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nrepresentation infinite algebras
"... From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes o ..."
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From the viewpoint of higher dimensional AuslanderReiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call nrepresentation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: npreprojective, npreinjective and nregular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext 1orthogonal families of modules. Moreover we give general constructions of nrepresentation infinite algebras. Applying Minamoto’s theory on Fano algebras in noncommutative algebraic geometry, we describe the category of nregular modules in terms of the corresponding preprojective algebra. Then we introduce nrepresentation tame algebras, and show that the category of nregular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra. This generalizes the classical description of regular modules over tame hereditary algebras. As an application, we show that the representation dimension of an nrepresentation tame algebra is at least n+2.
τ2stable tilting complexes over weighted projective lines. arXiv:1402.6036
, 2014
"... Abstract. Let X be a weighted projective line and cohX the associated categoy of coherent sheaves. We classify the tilting complexes T in Db(cohX) such that τ2T ∼ = T, where τ is the AuslanderReiten translation in Db(cohX). As an application of this result, we classify the 2representationfinite ..."
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Abstract. Let X be a weighted projective line and cohX the associated categoy of coherent sheaves. We classify the tilting complexes T in Db(cohX) such that τ2T ∼ = T, where τ is the AuslanderReiten translation in Db(cohX). As an application of this result, we classify the 2representationfinite algebras which are derivedequivalent to a canonical algebra. This complements IyamaOppermann’s classification of the iterated tilted 2representationfinite algebras. By passing to 3preprojective algebras, we obtain a classification of the selfinjective clustertilted algebras of canonicaltype. This complements Ringel’s classification of the selfinjective clustertilted algebras. 1.
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"... Abstract. Over any field of positive characteristic we construct 2CYtilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields o ..."
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Abstract. Over any field of positive characteristic we construct 2CYtilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2CYtilted even if they do not arise from potentials. In another direction, we compute the fractionally CalabiYau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2. Introduction A 2CYtilted algebra is an endomorphism algebra of a clustertilting object in a 2CalabiYau triangulated category. There are close connections between 2CYtilted algebras and Jacobian algebras of quivers with potentials as introduced by Derksen, Weyman and Zelevinsky On the other hand, by the work of Amiot [2], any finitedimensional Jacobian algebra is 2CYtilted. It is therefore natural to ask whether any 2CYtilted algebra is a Jacobian algebra of a quiver with potential [3, Question 2.20]. The purpose of this note is twofold. First, we provide a negative answer to this question over any field of positive characteristic. Our examples are given by certain selfinjective Nakayama algebras which are also known as truncated cycle algebras. Second, we show that it is actually possible to slightly extend the notion of a potential in order to exclude this kind of examples. Let us explain the motivation behind such extension. Since 2CYtilted algebras have some remarkable homological and structural properties Consider for example the algebra Λ K = K[x]/(x n−1 ) over a field K for some n > 2, which could be described as a quiver with one vertex, one loop x and a relation x n−1 . Date: March 26, 2014.
TORSION CLASSES AND tSTRUCTURES IN HIGHER HOMOLOGICAL ALGEBRA
"... Abstract. Higher homological algebra was introduced by Iyama. It is also known as nhomological algebra where n> 2 is a fixed integer, and it deals with ncluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequence ..."
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Abstract. Higher homological algebra was introduced by Iyama. It is also known as nhomological algebra where n> 2 is a fixed integer, and it deals with ncluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with n + 2 objects. This was recently formalised by Jasso in the theory of nabelian categories. There is also a derived version of nhomological algebra, formalised by Geiss, Keller, and Oppermann in the theory of (n+ 2)angulated categories (the reason for the shift from n to n + 2 is that angulated categories have triangulated categories as the “base case”). We introduce torsion classes and tstructures into the theory of nabelian and (n + 2)angulated categories, and prove several results to motivate the definitions. Most of the results concern the nabelian and (n+2)angulated categoriesM (Λ) and C (Λ) associated to an nrepresentation finite algebra Λ, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in M (Λ) and intermediate tstructures in C (Λ) which is a category one can reasonably view as the nderived category of M (Λ). We hint at the link