The Use of Ultraproducts in Commutative Algebra / by Hans Schoutens.

Format:

Book

Author/Creator:

Schoutens, Hans, author.

Contributor:

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics, 0075-8434 ; 1999.

Lecture Notes in Mathematics, 0075-8434 ; 1999

Language:

English

Subjects (All):

Algebra.

Geometry, Algebraic.

Commutative Rings and Algebras.

Algebraic Geometry.

Local Subjects:

Commutative Rings and Algebras.

Algebraic Geometry.

Physical Description:

1 online resource (X, 210 pages).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg, 2010.

System Details:

text file PDF

Summary:

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.

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.

Other Format:

Printed edition:

ISBN:

9783642133688

Access Restriction:

Restricted for use by site license.