The Use of Ultraproducts in Commutative Algebra

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.

List of contents

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.

Product details

• Authors: Hans Schoutens
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 31.07.2010
• EAN: 9783642133671
• ISBN: 978-3-642-13367-1
• No. of pages: 210
• Dimensions: 151 mm x 236 mm x 14 mm
• Weight: 342 g
• Illustrations: X, 210 p.
• Series: Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; B, Algebraische Geometrie, Mathematics and Statistics, Algebraic Geometry, Commutative algebra, Commutative rings, Commutative Rings and Algebras, uniform bounds