Commutative Algebra, Vol. 2

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