Exercise 6, Appendix 0

Let $A$ be a local ring with residue field $k$. Let $I$ be an index set. Now prove that the ring $A\mathbf{[}(X_i)_{i\in I}\mathbf{]}$ (cf. App., Example 2), which is a local ring blowing-up of $A$, has as residue field the purely transcendental extension $k((X_i)_{i\in I})$ of $k$.

Theorem of Chevalley: VIII, p. 101, exerc. 10. Theorem of Nagata: IX, p. 32.

Witt vector: IX, p. 7. Finite length Witt vector: IX, p. 12.

Table of Contents

CHAPTER VIII. — DIMENSION ............................................. VIII.1

§ 1. Krull dimension of a ring .................................. VIII.1 1. Krull dimension of a topological space ................. VIII.1 2. Codimension of a closed subset ............................... VIII.4 3. Dimension of a ring, height of an ideal .................... VIII.6 4. Dimension of a finitely generated module ............................ VIII.10 5. Cycles associated with a module ................................... VIII.11

§ 2. Dimension of algebras .......................................... VIII.13 1. Dimension and flatness .......................................... VIII.13 2. Dimension of an algebra of finite type ........................... VIII.16 3. Dimension of an integral algebra ............................... VIII.17 4. Algebras of finite type over a field ............................. VIII.19

§ 3. Dimension of Noetherian rings ............................. VIII.24 1. Dimension of a quotient ring ................................. VIII.24 2. Dimension and intersecting sequence ..................................... VIII.26 3. First applications .......................................... VIII.29 4. Changes of rings ........................................... VIII.32 5. Construction of intersecting sequences ............................... VIII.36

§ 4. Hilbert-Samuel series ........................................ VIII.37 1. The ring Z((T)) .................................................. VIII.37 2. Poincaré series of a graded module over a polynomial ring .................................................. VIII.39 3. Hilbert-Samuel series of a well-filtered module ............... VIII.43 4. Degree of the Hilbert-Samuel function ....................... VIII.47 5. Hilbert-Samuel series of a quotient module ................. VIII.48

§ 5. Regular local rings ......................................... VIII.51 1. Definition of regular local rings ....................... VIII.51 2. Associated graded ring of a regular local ring ............. VIII.53 3. Quotients of regular local rings .......................... VIII.55 4. Eisenstein polynomials ......................................... VIII.56 5. Structure of complete regular Noetherian local rings VIII.60

§ 6. Dimension of graded rings ........................................ VIII.62 1. Filtered ring associated with a graded ring ......................... VIII.62 2. Dimension and chains of graded ideals .......................... VIII.63 3. Dimension of graded modules .................................. VIII.65 4. Semi-continuity of dimension ................................ VIII.69 5. Regular graded algebras .................................... VIII.70

§ 7. Multiplicities ............................................................. VIII.71 1. Multiplicity of a module with respect to an ideal ............... VIII.71 2. Multiplicities and flat extensions ............................... VIII.73 3. Multiplicities and finite extensions ............................... VIII.74 4. Multiplicities and intersecting sequences ................................. VIII.76 5. Superficial elements ............................................... VIII.77 Exercises of § 1 ............................................................. VIII.81 Exercises of § 2 ............................................................. VIII.84 Exercises of § 3 ............................................................. VIII.86 Exercises of § 4 ............................................................. VIII.87 Exercises of § 5 ............................................................. VIII.94 Exercises of § 6 ............................................................. VIII.99 Exercises of § 7 ............................................................. VIII.103

CHAPTER IX. — COMPLETE NOETHERIAN LOCAL RINGS ............ IX.1

§ 1. Witt vectors ..................................................... IX.1 1. Witt polynomials .................................................. IX.1 2. The mappings f, v and Φ .......................................... IX.2 3. Construction of polynomials ......................................... IX.4 4. The ring W(A) of Witt vectors .............................. IX.6 5. The homomorphism F and the shift V ............................. IX.7 6. Filtration and topology of the ring W(A) ....................... IX.10 7. The rings W_n(A) of finite length Witt vectors ... IX.12 8. The ring of Witt vectors with coefficients in a ring of characteristic p ............................................. IX.15

§ 2. Cohen rings .................................................... IX.17 1. p-rings ............................................................ IX.17 2. Cohen rings .................................................. IX.20 3. Existence and uniqueness of p-rings ............................... IX.22 4. Multiplicative representatives ....................................... IX.24 5. Structure of complete Noetherian local rings .......... IX.26

§ 3. Residue fields of representatives ............................................. IX.28 1. Local rings of equal characteristics .................... IX.28 2. A lifting theorem .................................. IX.28 3. Fields of representatives ....................................... IX.29

§ 4. Integral closure of a complete local ring ................. IX.30 1. Japanese rings ............................................. IX.30 2. Nagata's theorem .......................................... IX.31 3. Some lemmas .............................................. IX.33 4. Nagata rings ............................................ IX.34

Appendix ............................................................. IX.37 1. Inductive limit of local rings ............................ IX.37 2. Enlargement of a local ring ................................. IX.37 3. Existence of $p$-rings ...................................... IX.41 Exercises of § 1 .................................................... IX.42 Exercises of § 2 .................................................... IX.68 Exercises of § 3 .................................................... IX.75 Exercises of § 4 .................................................... IX.78 Exercises of the Appendix ........................................... IX.84 Index of notations ................................................ 195 Terminological index ............................................... 196 Table of contents ................................................ 198

MASSON, Publisher. 120, boulevard Saint-Germain 75280 Paris Cedex 06 Legal deposit: June 1983

JOUVE 18, rue Saint-Denis 75001 Paris Legal deposit: June 1983 Printing number: 10714