Erratum to “Integral closure of ideals in excellent local rings”: [J. Algebra 187 (1997) 422–445]
✍ Scribed by Donatella Delfino; Irena Swanson
- Publisher
- Elsevier Science
- Year
- 2004
- Tongue
- English
- Weight
- 171 KB
- Volume
- 274
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
We are grateful to Ray Heitmann for pointing out that Theorem 2.7 in the published version is wrong. The proof of the main theorem of the published paper used Theorem 2.7.
Here we give new proofs of the main theorem as well as of some intermediate results. We point out that the main results still prove special cases of the Linear Artin Approximation theorem.
The main theorem. Let (R, m) be an excellent local ring. Let I be an ideal of R. Then there exists a positive integer c such that
As in [1, Section 2], the proof of this theorem reduces to the case where (R, m) is a complete local normal integral domain and I is principal. However, contrary to the claims in [1], we may not assume that I is a radical ideal. In fact, Theorems 2.7 and 2.8 should be cut out of [1].
The following is a slight (and needed) generalization of [1, Theorem 3.9]. The proof here is essentially the same as the one in [1], only more direct. Theorem 3.9. Let (R, m) be a complete normal local domain and f R a non-zero principal ideal. In the case when R does not contain a field, we let p be a generator of the maximal