The Use of Ultraproducts in Commutative Algebra

1. Auflage. 2010
ISBN: 978-3-642-13367-1
Verlag: Springer

Buch, Englisch, Band 1999, 210 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 700 g

Reihe: Lecture Notes in Mathematics

In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.

Zielgruppe

Research

Autoren/Hrsg.

Schoutens, Hans

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

Ultraproducts and ?o?’ Theorem.- Flatness.- Uniform Bounds.- Tight Closure in Positive Characteristic.- Tight Closure in Characteristic Zero. Affine Case.- Tight Closure in Characteristic Zero. Local Case.- Cataproducts.- Protoproducts.- Asymptotic Homological Conjectures in Mixed Characteristic.