scholarly journals What makes a complex a virtual resolution?

2021 ◽  
Vol 8 (28) ◽  
pp. 885-898
Author(s):  
Michael Loper
Keyword(s):  
Classical Theory ◽  
Toric Variety ◽  
Chain Complex ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Content Type ◽  
Graded Modules ◽  
Cox Ring ◽  
A Chain ◽  
Free Modules

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

scholarly journals FINITE DOMINATION AND NOVIKOV RINGS. ITERATIVE APPROACH

2012 ◽  
Vol 55 (1) ◽  
pp. 145-160 ◽  
Author(s):  
THOMAS HÜTTEMANN ◽  
DAVID QUINN
Keyword(s):  
Laurent Series ◽  
Chain Complex ◽  
Laurent Polynomial ◽  
Projective Modules ◽  
Finitely Generated ◽  
Formal Laurent Series ◽  
Chain Complexes ◽  
Laurent Polynomial Ring ◽  
Bounded Below ◽  
Free Modules

AbstractSuppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x−1]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ⊗LR((x)) and C ⊗LR((x−1)) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology34(3) (1995), 619–632). Here R((x)) = R[[x]][x−1] and R((x−1)) = R[[x−1]][x] are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.

On coseparable and 𝔪-coseparable modules

2020 ◽  
pp. 2150031
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Rings ◽  
Direct Products ◽  
Finitely Generated ◽  
Direct Sums ◽  
Direct Sum ◽  
Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

scholarly journals Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rosa M. Miró-Roig ◽  
Martí Salat-Moltó
Keyword(s):  
Polynomial Ring ◽  
Toric Variety ◽  
Hilbert Function ◽  
Hilbert Polynomial ◽  
Lattice Polytopes ◽  
Sharp Bounds ◽  
Graded Modules ◽  
Cox Ring ◽  
Regularity Index

Abstract In this paper, we consider Z r \mathbb{Z}^{r} -graded modules on the Cl ⁡ ( X ) \operatorname{Cl}(X) -graded Cox ring C ⁢ [ x 1 , … , x r ] \mathbb{C}[x_{1},\dotsc,x_{r}] of a smooth complete toric variety 𝑋. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive Z s + r + 2 \mathbb{Z}^{s+r+2} -graded modules over any non-standard bigraded polynomial ring C ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\dotsc,x_{s},\allowbreak y_{0},\dotsc,y_{r}] . In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

Supervisors as recipients and implementers of organizational change: evidence from an Indian chain hospital

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Arash Mashhady
Keyword(s):  
Organizational Change ◽  
Organizational Support ◽  
Professional Relationships ◽  
Organizational Status ◽  
Content Type ◽  
Organizational Policies ◽  
Organizational Policies And Practices ◽  
A Chain ◽  
Change Attitude ◽  
Policies And Practices

Purpose Supervisors play an important role in the implementation of organizational policies and practices. This study aims to examine the role of supervisors as both recipients and main implementers of organizational change by investigating how supervisors’ relationship with organization would affect their attitude toward change (ATC) and how employees–supervisor relationship, as perceived by employees, would influence their reaction to change. Design/methodology/approach The influence of participation, perceived organizational support (POS) and mutual expectation clarity (MEC) on supervisors’ ATC was examined, along with the influence of leader–member exchange, perception of supervisor’s expressed ATC and also supervisors’ organizational status on employees’ ATC. Two studies were conducted in a chain hospital in India. Findings The findings suggest that supervisors’ ATC improved by higher participation, POS and MEC. Also, while employees’ change attitude was predicted by how they perceived their supervisors’ status, expressed reaction toward change and perception of employee–supervisor relationship, for employees who either perceived highly negative change attitude of their supervisors or believed that their supervisors had low organizational status, the employee–supervisor relationship had almost no effect on improving employees’ attitude. Originality/value Considering that supervisors often tend to engage in professional relationships with their subordinate employees, little is investigated on how, through the lens of relationships, supervisors may affect employees’ ATC. This paper attempts to make a difference by conducting two connected studies in a chain hospital to examine how supervisors – as recipients and implementers of organizational policies and practices – could influence employees’ ATC. The findings suggest managerial implications that could inform practitioners toward improvement of employee buy-ins for change programs.

scholarly journals Constructing Gröbner bases for Noetherian rings

2013 ◽  
Vol 24 (2) ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Gröbner Basis ◽  
Finite Depth ◽  
Commutative Rings ◽  
Polynomial Ideal ◽  
Constructive Proof ◽  
Noetherian Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Prime Decomposition ◽  
Separate Paper

