摘要 :
The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module M is a weakly...
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The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module M is a weakly Koszul-like module if and only if it can be approximated by Koszul-like graded submodules, which is equivalent to the fact that G(M) is a Koszul-like module, where G(M) denotes the associated graded module of M. As applications, the relationships between minimal graded projective resolutions of M and G(M), and Koszul-like submodules are established. Moreover, the Koszul dual of a weakly Koszul-like module is proved to be generated in degree 0 as a graded E(A) -module.
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摘要 :
In this paper, the notion of Koszul-like algebra is introduced; this notion generalizes the notion of Koszul algebra and includes some Artin-Schelter regular algebras of global dimension 5 as special examples. Basic properties of ...
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In this paper, the notion of Koszul-like algebra is introduced; this notion generalizes the notion of Koszul algebra and includes some Artin-Schelter regular algebras of global dimension 5 as special examples. Basic properties of Koszul-like modules are discussed. In particular, some necessary and sufficient conditions for ΚL(A) = L(A) are provided, where ΚL(A) and L(A) denote the categories of Koszul-like modules and modules with linear presentations (see [1]-[3], etc.) respectively, and A is a Koszul-like algebra. We construct new Koszul-like algebras from the known ones by the "one-point extension." Some criteria for a graded algebra to be Koszul-like are provided. Finally, we construct many classical Koszul objects from the given Koszul-like objects.
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The δ-Koszulity of finitely generated graded modules is discussed and the notion of weakly δ-Koszul module is introduced. Let M ∈ gr(A) and {S_(d1), S_(d2),..., S_(dm)} denote the set of minimal homogeneous generating spaces of...
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The δ-Koszulity of finitely generated graded modules is discussed and the notion of weakly δ-Koszul module is introduced. Let M ∈ gr(A) and {S_(d1), S_(d2),..., S_(dm)} denote the set of minimal homogeneous generating spaces of M where S_(di) consists of homogeneous elements of M of degree d_i. Put ?_1 = 〈S_(d1)〉, ?_2 = 〈S_(d1), S_(d2)〉,..., ?_m = 〈S_(d1), S_(d2),..., S_(dm)〉. Then M admits a chain of graded submodules: 0 = ?_0 ? ?_1 ? ?_2 ? ··· ? ?_m = M. Moreover, it is proved that M is a weakly δ-Koszul module if and only if all ?_i/?_(i-1)[-d_i] are δ-Koszul modules, if and only if the associated graded module G(M) is a δ-Koszul module. Further, as applications, the relationships of minimal graded projective resolutions among M, G(M) and these quotients ?_i/?_(i-1) are established. The Ext module ?_(i≥0) Ext_A ~i(M, A_0) of a weakly δ-Koszul module M is proved to be finitely generated in degree zero.
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In order to study the finite generation property of the Yoneda- Ext algebras of positively graded algebras, Green and Marcos introduced δ-Koszul objects in 2005 [2]. Motivated by ([7]-[11]), we are interested in the δ-Koszulity ...
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In order to study the finite generation property of the Yoneda- Ext algebras of positively graded algebras, Green and Marcos introduced δ-Koszul objects in 2005 [2]. Motivated by ([7]-[11]), we are interested in the δ-Koszulity of a finitely generated graded module over a δ-Koszul algebra and introduce the notion of weakly δ-Koszul module in this paper. The following are proved to be equivalent and are the main results of this paper: ? M is a weakly δ-Koszul module; ? the associated graded module G(M) is a δ-Koszul module; ? M admits a chain of graded submodules: 0= M 0_? M1 ? M2 ? ... ? M_m= M, such that all M_i/M_(i-1) are δ-Koszul modules.
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摘要 :The main purpose of this paper is to provide some new criteria for a standard graded algebra A = ⊕ i≥0 A i to be a λ-Koszul algebra, which was first introduced in [12] and was another class of “Koszul-type” algebras includin...
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The main purpose of this paper is to provide some new criteria for a standard graded algebra A = ⊕ i≥0 A i to be a λ-Koszul algebra, which was first introduced in [12] and was another class of “Koszul-type” algebras including Koszul and d-Koszul algebras as special examples.
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Generalized d-Koszul modules are introduced to solve an open problem: the odd Ext-module E~ (odd)(M) of a d-Koszul module M over a d-Koszul algebra ∧ is a Koszul module over the even Yoneda algebra Eev(∧).
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In this paper, the notions of nonpure piecewise-Koszul algebra and nonpure piecewise-Koszul module are introduced, which are the “nonpure” version of piecewise-Koszul algebras and modules first introduced in [19]. Some criteria ...
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In this paper, the notions of nonpure piecewise-Koszul algebra and nonpure piecewise-Koszul module are introduced, which are the “nonpure” version of piecewise-Koszul algebras and modules first introduced in [19]. Some criteria for a standard graded algebra to be nonpure piecewise-Koszul are given. We also discuss some basic properties of nonpure piecewise-Koszul modules. Further more, we give a sufficient condition for the questions raised in [20] to be true in terms of nonpure piecewise-Koszul modules.
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摘要 :
Let Lambda be a Koszul algebra, and let M be a graded Lambda-bimodule. We prove that the trivial extension algebra of Lambda by M is also a Koszul algebra whenever M is Koszul as a left Lambda-module. Applications and examples are also provided.
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We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give a...
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We study the so-called weakly Koszul modules and characterise their Koszul duals. We show that the (adjusted) associated graded module of a weakly Koszul module exactly determines the homology modules of the Koszul dual. We give an example of a quasi-Koszul module which is not weakly Koszul. (c) 2007 Elsevier Inc. All rights reserved.
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I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of ...
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I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.
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