This monograph provides an exhaustive treatment of several classes of Noetherian rings and morphisms of Noetherian local rings. Chapters carefully examine some of the most important topics in the area, including Nagata, F-finite and excellent rings, Bertini’s Theorem, and Cohen factorizations. Of particular interest is the presentation of Popescu’s Theorem on Neron Desingularization and the structure of regular morphisms, with a complete proof. Classes of Good Noetherian Rings will be an invaluable resource for researchers in commutative algebra, algebraic and arithmetic geometry, and number theory.
Author(s): Cristodor Ionescu
Series: Frontiers in Mathematics
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 486
City: Cham
Preface
Conventions
Contents
1 Fibres of Noetherian Rings
1.1 Properties of Noetherian Local Rings
1.2 The P-Locus of a Noetherian Ring
1.3 Some Useful Results on Inductive Limits
1.4 Algebras over a Field and Base Change
1.5 Preparing for Bad Examples
1.6 P-Morphisms and P-Rings
2 Nagata Rings and Reduced Morphisms
2.1 Japanese Rings
2.2 Nagata Rings
2.3 The Zariski-Nagata Theorem and Applications
3 Excellent Rings and Regular Morphisms
3.1 Chain Conditions
3.2 Regular Morphisms and André's Theorem
3.3 Excellent Rings
3.4 Criteria of Excellence Using the Frobenius Morphism
3.5 F-Finite Rings
3.6 Universally Finite Module of Differentials and Excellent Rings
3.7 Inductive Limits of P-Rings and Applications
4 Localization and Lifting Theorems
4.1 Some Geometric Preliminaries
4.2 Localization Theorems
4.3 Lifting Theorems in the Semilocal Case
4.4 Lifting of the Property Reg-2
4.5 The Second Theorem of Bertini for Local Rings
4.6 Lifting in the General Case
5 Structure of Regular Morphisms
5.1 Rings with Artin Approximation Property
5.2 Néron Desingularization
5.3 Excellent Henselian Rings
6 Further Results on Classes of Good Rings
6.1 Structure of Local Homomorphisms: Cohen Factorizations
6.2 Morphisms with Complete Intersection Fibres and Around
6.3 Hochschild Homology and Regular Morphisms
6.4 About Normality and Formal Fibres
References
Index of Terminology
Authors Index
List of Examples
