Modules of generalized fractions and general local cohomology modules
type: Item
instance of: scholarly article
title: Modules of generalized fractions and general local cohomology modules
zbMATH Open document ID: 0601.13007
DOI: 10.1007/BF01196355
author: B. George
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let A be a commutative Noetherian ring with identity. Following the author [J. Lond. Math. Soc., II. Ser. 19, 402-410 (1979; Zbl 0396.13016)] a non-empty set of ideals of A, say \(\Phi\), is called a system of ideals of whenever \({\mathfrak a},{\mathfrak b}\in \Phi\), then there is an ideal \({\mathfrak c}\) in \(\Phi\) such that \({\mathfrak c}\subseteq {\mathfrak a}\cdot {\mathfrak b}\). For such a system, for every A-module M we define \(L_{\Phi}(M)=\{m\in M| \quad {\mathfrak a}\cdot m=0\) for some \({\mathfrak a}\in \Phi\}.\) This functor is called the general local cohomology functor with respect to (w.r.t.) \(\Phi\). For any A-module M, the modules \(H^ i_{\Phi}(M)\) are called general local cohomology modules of M, where \(H^ i_{\Phi}\) is the i-th right derived functor of \(L_{\Phi}\). Also following \textit{R. Y. Sharp} and \textit{H. Zakeri} [Mathematika 29, 32-41 (1982; Zbl 0497.13006)], let \(D_ n(A)\) denote the set of all \(n\times n\) lower triangular matrices over A. A triangular subset of \(A^ n\) is a non- empty subset U of \(A^ n\) such that (i) whenever \((u_ 1,...,u_ n)\in U\), then \((u_ 1^{\alpha _ 1},...,u_ n^{\alpha _ n})\in U\) for all choices of positive integers \(\alpha _ 1,...,\alpha _ n\), and (ii) whenever \((u_ 1,...,u_ n),(v_ 1,...,v_ n)\in U\), then there exists an element \((w_ 1,...,w_ n)\in U\) and matrices \(H,K\in D_ n(A)\) such that \(H[u_ 1,...,u_ n]^ T=[w_ 1,...,w_ n]^ T=K[v_ 1,...,v_ n]^ T\) where T means transpose operation on matrices. Now let U be a triangular subset of \(A^ n\) and M an A-module. The module of generalized fractions of M w.r.t. U, which is denoted by \(U^{-n}M\), is the set of elements of the form \(m/(u_ 1,..,u_ n)\) for \(m\in M\) and \((u_ 1,...,u_ n)\in U\), in which \(m/(u_ 1,...,u_ n)\) denotes the equivalence class of \((m,(u_ 1,...,u_ n))\in M\times U\) under the equivalence relation \(\sim\) defined as follows. For elements \((a,(u_ 1,...,u_ n))\) and \((b,(v_ 1,...,v_ n))\) of \(M\times U\) w…
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13C12
Mathematics Subject Classification ID: 13B30
zbMATH DE Number: 3968733
zbMATH Keywords: general local cohomology modules
zbMATH Keywords: triangular subset
zbMATH Keywords: module of generalized fractions
zbMATH Keywords: system of parameters
zbMATH Keywords: system of ideals
zbMATH Keywords: torsion theory
zbMATH Keywords: local ring
MaRDI profile type: Publication
cites work: Torsion Theories and Local Cohomology over Commutative Noetherian Rings
cites work: A common generalization of local cohomology theories
cites work: Q5813687
cites work: Modules of Generalized Fractions, Matrices and Determinantal Maps
cites work: Modules of generalized fractions
cites work: Local cohomology and modules of generalized fractions
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description: scientific article; zbMATH DE number 3968733