© 2010

The Use of Ultraproducts in Commutative Algebra


  • Novel use of ultraproducts in algebra

  • Provides a gentle introduction to tight closure in characteristic zero

  • Contains a survey chapter on various flatness criteria


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

Table of contents

  1. Front Matter
    Pages i-x
  2. Hans Schoutens
    Pages 1-6
  3. Hans Schoutens
    Pages 7-27
  4. Hans Schoutens
    Pages 29-50
  5. Hans Schoutens
    Pages 51-63
  6. Hans Schoutens
    Pages 65-80
  7. Hans Schoutens
    Pages 113-125
  8. Hans Schoutens
    Pages 127-148
  9. Back Matter
    Pages 171-210

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.


algebra flatness homological conjectures tight closure ultraproduct uniform bounds

Authors and affiliations

  1. 1.CUNY Graduate Center, MathematicsCity University of New YorkNew YorkUSA

Bibliographic information