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Shiro Goto, Satoru Kimura, Naoyuki Matsuoka, and Tran Thi Phuong (2008). Quasi-Socle ideals in local rings with gorenstein tangent cones. Journal of Commutative Algebra, arXiv: 0710. 1387v2 [math.AC] 29 jul 2008. 16 pages.

Link: http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.1387v2.pdf

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 $\mathrm{G}(\mathfrak{m}) = \bigoplus_{n \geq 0}{\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$ are explored, where $q \geq 1$ 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 $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ and $\mathrm{F}(I) = \bigoplus_{n \ge 0}I^n/\mathfrak{m} I^n$ are Cohen-Macaulay are answered. Criteria for $\mathrm{G} (I)$ and $\mathcal{R} (I) = \bigoplus_{n \geq 0}I^n$ to be Gorenstein rings are given.