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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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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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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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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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The main aim of this article is to discuss the piecewise-Koszul property of finitely generated modules in the nongraded case. In particular, the notion of quasi-piecewise-Koszul module, a natural extension of quasi-Koszul modules ...
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The main aim of this article is to discuss the piecewise-Koszul property of finitely generated modules in the nongraded case. In particular, the notion of quasi-piecewise-Koszul module, a natural extension of quasi-Koszul modules (see [2]), quasi-d-Koszul modules (see [12] or [6]) and piecewise-Koszul modules (see [7]), is introduced. Let R be a Noetherian semiperfect algebra with Jacobson radical J and M a finitely generated R-module. The structure of the graded E(R) = ? _(i≥0) Ext _R ~i(R/J, R/J)-module E(M) = ? _(i≥0) Ext _r ~i(M, R/J) is studied in detail and some necessary and sufficient conditions for a finitely generated R-module to be quasi-piecewise-Koszul are provided. Moreover, as an application of quasi-piecewise-Koszul modules, we give a necessary and sufficient condition for the minimal Horseshoe Lemma to be true in the category of quasi-piecewise-Koszul modules, which perfects Theorem 2.8 of [13] and Theorem 3.1 of [9]. Finally, some applications of the minimal Horseshoe Lemma are also given.
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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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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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Let A be a δ-Koszul algebra, and let K~δ(A) and L(A) denote the categories of δ-Koszul modules and modules with linear presentations. Some necessary and sufficient conditions for K~δ(A) = L(A) are given. Set E(A) :=⊕i≥0Ext_A...
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Let A be a δ-Koszul algebra, and let K~δ(A) and L(A) denote the categories of δ-Koszul modules and modules with linear presentations. Some necessary and sufficient conditions for K~δ(A) = L(A) are given. Set E(A) :=⊕i≥0Ext_A~i(A_0,A_0) and B(A) := sup{i ∈ N| Ext_A~i(A_0, A_0) ∩ V ≠ 0}, where V is a minimal graded generating space of E(A). In the present paper, we prove that {B(A)| A is δ - Koszul} = N. Finally, the Koszulity of the graded Hopf Galois extension of δ-Koszul algebras is studied.
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We give different definitions of torsion and completion functors for DG-modules, explicit these constructions using the stable Koszul DG-module and the telescope DG-module over good enough DG-rings. We prove that over a non-negati...
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We give different definitions of torsion and completion functors for DG-modules, explicit these constructions using the stable Koszul DG-module and the telescope DG-module over good enough DG-rings. We prove that over a non-negative commutative noetherian DG-ring, via Koszul homology, via RHom homology, and via local cohomology, all yield the same invariant. As applications of this result, new notions of depth and width for DG-modules are introduced, and the classical Auslander-Buchsbaum formula and amplitude inequalities for DG-modules are established.
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