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

Research on risk spread model of industrial chain

2014 ◽  
Vol 4 (2) ◽  
pp. 328-338 ◽  
Author(s):  
Yanfeng Chu ◽  
Mei-Mei Dai
Keyword(s):  
Risk Communication ◽  
External Environment ◽  
Chain Complex ◽  
Propagation Model ◽  
Normal Operation ◽  
Design Algorithm ◽  
Content Type ◽  
Industry Chain ◽  
Industrial Chain ◽  
Spread Model

Purpose – The industrial chain network is a complex system consisted by many members of the enterprise, and the complex relationship and the interaction with the external environment among the node enterprises and the existence of various uncertainty all increase the risk of the industry chain. The risk of some individual node enterprises will not only affect the normal operation but also spread the risk to other enterprises by network relationship because of their own mismanagement or deterioration of the external environment. The purpose of this paper is to make an attempt to establish the risk spread model of the industrial chain based on complex networks. Design/methodology/approach – By improving Lobos disaster diffuse model, the paper introduces two indexes: the risk spread range and the risk propagation velocity to measure of industrial chain risk communication effects, and design algorithm for industrial chain complex network structure. The risk spread range can be used to measure the coverage of the risk communication influence produced by the propagation enterprises in the industry chain and to analyse the risk spread breadth on the industrial chain network .The speed index of risk communication represents the total numbers of infection enterprises in unit simulation time. Findings – This paper proposes the universal industrial chain risk propagation model. Originality/value – Through proposed algorithm constructs industrial chain network, and enterprise class divide, the importance of the product chain enterprises in the industry chain is strengthened.

scholarly journals CATEGORIFICATION OF QUANTUM GENERALIZED KAC–MOODY ALGEBRAS AND CRYSTAL BASES

2012 ◽  
Vol 23 (11) ◽  
pp. 1250116 ◽  
Author(s):  
SEOK-JIN KANG ◽  
SE-JIN OH ◽  
EUIYONG PARK
Keyword(s):  
Crystal Structures ◽  
Grothendieck Group ◽  
Integral Form ◽  
Cartan Matrix ◽  
Algebra Homomorphism ◽  
Finitely Generated ◽  
Moody Algebra ◽  
Graded Modules ◽  
Isomorphism Classes ◽  
Highest Weight

We construct and investigate the structure of the Khovanov-Lauda–Rouquier algebras R and their cyclotomic quotients Rλ which give a categorification of quantum generalized Kac–Moody algebras. Let U𝔸(𝔤) be the integral form of the quantum generalized Kac–Moody algebra associated with a Borcherds–Cartan matrix A = (aij)i, j ∈ I and let K0(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism [Formula: see text] and that Φ is an isomorphism if aii ≠ 0 for all i ∈ I. Let B(∞) and B(λ) be the crystals of [Formula: see text] and V(λ), respectively, where V(λ) is the irreducible highest weight Uq(𝔤)-module. We denote by 𝔅(∞) and 𝔅(λ) the isomorphism classes of irreducible graded modules over R and Rλ, respectively. If aii ≠ 0 for all i ∈ I, we define the Uq(𝔤)-crystal structures on 𝔅(∞) and 𝔅(λ), and show that there exist crystal isomorphisms 𝔅(∞) ≃ B(∞) and 𝔅(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac–Moody algebras.

Supply chain structures for distributing surplus food

2020 ◽  
Vol 31 (4) ◽  
pp. 865-883
Author(s):  
Caroline Sundgren
Keyword(s):  
Supply Chain ◽  
Service Provider ◽  
Closed Loop ◽  
Food Distribution ◽  
Weak Links ◽  
Content Type ◽  
Different Types ◽  
Multiple Case ◽  
A Chain

PurposeNew actors have emerged in the food supply chain in response to the increased awareness of food waste and the need to distribute surplus food. The purpose of this study is to analyse the different supply chain structures that have emerged to make surplus food available to consumers.Design/methodology/approachThis study adopts a qualitative multiple-case study of three new surplus food actors: a surplus food platform, an online retailer and a surplus food terminal. Data sources included interviews, documentary evidence and participatory observations.FindingsThree different types of actor constellations in surplus food distribution have been identified: a triad, a tetrad and a chain. Both centralised (for ambient products) and decentralised supply chain structures (for chilled products) have emerged. The analysis identified weak links amongst new actors and surplus food suppliers. The new actors have adopted the roles of connector, service provider and logistics service provider and the sub-roles of mediator, auditor and consultant.Originality/valueThis paper contributes to research on closed-loop or circular supply chains for the reuse of products in the context of surplus food distribution.

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