Quasi-socle ideals and Goto numbers of parameters
Let \((A, \mathfrak m)\) be a commutative Noetherian local ring and \(Q\) a parameter ideal. Denote the Goto number by \(g(Q):=\max\{q\mid (Q:\mathfrak m^q)\subseteq\overline{Q}\}\), where \(\overline{Q}\) is the integral closure of \(Q\). \textit{W. Heinzer} and \textit{I. Swanson} [J. Algebra 321, No. 1, 152--166 (2009; Zbl 1166.13018)] proved that if \(\dim A=1\), then there exists an integer \(k\gg0\) such that \(g(Q)\) is constant for every parameter ideal \(Q\) contained in \(\mathfrak m^k\). One of the main results of the paper under review extend this result in the following sense: Let \(R=k[R_1]\) be a homogeneous Gorenstein ring over a field \(k\) with \(d:=\dim R > 0\). Choose a homogeneous system \(f_1,\ldots,f_d\) of parameters of \(R\) and put \(q=(f_1,\ldots, f_d)\). Define \(g(q)\) by \(\max\{n\mid (q:R_+^n)\subseteq\overline{q}\}\). Denote the a-invariant of \(R\) by \(a(R)\) and degree of \(f_i\) by \(v_i\). Then \(g(q)=a(R)+\sum v_i -\max\{v_i\}\). By definition, a quasi-socle ideal is an ideal of type \((Q:_A\mathfrak m^q)\) where \(Q\) is a parameter ideal and \(q\) is an integer. In the paper there are several results about quasi-socle ideals of rings of type \(B/yB\) where \(B\) is a two dimensional ring with some additional assumptions.
scientific article; zbMATH DE number 5669119
scientific article