Iterated socles and integral dependence in regular rings

Let R be a formal power series ring over a field, with maximal ideal m, and let I be an ideal of R. We study iterated socles of I, that is, ideals of the form I:_R 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 ≤ o(I₁ (ϕ_d)) − 1, where o(I₁ (ϕ_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 ≤ o(I₁ (ϕ_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.