Abelian hereditary fractionally Calabi-Yau categories
TLDR
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].Abstract:
As a generalization of a Calabi-Yau category, we will say a k-linear Hom-finite triangulated category is fractionally Calabi-Yau if it admits a Serre functor S and there is an n > 0 with S^n = [m]. An abelian category will be called fractionally Calabi-Yau is its bounded derived category is. We provide a classification up to derived equivalence of abelian hereditary fractionally Calabi-Yau categories (for algebraically closed k). They are: the category of finite dimensional representations of a Dynkin quiver, the category of finite dimensional nilpotent representations of a cycle, and the category of coherent sheaves on an elliptic curve or a weighted projective line of tubular type. To obtain this classification, we introduce generalized 1-spherical objects and use them to obtain results about tubes in hereditary categories (which are not necessarily fractionally Calabi-Yau).read more
Citations
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Discrete derived categories I: homomorphisms, autoequivalences and t-structures
TL;DR: In this paper, the homomorphism hammocks and autoequivalences on discrete derived categories of t-structures have been studied, and they have been used to classify silting objects and bounded tstructures.
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2-CY-tilted algebras that are not Jacobian
TL;DR: In this paper, an extension of the notion of a potential, called hyperpotential, was proposed to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials.
Journal ArticleDOI
Abstract representation theory of Dynkin quivers of type A
Moritz Groth,Jan Šťovíček +1 more
TL;DR: The representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra, was studied in this article.
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Tilting theory via stable homotopy theory
Moritz Groth,Jan Stovicek +1 more
TL;DR: In this paper, it was shown that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory.
Posted Content
Discrete derived categories I: homomorphisms, autoequivalences and t-structures
TL;DR: In this paper, the homomorphism hammocks and autoequivalences on these categories were studied, and they were used to classify silting objects and bounded t-structures.
References
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Book
Representation Theory of Artin Algebras
TL;DR: Artin rings as mentioned in this paper have been used to represent morphisms in the Auslander-Reiten-quiver and the dual transpose and almost split sequences, and they have been shown to be stable equivalence.
Book
Tame Algebras and Integral Quadratic Forms
TL;DR: In this article, the construction of stable separating tubular families and tubular algebras are discussed. But they do not discuss the relation between tubular extensions and directed algesbras.
Journal ArticleDOI
Des catégories abéliennes
TL;DR: The Bulletin de la S. M. F. as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.html).
Journal ArticleDOI
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.
Book
Categories and Sheaves
正樹 柏原,Pierre Schapira +1 more
TL;DR: The Language of Categories as mentioned in this paper is a language of categories that includes the following features: limits, filters and filters, generators and representability, localization, and localisation.