Title page
Date-line
Preface
Instructions to the reader
TABLE OF CONTENTS
VI. VALUATION THEORY
§ 1. Introductory remarks
§ 2. Places
§ 3. Specialization of places
§ 4. Existence of places
§ 5. The center of a place in a subring
§ 5bis. The notion of the center of a place in algebraic geometry
§ 6. Places and field extensions
§ 7. The case of an algebraic field extension
§ 8. Valuations
§ 9. Places and valuations
§ 10. The rank of a valuation
§ 11. Valuations and field extensions
§ 12. Ramification theory of general valuations
§ 13. Classical ideal theory and valuations
§ 14. Prime divisors in fields of algebraic functions
§ 15. Examples of valuations
§ 16. An existence theorem for composite centered valuations
§ 17. The abstract Riemann surface of a field
§ 18. Derived normal models
VII. POLYNOMIAL AND POWER SERIES RINGS
§ 1. Formal power series
§ 2. Graded rings and homogeneous ideals
§ 3. Algebraic varieties in the affine space
§ 4. Algebraic varieties in the projective space
§ 4bis. Further properties of projective varieties
§ 5. Relations between non-homogeneous and homogeneous ideals
§ 6. Relations between affine and projective varieties
§ 7. Dimension theory in finite integral domains
§ 8. Special dimension-theoretic properties of polynomial rings
§ 9. Normalization theorems
§ 10. Dimension theory in power series rings
§ 11. Extension of the ground field
§ 12. Characteristic functions of graded modules and homogeneous ideals
§ 13. Chains of syzygies
VIII. LOCAL ALGEBRA
§ 1. The method of associated graded rings
§ 2. Some topological notions. Completions
§ 3. Elementary properties of complete modules
§ 4. Zariski rings
§ 5. Comparison of topologies in a noetherian ring
§ 6. Finite extensions
§ 7. Hensel's lemma and applications
§ 8. Characteristic functions
§ 9. Dimension theory. Systems of parameters
§ 10. Theory of multiplicities
§ 11. Regular local rings
§ 12. Structure of complete local rings and applications
§ 13. Analytical irreducibility and analytical normality of normal varieties
APPENDIX
1. Relations between prime ideals in a noetherian domain $\mathfrak{o}$ and in a simple ring extension $\mathfrak{o}[t]$ of $\mathfrak{o}$
2. Valuations in noetherian domains
3. Valuation ideals
4. Complete modules and ideals
5. Complete ideals in regular local rings of dimension 2
6. Macaulay rings
7. Unique factorization in regular local rings
INDEX OF DEFINITIONS

lgrsnf/Zariski O., Samuel P. Commutative algebra. Vol.2 (Van Nostrand, 1960)(ASIN B000OG52ZI)(600dpi)(K)(T)(O)(422s)_MAa_.djvu

2024-08-04