Iterated socles and integral dependence in regular rings

Abstract

Let R R be a formal power series ring over a field, with maximal ideal m \mathfrak {m} , and let I I be an ideal of R R . We study iterated socles of I I , that is, ideals of the form I : R m s I :_R {\mathfrak m}^s for positive integers s 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 I , with reduction number at most one, provided s ≤ o ( I 1 ( φ d ) ) − 1 s \leq \text {o}(I_1(\varphi _d))-1 , where o ( I 1 ( φ d ) ) \text {o}(I_1(\varphi _d)) is the order of the ideal of entries of the last map in a minimal free R R -resolution of R / I R/I . In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o ( I 1 ( φ d ) ) 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 I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R R -resolution of R / I 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.