The use of ultraproducts in commutative algebra
type: Item
instance of: scholarly article
title: The use of ultraproducts in commutative algebra
zbMATH Open document ID: 1205.13002
DOI: 10.1007/978-3-642-13368-8
author: Hans Schoutens
published in: Lecture Notes in Mathematics
publication date: 2010-08-11T00:00:00Z
review text: Although the first important application of ultraproducts to algebra and number theory has been made already in the sixties by \textit{J. Ax} and \textit{S. Kochen} [Am. J. Math. 87, 605--630, 631--648 (1965; Zbl 0136.32805)] who used them to obtain essential progress in a problem of Artin, it took some time before this method has been applied to other algebraic problems. The aim of this book is to show several important applications of ultraproducts and their modifications (cataproducts and protoproducts) in the theory of commutative rings without using the language of model theory. The first few chapters introduce the principal notions and results, including a version of \textit{J. Łoś}'s theorem [in: Math. interpretation of formal systems, 98--113 (1955; Zbl 0068.24401)] and a kind of Lefschetz principle. In Chapter 3 flat modules and homomorphisms are considered, and a series of flatness criteria are given. The first applications of ultraproducts appear in the next chapter, where the existence of uniform bounds over polynomial rings is considered, following the approach of \textit{K. Schmidt} and \textit{L. van den Dries} [Invent. Math. 76, 77--91 (1984; Zbl 0539.13011)]. In Chapter 5 one finds the theory of tight closure in characteristic \(p>0\), exposed in an axiomatic way, and this is applied in the next two chapters to construct this theory in characteristic zero. In chapters 8 and 9 cataproducts and protoproducts are defined, studied and applied to uniform bounds (cataproducts are certain factors of ultraproducts of local Noetherian rings and have the nice property that if these rings have bounded embedding dimension, then their cataproduct is Noetherian, and protoproducts are certain subrings of ultraproducts). The last chapter deals with homological conjectures in the mixed characteristic case and brings solutions for some of them when the residual characteristic is sufficiently large. In his Introduction the author writes: ``[The ultraproducts] were con…
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 03C20
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13D22
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13L05
zbMATH DE Number: 5768468
zbMATH Keywords: ultraproduct
zbMATH Keywords: flatness
zbMATH Keywords: tight closure
zbMATH Keywords: Artin approximation
zbMATH Keywords: Lefschetz principle
zbMATH Keywords: homological conjectures
Wikidata QID: Q56038143
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-13368-8
OpenAlex ID: W597691298
description: scientific article; zbMATH DE number 5768468