},
pages = {435-474},
publisher = {Société mathématique de France},
title = {On the structure of triangulated categories with finitely many indecomposables},
url = {http://eudml.org/doc/272357},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Amiot, Claire
TI - On the structure of triangulated categories with finitely many indecomposables
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 3
SP - 435
EP - 474
AB - We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $\mathcal {T}$ 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 $\mathcal {T}$ is standard,i.e.$k$-linearly equivalent to an orbit category $\mathcal {D}^b(~mod \;k\Delta )/\Phi $. This happens in particular when $\mathcal {T}$ is maximal $d$-Calabi-Yau with $d\ge 2$. Moreover, if $\mathcal {T}$ is standard and algebraic, we can even construct a triangle equivalence between $\mathcal {T}$ 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.
LA - eng
KW - locally finite triangulated category; Calabi-Yau category; Dynkin diagram; Auslander-Reiten quiver; orbit category
UR - http://eudml.org/doc/272357
ER -

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