Generalized fractions, determinantal maps, and top cohomology modules
This paper contains some interesting additions to the theory of modules of generalized fractions begun by the reviewer and \textit{H. Zakeri} in Mathematika 29, 32-41 and 296-306 (1982; Zbl 0497.13006 and 523.13001) and in Commutative algebra, Symp. Durham 1981, Lond. Math. Soc. Lect. Notes Ser. 72, 61-82 (1982; Zbl 0501.13004) and developed by the author in J. Lond. Math. Soc. 28, 417-427 (1983; Zbl 0497.13007). - Let A be a commutative ring (with identity), let n be a positive integer, let \(D_ n(A)\) (respectively \(M_ n(A))\) denote the set of \(n\times n\) lower triangular matrices (respectively \(n\times n\) matrices) over A, and let M be an A-module. If \(x_ 1,...,x_ n\), \(y_ 1,...,y_ n\in A\) and there exists \(H\in M_ n(A)\) such that \(H(x_ 1,...,x_ n)^ T=(y_ 1,...,y_ n)^ T,\) then multiplication by \(| H|\) (the determinant of H) induces an A-homomorphism \(\alpha_ H:M/\sum^{n}_{i=1}x_ iM\to M/\sum^{n}_{i=1}y_ iM. \alpha_ H\) is called the determinantal map associated with H; such maps play an important role in this paper. For instance, if we let U be a triangular subset of \(A^ n\) and set \(U[1]=U\times \{1\},\) a triangular subset of \(A^{n+1}\), then the author's main result of {\S}2 shows that \(U[1]^{-n-1}M\) may be viewed as a direct limit (over the directed set U) of the modules \(M/\sum^{n}_{i=1}x_ iM\) for \((x_ 1,...,x_ n)\in U\), the constituent maps of the direct system being determinantal maps associated with lower triangular matrices in \(D_ n(A)\). Next, {\S}3 begins with two technical lemmas; the second of these, although too complicated to be stated here, could well, in the reviewer's opinion, turn out to be very useful. This is then used to show that, if A is Noetherian, M is finitely generated, \(x_ 1,...,x_ n\) and \(y_ 1,...,y_ n\) are poor M- sequences and \(H\in M_ n(A)\) is as above then \(\alpha_ H\) is injective. The author also uses his ideas to provide a different approach to results of the reviewer and Zakeri in their second cited Mathematika paper which connect modules of generalized fractions and certain local cohomology modules. He ends with a generalization of one of the Matlis' results concerning duality [\textit{E. Matlis}, Commun. Algebra 1, 87-114 (1974; Zbl 0277.13011)] and some comments about the local Lichtenbaum-Hartshorne theorem.
scientific article; zbMATH DE number 3853225
scientific article