Commutative Algebra: Volume IIThis second volume of our treatise on commutative algebra deals largely with three basic topics, which go beyond the more or less classical material of volume I and are on the whole of a more advanced nature and a more recent vintage. These topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra. Because most of these topics have either their source or their best motivation in algebraic geom etry, the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered through out the exposition. Thus, this volume can be used in part as an introduc tion to some basic concepts and the arithmetic foundations of algebraic geometry. The reader who is not immediately concerned with geometric applications may omit the algebro-geometric material in a first reading (see" Instructions to the reader," page vii), but it is only fair to say that many a reader will find it more instructive to find out immediately what is the geometric motivation behind the purely algebraic material of this volume. The first 8 sections of Chapter VI (including § 5bis) deal directly with properties of places, rather than with those of the valuation associated with a place. These, therefore, are properties of valuations in which the value group of the valuation is not involved. |
Contents
| vi | |
| 3 | |
| 7 | |
| 11 | |
| 15 | |
5bis The notion of the center of a place in algebraic geometry | 21 |
Places and field extensions | 24 |
The case of an algebraic field extension | 27 |
Normalization theorems | 209 |
CHAPTER PAGE 10 Dimension theory in power series rings | 217 |
Extension of the ground field | 221 |
Characteristic functions of graded modules and homogeneous ideals | 230 |
Chains of syzygies | 237 |
LOCAL ALGEBRA 1 The method of associated graded rings | 248 |
Some topological notions Completions | 251 |
3 Elementary properties of complete modules | 258 |
Valuations | 32 |
Places and valuations | 35 |
The rank of a valuation | 39 |
Valuations and field extensions | 50 |
Ramification theory of general valuations | 67 |
Classical ideal theory and valuations | 82 |
Prime divisors in fields of algebraic functions | 88 |
Examples of valuations | 99 |
An existence theorem for composite centered valuations | 106 |
The abstract Riemann surface of a field | 110 |
Derived normal models | 123 |
POLYNOMIAL AND POWER SERIES RINGS 1 Formal power series | 129 |
Graded rings and homogeneous ideals | 149 |
Algebraic varieties in the affine space | 160 |
Algebraic varieties in the projective space | 168 |
4bis Further properties of projective varieties | 173 |
Relations between nonhomogeneous and homogeneous ideals | 179 |
Relations between affine and projective varieties | 187 |
Dimension theory in finite integral domains | 192 |
Special dimensiontheoretic properties of polynomial rings | 203 |
Zariski rings | 261 |
Comparison of topologies in a noetherian ring | 270 |
Finite extensions | 276 |
Hensels lemma and applications | 278 |
Characteristic functions | 283 |
Dimension theory Systems of parameters | 288 |
Theory of multiplicities | 294 |
Regular local rings | 301 |
Structure of complete local rings and applications | 304 |
Analytical irreducibility and analytical normality of normal varieties | 313 |
Relations between prime ideals in a noetherian domain o and in a simple ring extension o t of o | 321 |
Valuations in noetherian domains | 330 |
Valuation ideals | 340 |
Complete modules and ideals | 347 |
Complete ideals in regular local rings of dimension 2 | 362 |
Macaulay rings | 394 |
Unique factorization in regular local rings | 404 |
| 409 | |
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Common terms and phrases
a₁ affine variety algebraic extension assertion associated prime associated prime ideal assume belongs closure coefficients complete ideal completes the proof contains contracted ideal coördinate Corollary decomposition defined denote dimension direct sum directional form exists an element F₁ fact finite A-module finite integral domain finite number follows graded ring Hausdorff space Hence homogeneous elements homogeneous ideal homomorphism implies integral domain integrally closed intersection irreducible irredundant isolated prime ideal isomorphic kernel Lemma m-topology mapping maximal ideal module noetherian ring non-negative P₁ polynomial ring power series ring primary prime divisor prime sequence Proposition prove quotient field quotient ring rank regular local ring relation residue field respect satisfies semi-local ring submodule subring subset system of parameters Theorem 19 topology transcendence degree v-ideal v₁ valuation ideals valuation ring value group whence X₁ Y₁ zero divisor