On two theorems concerning reductions in local rings
Let \((R,m,k)\) denote a local ring. Let I and J be ideals in R with \(J\subseteq I\). Then J is called a reduction of I if \(JI^ n=I^{n+1}\) for some positive integer n. This and the related concept of analytic spread of I were introduced by \textit{D. G. Northcott} and \textit{D. Rees} [Proc. Camb. Philos. Soc. 50, 145-158 (1954; Zbl 0057.026)]. Both of them have proved extremely useful in local algebra. In the paper under review the author pursues two aims. First he generalizes a recent theorem of \textit{D. Rees} [J. Lond. Math. Soc., II. Ser. 29, 397-414 (1984; Zbl 0572.13005); theorem 1.3] on the existence of so-called complete reductions of a finite set of ideals of R. Secondly he gives a proof of the basic theorem of \textit{P. Eakin} and \textit{A. Sathaye} [J. Algebra 41, 439-454 (1976; Zbl 0348.13012) theorem 1] on the existence of reductions which avoids the adding on of an infinite set of indeterminates and the resulting action of a permutation group. In fact the proofs are rather short and ingenious. In the proof of the Rees theorem the author uses a general position result in a singly-graded ring in contrast to the original argument which works in a multi-graded Rees ring. The essential object in the paper is the fibre ring over the closed point in the blow-up of the ring along the given ideal.
complete reductions of a finite set of ideals
scientific article; zbMATH DE number 4043976