Reduction numbers for ideals of analytic spread one
Let R be a local (Noetherian) ring with infinite residue field, and let I be an ideal of R. An ideal \(J\subseteq I\) is called a reduction provided there exists a nonnegative integer r such that \(JI^ r=I^{r+1}\), and a reduction is said to be minimal if it contains no proper reduction of itself. Northcott and Rees proved that minimal reductions exist, and introduced the related notion of analytic spread [\textit{D. G. Northcott} and \textit{D. Rees}, Proc. Camb. Philos. Soc. 50, 145-158 (954; Zbl 0057.026)]. The analytic spread of I is defined to be the number \(\dim (R/M\oplus I/I^ 2M\oplus I^ 2/I^ 3M\oplus...)\), and Northcott and Rees showed that this number is equal to the number of elements in a minimal generating set for a minimal reduction of I. The reduction number of I is defined as \(r(I)=\min \{n\in {\mathbb{Z}}^+\cup \{0\}: there\quad exists\) a minimal reduction J of I with \(JI^ n=I^{n+1}\}.\) The main result deals with the reduction number of ideals of analytic spread one. In particular, the following theorem is proved: Let (R,M) be a quasi-unmixed analytically unramified local ring having infinite residue field. Assume that R contains a field of characteristic different than two and that the multiplicity of R is less than or equal to two. If I is a regular ideal of analytic spread one, then r(I)\(\leq 1\).
scientific article; zbMATH DE number 4010654