The Use of Ultraproducts in Commutative Algebra (Lecture Notes in Mathematics Book 1999)

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. Front Matter....Pages i-x Introduction....Pages 1-6 Ultraproducts and Łoś’ Theorem....Pages 7-27 Flatness....Pages 29-50 Uniform Bounds....Pages 51-63 Tight Closure in Positive Characteristic....Pages 65-80 Tight Closure in Characteristic Zero. Affine Case....Pages 81-95 Tight Closure in Characteristic Zero. Local Case....Pages 97-112 Cataproducts....Pages 113-125 Protoproducts....Pages 127-148 Asymptotic Homological Conjectures in Mixed Characteristic....Pages 149-169 Back Matter....Pages 171-210 Ultraproducts And Łoś' 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 -- Henselizations -- Boolean Rings. Hans Schoutens. Includes Bibliographical References (p. 193-197) And Index.