Non-standard tight closure for affine \(\mathbb C\)-algebras
type: Item
instance of: scholarly article
title: Non-standard tight closure for affine \(\mathbb C\)-algebras
zbMATH Open document ID: 1082.13005
author: Hans Schoutens
published in: Manuscripta Mathematica
publication date: 2003-08-17T00:00:00Z
review text: The paper under review proposes another approach to tight closure in characteristic zero based on methods from model theory. Tight closure is a closure operation for ideals in a Noetherian ring \(R\) containing a field of positive characteristic, and the containment \(z \in I^*\) is given by the condition that there exists a \(c \in R\), not in any minimal prime, such that \(c z^q \in I^{[q]}\) holds for all prime powers. By reduction to positive characteristic, one also obtains a notion of tight closure in characteristic zero, which means in the easiest case that for a finitely generated \(\mathbb{Z}\)-algebra \(S\) the containment \(z \in I^*\) in \(S_\mathbb{Q}\) is by definition the containment \(z \in I^*_p\) in \(S_{k(p)}\) for almost all prime numbers. The approach of the paper is that there is an isomorphism \(\mathbb{C} = \lim \overline{\mathbb F}_p\), where the limit runs over an ultrafilter of sets of prime numbers. In this sense a complex number is represented by a sequence of numbers in \(\overline{\mathbb F}_p\), and in the same manner polynomials \(f_i\) over \(\mathbb{C}\) may be considered as a sequence of polynomials over \(\overline{\mathbb F}_p\). The corresponding factor rings \(R= \mathbb{C}[X_j]/(f_i)\) and \(R_p= \overline{\mathbb F}_p[X_j]/(f_i)\) are related in such a way that the limit of the \(R_p\), denoted by \(R_\infty\), is a (non-Noetherian) faithfully flat extension of \(R\). Moreover, the characteristic\,\(p\) Frobenius endomorphisms of \(R_p\) yield a non-standard Frobenius endomorphism \(F_\infty\) of \(R_\infty\), and with this characteristic zero Frobenius the author defines the non-standard tight closure containment \(z \in I^*\) (\(I \subseteq R\)) by the requirement that \(cF^m_\infty(z) \in F^m _\infty(I) R_\infty\) holds for some \(c\) not in any minimal prime of \(R\) and all \(m\). The first half of the paper (sections 2--5) gives the logical foundations for this closure operation and several variants, while the second h…
reviewed by: Holger Brenner
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 12L10
Mathematics Subject Classification ID: 12L15
zbMATH DE Number: 1965511
zbMATH Keywords: tight closure
zbMATH Keywords: ultraproduct
zbMATH Keywords: model theory
zbMATH Keywords: non-standard Frobenius endomorphism
MaRDI profile type: Publication
full work available at URL: s00229-003-0380-6
OpenAlex ID: W1591841954
DOI: 10.1007/S00229-003-0380-6
description: scientific article; zbMATH DE number 1965511