Errata and Minor Additions for Integral Closure of Ideals, Rings, and Modules by

Abstract

p. 8, lines 6–7 of proof of 1.3.3: replace “The lowest degree component” by “The lowest degree component monomial”; replace “components” by “component monomials”. p. 12, in Definition 1.4.7, now allow the Newton polyhedron to be the convex hull to be either in R or in Q. This harmonizes with the subsequent general definition of the Newton polyhedron in 18.4.1. p. 13, Theorem 1.4.10: The following proof is clearer: Proof: Let n ≥ d. It suffices to prove that I is integrally closed under the assumption that I, I, . . . , In−1 are integrally closed. For this it suffices to prove that every monomial X1 1 · · ·X cd d in the integral closure of I n lies in I. Let {X1 , . . . , Xt} be a monomial generating set of I. By the form of the integral equation of a monomial over a monomial ideal there exist non-negative rational numbers ai such that ∑ ai = n and the vector (c1, . . . , cd) is componentwise greater than or equal to ∑ aivi. By Carathéodory’s Theorem A.2.1 (new version in errata!), by possibly reindexing the generators of I, there exist non-negative rational numbers b1, . . . , bd such that ∑d i=1 bi ≥ n and (c1, . . . , cd) ≥ ∑d i=1 bivi (componentwise). As n ≥ d, there exists j ∈ {1, . . . , d} such that bj ≥ 1. Then (c1, . . . , cd) − vj ≥ ∑ i(bi − δij)vi says that the monomial corresponding to the exponent vector (c1, . . . , cd)− vj is integral over In−1. Since by assumption In−1 is integrally closed, the monomial corresponding to (c1, . . . , cd) − vj is in In−1. Thus X1 1 · · ·X cd d ∈ In−1X v j ⊆ I.

Cite this paper

@inproceedings{Swanson2013ErrataAM, title={Errata and Minor Additions for Integral Closure of Ideals, Rings, and Modules by}, author={Irena Swanson and Craig Huneke and R{\"{u}diger Achilles and Carles Bivi{\`a}-Ausina and Trung Dinh and Florian Enescu and Darij Grinberg and William J. Heinzer and Jo{\~a}o H{\'e}lder and Cristodor Ionescu and Dan Katz and Youngsu Kim and Karl-Heinz Kiyek and Manoj Kummini and Giulio Peruginelli and Janet Striuli and Bernd Ulrich}, year={2013} }