Iterated socles and integral dependence in regular rings

Corso, Alberto; Goto, Shiro; Huneke, Craig; Polini, Claudia; Ulrich, Bernd

doi:10.1090/tran/6926

Iterated socles and integral dependence in regular rings
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by Alberto Corso, Shiro Goto, Craig Huneke, Claudia Polini and Bernd Ulrich PDF

Trans. Amer. Math. Soc. 370 (2018), 53-72 Request permission

Abstract:

Let $R$ be a formal power series ring over a field, with maximal ideal $\mathfrak {m}$, and let $I$ be an ideal of $R$. We study iterated socles of $I$, that is, ideals of the form $I :_R {\mathfrak m}^s$ for positive integers $s$. We are interested in iterated socles in connection with the notion of integral dependence of ideals. In this article we show that iterated socles are integral over $I$, with reduction number at most one, provided $s \leq \text {o}(I_1(\varphi _d))-1$, where $\text {o}(I_1(\varphi _d))$ is the order of the ideal of entries of the last map in a minimal free $R$-resolution of $R/I$. In characteristic zero, we also provide formulas for the generators of iterated socles whenever $s\leq \text {o}(I_1(\varphi _d))$. This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal $I$ in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free $R$-resolution of $R/I$. Applications are given to iterated socles of determinantal ideals with generic height. In particular, we give surprisingly simple formulas for iterated socles of height two ideals in a power series ring in two variables. The generators of these socles are suitable determinants obtained from the Hilbert-Burch matrix.

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