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In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of ...
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In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra A considered as an A-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space V to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.
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摘要 :
We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twist...
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We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.
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Discrete Koszul algebra, another extension of Koszul algebras, is introduced in this paper. The Yoneda algebra of a discrete Koszul algebra is investigated in detail. As an application, we give an answer to a question proposed by ...
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Discrete Koszul algebra, another extension of Koszul algebras, is introduced in this paper. The Yoneda algebra of a discrete Koszul algebra is investigated in detail. As an application, we give an answer to a question proposed by Green and Marcos (Commun Algebra 33:1753-1764, 2005). In particular, the relationship between discrete Koszul algebras and Koszul algebras is established. Further, we construct new discrete Koszul algebras from the given ones in terms of one-point extension.
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The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K-2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K-2 algebra would be another K-2 algebra. We sho...
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The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K-2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K-2 algebra would be another K-2 algebra. We show that this is not necessarily the case by constructing a monomial K-2 algebra for which the corresponding Yoneda algebra is not K-2.
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The notion of a K_2-algebra was recently introduced by Cassidy and Shelton as a generalization of the notion of a Koszul algebra. The Yoneda algebra of any connected graded algebra admits a canonical A∞-algebra structure. This st...
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The notion of a K_2-algebra was recently introduced by Cassidy and Shelton as a generalization of the notion of a Koszul algebra. The Yoneda algebra of any connected graded algebra admits a canonical A∞-algebra structure. This structure is trivial if the algebra is Koszul. We study the A∞-structure on the Yoneda algebra of a K_2-algebra. For each non-negative integer n we prove the existence of a K_2-algebra B and a canonical A∞-algebra structure on the Yoneda algebra of B such that the higher multiplications mi are nonzero for all 3≤i≤n+3. We also provide examples which show that the K_2 property is not detected by any obvious vanishing patterns among higher multiplications.
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We show that if A is a n-Koszul algebra and E = E(Lambda) is its Yoneda algebra, then there is a full subcategory L-E of the category Gr(E) of graded E-modules, which contains all the graded E-modules presented in even degrees, th...
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We show that if A is a n-Koszul algebra and E = E(Lambda) is its Yoneda algebra, then there is a full subcategory L-E of the category Gr(E) of graded E-modules, which contains all the graded E-modules presented in even degrees, that embeds fully faithfully in the category C(Gr(Lambda)) of cochain complexes of graded Lambda-modules. That extends the known equivalence, for Lambda Koszul (i.e. n = 2), between GrE and the category of linear complexes of graded A-modules. (C) 2007 Elsevier Inc. All rights reserved.
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In this paper, we study the derived categories of a Koszul algebra and its Yoneda algebra to determine when those categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to...
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In this paper, we study the derived categories of a Koszul algebra and its Yoneda algebra to determine when those categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We have considered discrete Koszul algebras and we gave necessary and sufficient conditions for those Koszul algebras to be derived equivalent to their Yoneda algebras. We also study the class of Koszul algebras which are derived equivalent to hereditary algebras. For the case where the hereditary algebra is tame, we characterized the derived equivalence between those Koszul algebras and their Yoneda algebras.
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The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating th...
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The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let I > (t) be the Yoneda algebra of a reconstruction algebra of type A (1) over a field k. In this paper, a minimal projective bimodule resolution of I > (t) is constructed, and the k-dimensions of all Hochschild homology and cohomology groups of I > (t) are calculated explicitly.
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Let A be a connected-graded algebra with trivial module k, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A):= Ext _A(k,k) to E(B). Cassidy and Shelton have shown that when A satisfies th...
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Let A be a connected-graded algebra with trivial module k, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A):= Ext _A(k,k) to E(B). Cassidy and Shelton have shown that when A satisfies their K _2 property, B will also be K _2·. We prove the converse of this result.
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The main purpose of this paper is to study a concrete example of δ-Koszul algebras, which is related to three questions raised by Green and Marcos.
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