Modules of generalized fractions and general local cohomology modules
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 \(M\times U\) we write \((a,(u_ 1,...,u_ n))\sim (b,(v_ 1,...,v_ n))\) if and only if there exists \((w_ 1,...,w_ n)\in U\) and \(H,K\in D_ n(A)\) such that \(H[u_ 1,...,u_ n]^ T=[w_ 1,...,w_ n]^ T=K[v_ 1,...,v_ n]^ T\) and \(| H| a-| K| b\in (\sum ^{n- 1}_{i=1}Aw_ i)M.\) The module operations for \(U^{-n}M\) are as follows. If a,b\(\in M\) and \((u_ 1,...,u_ n),(v_ 1,...,v_ n)\in U\). then \(a/(u_ 1,...,u_ n)+b/(v_ 1,...,v_ n)=(Ha+Kb)/(w_ 1,...,w_ n)\) for any choice of \((w_ 1,...,w_ n)\in U\) and \(H,K\in D_ n(A)\) for which \(Hu=Kv\). (Here for any element \((s_ 1,...,s_ n)\in U\), the matrix \([s_ 1,...,sn]^ T\) is denoted by s.) Also for \(\lambda\) \(\in A:\) \(\lambda (a/(u_ 1,...,u_ n))=\lambda a/(u_ 1,...,u_ n)\). If U is a triangular subset of \(A^ n\), the set \(U[1]=\{(u_ 1,...,u_ n,1)| (u_ 1,...,u_ n)\in U\}\) is a triangular subset of \(A^{n+1}\) also. Now suppose A is a Noetherian local ring of dimension n. Then \(U=\{(u_ 1,...,u_ n)| u_ 1,...,u_ n\) is a system of parameters for \(A\}\) is a triangular subset of \(A^ n\). - It is proved [\textit{R. Y. Sharp} and \textit{H. Zakeri}, Mathematika 29, 296-306 (1982; Zbl 0523.13001)] that \(U[1]^{-n-1}M\cong H^ n_{{\mathfrak m}}(M)\) where \(H^ n_{{\mathfrak m}}(M)\) is the n-th local cohomology module of M w.r.t. the unique maximal ideal \({\mathfrak m}\) of A, \(n=\dim A\), and U is the above triangular subset of \(A^ n\) consisting of systems of parameters. Then as a generalization of this, let \({\mathfrak a}\) be an ideal of A which is generated by \(x_ 1,...,x_ n\), and \(U_ x\) be the triangular subset \(U_ x=\{x_ 1^{\alpha _ 1},...,x_ n^{\alpha _ n})| \quad \alpha _ 1,...,\alpha _ n\in {\mathbb{N}}\}\) of \(A^ n\). Then [see \textit{G. J. Gibson}, J. Lond. Math. Soc., II. Ser. 33, 245-252 (1986; Zbl 0562.13009)] \(U_ x[1]^{-n-1}M\cong H^ n_{{\mathfrak a}}(M).\) In this paper is proved a further generalization of these results, which is in terms of general local cohomology modules: Let U be an triangular subset of \(A^ n\). Take \(\Phi\) as the set of all ideals which are generated by an element of U. Then \(\Phi\) is a system of ideals of A and moreover we have the isomorphism \(U[1]^{-n-1}M\cong H^ n_{\Phi}(M)\). Moreover any system of ideals defines a torsion theory over A. This is the theory with the filter D generated by \(\Phi\) as follows \(D=\{{\mathfrak b},\) an ideal of \(A| \quad {\mathfrak b}\supseteq {\mathfrak a}\) for some \({\mathfrak a}\in \Phi \}.\) Thus any triangular subset U of \(A^ n\) defines a torsion theory \({\mathcal F}\) over A in a natural way. At the end, as an application, it is proved that if \({\mathcal F}\) is a torsion theory in which its idempotent filter has a cofinal subset of n- generated ideals (i.e. ideals generated by n elements), and that T is the corresponding torsion radical, then for any A-modules M and N\ \(T_ n(M\otimes _ AN)\simeq T_ n(M)\otimes _ AN\) where \(T_ n\) is the n-th right derived functor of T.
scientific article; zbMATH DE number 3968733
scientific article