The Use of Ultraproducts in Commutative Algebra

1. Front Matter

   Pages i-x

   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

    PDF

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

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

  Hans Schoutens

Policies and ethics