Commutative Algebra: Chapters 1-7, Volume 1Springer Science & Business Media, 3 août 1998 - 625 pages This is the softcover reprint of the English translation of 1972 (available from Springer since 1989) of the first 7 chapters of Bourbaki's 'Algèbre commutative'. It provides a very complete treatment of commutative algebra, enabling the reader to go further and study algebraic or arithmetic geometry. The first 3 chapters treat in succession the concepts of flatness, localization and completions (in the general setting of graduations and filtrations). Chapter 4 studies associated prime ideals and the primary decomposition. Chapter 5 deals with integers, integral closures and finitely generated algebras over a field (including the Nullstellensatz). Chapter 6 studies valuation (of any rank), and the last chapter focuses on divisors (Krull, Dedekind, or factorial domains) with a final section on modules over integrally closed Noetherian domains, not usually found in textbooks. Useful exercises appear at the ends of the chapters. |
Table des matières
Description of formal mathematics 2 Theory of sets 3 Ordered sets | 1 |
Flat modules | 9 |
Faithfully flat rings and finiteness conditions | 34 |
CHAPTER II | 51 |
Local rings Passage from the local to the global | 80 |
Spectra of rings and supports of modules | 94 |
ideals | 108 |
Exercises for 1 | 121 |
Exercises for 1 | 355 |
Exercises for 2 | 362 |
Exercises for 3 | 370 |
VALUATIONS | 375 |
Places | 381 |
The height of a valuation | 393 |
The topology defined by a valuation | 399 |
Approximation theorem | 412 |
Exercises for 3 | 136 |
Exercises for 5 | 146 |
GRADUATIONS FILTRATIONS AND TOPOLOGIES | 155 |
General results on filtered rings and modules | 162 |
Associated prime ideals and primary decomposition 5 Integers | 168 |
Lifting in complete rings | 209 |
Flatness properties of filtered modules | 226 |
Exercises for 1 | 232 |
Exercises for 3 | 245 |
Exercises for 4 | 255 |
ASSOCIATED PRIME IDEALS AND PRIMARY DECOMPOSITION | 261 |
Primary decomposition | 267 |
Primary decomposition in graded modules | 283 |
Exercises for 2 | 290 |
Exercises for 3 | 301 |
The lift of prime ideals | 325 |
Finitely generated algebras over a field | 344 |
The relation | 421 |
The relation Σe ƒ₁ | 427 |
Extensions of a valuation to a transcendental extension | 434 |
Exercises for 1 | 441 |
Exercises for 4 | 449 |
Exercises for 6 | 459 |
Exercises for 9 | 470 |
Dedekind domains | 493 |
Factorial domains | 502 |
Modules over integrally closed Noetherian domains | 512 |
Exercises for 1 | 545 |
Exercises for 2 | 556 |
Exercises for 3 | 563 |
Historical note Chapters I to VII | 579 |
Bibliography | 603 |
Table of implications | 621 |
Expressions et termes fréquents
A-algebra A₁ Algebra b₁ bijective canonical homomorphism canonical image canonical mapping canonically identified canonically isomorphic closure coefficients commutative ring contained Corollary to Proposition Dedekind domain Deduce defined definition denote diagram direct sum divisor equivalent exact sequence Exercise extension faithfully flat field of fractions filtered filtration finite number finitely generated A-module finitely presented follows formal power series free A-module graded Hausdorff hence hypothesis implies integral domain integrally closed invertible irreducible isomorphic Jacobson radical kernel Krull domain lattice left A-module left ideal Lemma local ring m-adic topology M₁ maximal ideal module morphism multiplicative subset necessary and sufficient nilpotent Noetherian ring non-zero ordered group p₁ polynomial ring prime ideal principal ideal domain projective A-module Proposition 11 Proposition 13 prove quotient residue field resp right A-module S-¹A S-¹M Show Spec(A sub-A-module subgroup submodule subring Suppose surjective Theorem valuation ring whence