Skip to main content
SpringerLink
Log in
Book cover

The Use of Ultraproducts in Commutative Algebra

  • Book
  • © 2010

Overview

Authors:
  1. Hans Schoutens

    1. CUNY Graduate Center, Mathematics, City University of New York, New York, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Novel use of ultraproducts in algebra
  • Provides a gentle introduction to tight closure in characteristic zero
  • Contains a survey chapter on various flatness criteria
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 1999)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (10 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • Hans Schoutens
    Pages 1-6

  3. Ultraproducts and Łoś’ Theorem

    • Hans Schoutens
    Pages 7-27

  4. Flatness

    • Hans Schoutens
    Pages 29-50

  5. Uniform Bounds

    • Hans Schoutens
    Pages 51-63

  6. Tight Closure in Positive Characteristic

    • Hans Schoutens
    Pages 65-80

  7. Tight Closure in Characteristic Zero. Affine Case

    • Hans Schoutens
    Pages 81-95

  8. Tight Closure in Characteristic Zero. Local Case

    • Hans Schoutens
    Pages 97-112

  9. Cataproducts

    • Hans Schoutens
    Pages 113-125

  10. Protoproducts

    • Hans Schoutens
    Pages 127-148

  11. Asymptotic Homological Conjectures in Mixed Characteristic

    • Hans Schoutens
    Pages 149-169

  12. Back Matter

    Pages 171-210
    Download chapter PDF
Back to top

Keywords

About this book

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.

Authors and Affiliations

  • CUNY Graduate Center, Mathematics, City University of New York, New York, USA

    Hans Schoutens

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation