Quasi-socle ideals in local rings with Gorenstein tangent cones

Abstract

Quasi-socle ideals, that is the ideals $I$ of the form $I=Q:{\mathfrak{m}}^q$ in a Noetherian local ring $(A,\mathfrak{m})$ with the Gorenstein tangent cone $\mathrm{G}(\mathfrak{m})={⨁}_{n\ge 0}{\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$ are explored, where $q\ge 1$ is an integer and $Q$ is a parameter ideal of $A$ generated by monomials of a system $x_1,x_2,...,x_d$ of elements in $A$ such that $(x_1,x_2,...,x_d)$ is a reduction of $\mathfrak{m}$. The questions of when $I$ is integral over $Q$ and of when the graded rings $\mathrm{G}(I)={⨁}_{n\ge 0}{I}^n/I^{n+1}$ and $\mathrm{F}(I)={⨁}_{n\ge 0}{I}^n/\mathfrak{m}{I}^n$ are Cohen-Macaulay are answered. Criteria for $\mathrm{G}(I)$ and $\mathcal{R}(I)={⨁}_{n\ge 0}{I}^n$ to be Gorenstein rings are given.

16 pages, to appear in Journal of Commutative Algebra

References in corpus (1)

• Quasi-socle ideals in Gorenstein numerical semigroup rings