Index of reducibility of parameter ideals in a local ring
Let (A,m) be a local ring. If a is an m-primary ideal of A, then the number N(a) (called the index of reducibility of a) of irreducible ideals that appear in an irredundant decomposition of a is S(A/a) where S(M) denotes the dimension of the socle of an A-module M considered as an A/m- vector space. If a is a parameter ideal, certain properties of N(a) played an important role in the early development of Gorenstein and Cohen-Macaulay rings. For example, if A is Cohen-Macaulay, N(a) is an invariant of the ring. The purpose of this paper is to study N(a) for A not necessarily Cohen-Macaulay. For a finitely generated A-module M, define \(r(M)=\sup \{S(M/qM)| q\quad is\quad a\quad parameter\quad ideal\quad of\quad M\}\). Upper and lower bounds for r(M) involving local cohomologies for M are given (which become equalities if M is a quasi- Buchsbaum module). It is shown that if dim \(A\leq 3\), then \(r(A)<\infty\), but for any integer \(d>3\) there is a local ring A with \(\dim A=3\) and \(r(A)=\infty\). Rings with constant index of reducibility of parameter ideals are also studied.
scientific article; zbMATH DE number 3855253
scientific article