Asymptotic resurgence via integral closures

Integral Closures of Cohen-macaulay Monomial Ideals

The purpose of this paper is to present a family of CohenMacaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.

Normalization via Rewrite Closures

Toric Ideals of Integral Closures from Graphs

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any polynomial ring generated by quadratic monomials. The toric ideal of relations between these generators is generated by a set of graphically described binomials. The spectra of the original ring and of its integral closure turn out to be canonicall...

Integral closures of ideals and rings

I assume some background from Atiyah–MacDonald [2] (especially the parts on Noetherian rings, primary decomposition of ideals, ring spectra, Hilbert’s Basis Theorem, completions). In the first lecture I will present the basics of integral closure with very few proofs; the proofs can be found either in Atiyah–MacDonald [2] or in Huneke–Swanson [13]. Much of the rest of the material can be found ...

Integral Closures of Ideals in Completions of Regular Local Domains

In this paper we exhibit an example of a three-dimensional regular local domain (A,n) having a height-two prime ideal P with the property that the extension P of P to the n-adic completion  of A is not integrally closed. We use a construction we have studied in earlier papers: For R = k[x, y, z], where k is a field of characteristic zero and x, y, z are indeterminates over k, the example A is...