Abelian 1-Calabi-Yau Categories
TLDR
In this article, all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k.Abstract:
In this paper, we show all k-linear abelian 1-Calabi-Yau categories over an algebraically closed field k are derived equivalent to either the category of coherent sheaves on an elliptic curve, or to the finite dimensional representations of k[[t]]. Since all abelian categories derived equivalent with these two are known, we obtain a classification of all k-linear abelian 1-Calabi-Yau categories up to equivalence.read more
Citations
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Abelian hereditary fractionally Calabi-Yau categories
TL;DR: In this article, a generalization of Calabi-Yau categories is proposed, where a k-linear hom-finite triangulated category is fractionally CalabiYau if it admits a Serre functor S and there is an n > 0 with S^n = [m].
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On equivariant derived categories
TL;DR: In this paper, the equivariant category associated to a finite group action on the derived category of coherent sheaves of a smooth projective variety was studied, and the existence of a Serre functor was proved.
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Frobenius-Perron Theory of Endofunctors
TL;DR: In this article, the Frobenius-Perron dimension of an endofunctor of a k-linear category is introduced, and some applications are provided for its use.
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Frobenius–Perron theory of endofunctors
TL;DR: In this paper, the Frobenius-Perron dimension of an endofunctor of a k-linear category is introduced and some applications are discussed. But none of these are in this paper.
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Numerically finite hereditary categories with Serre duality
TL;DR: In this article, the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank with Serre duality, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.
References
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Vector Bundles Over an Elliptic Curve
TL;DR: In this article, the authors studied vector bundles over an algebraically closed field k and proved that the vector bundles are a direct sum of line-bundles over the field k. The results are valid in both characteristic 0 and p.
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Braid group actions on derived categories of coherent sheaves
Paul Seidel,Richard P. Thomas +1 more
TL;DR: In this paper, the authors give a construction of braid group actions on coherent sheaves on a variety of manifolds and show that these actions are always faithful when the manifold is smooth.
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Braid group actions on derived categories of coherent sheaves
Paul Seidel,Richard P. Thomas +1 more
TL;DR: In this article, the authors give a construction of braid group actions on coherent sheaves on a variety of manifolds and show that these actions are always faithful when the manifold is an elliptic curve.
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Noetherian hereditary abelian categories satisfying Serre duality
Idun Reiten,M. Van den Bergh +1 more
TL;DR: Reiten, I, Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway as mentioned in this paper, B-3590 Diepenbeek, Belgium.
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Noetherian hereditary abelian categories satisfying Serre duality
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