Commutative algebra

To the Reader . v
Contents of the Elements of Mathematics Series . ix
Introduction . xix
Chapter I. Flat Modules . 1
§ 1. Diagrams and exact sequences . 1
1. Diagrams. 1
2. Commutative diagrams. 2
3. Exact sequences . 3
4. The snake diagram . 4
§ 2. Flat modules . 9
1. Revision of tensor products. 9
2. M-flat modules. 10
3. Flat modules. 12
4. Examples of flat modules . 14
5. Flatness of quotient modules . 15
6. Intersection properties. 17
7. Tensor products of flat modules. 19
8. Finitely presented modules. 20
9. Extension of scalars in homomorphism modules . 22
10. Extension of scalars: case of commutative rings . 22
11. Interpretation of flatness in terms of relations. 25
§ 3. Faithfully flat modules . 27
1. Definition of faithfully flat modules. 27
2. Tensor products of faithfully flat modules . 30
3. Change of ring . 31
4. Restriction of scalars . 31
5. Faithfully flat rings . 32
6. Faithfully flat rings and finiteness conditions. 34
7. Linear equations over a faithfully flat ring . 35
§ 4. Flat modules and “Tor” functors. 37
Exercises for § 1 . 39
Exercises for § 2. 41
Exercises for § 3. 49
Exercises for § 4. 50
Chapter II. Localization. 51
§ 1. Prime ideals. 51
1. Definition of prime ideals. 51
2. Relatively prime ideals . 53
§ 2. Rings and modules of fractions. 55
1. Definition of rings of fractions. 55
2. Modules of fractions. 60
3. Change of multiplicative subset. 64
4. Properties of modules of fractions. 67
5. Ideals in a ring of fractions. 70
6. Nilradical and minimal prime ideals. 73
7. Modules of fractions of tensor products and homomorphism
modules. 75
8. Application to algebras. 77
9. Modules of fractions of graded modules. 78
§ 3. Local rings. Passage from the local to the global . 80
1. Local rings . 80
2. Modules over a local ring. 82
3. Passage from the local to the global . 87
4. Localization of flatness . 91
5. Semi-local rings . 92
§ 4. Spectra of rings and supports of modules. 94
1. Irreducible spaces. 94
2. Noetherian topological spaces. 97
3. The prime spectrum of a ring . 98
4. The support of a module. 104
§ 5. Finitely generated projective modules. Invertible fractional
ideals. j Qg
1. Localization with respect to an element. 108
2. Local characterization of finitely generated projective
modules. 109
3. Ranks of projective modules. Ill
4. Projective modules of rank 1 . 114
5. Non-degenerate submodules. 116
6. Invertible submodules. 117
7. The group of classes of invertible modules. 119
Exercises for §1 . 121
Exercises for § 2. 123
Exercises for § 3. 136
Exercises for § 4. 140
Exercises for § 5. 146
Chapter III. Graduations, Filtrations and Topologies . 155
§ 1. Finitely generated graded algebras. 155
1. Systems of generators of a commutative algebra. 155
2. Criteria of finiteness for graded rings . 156
3. Properties of the ring A (d:> . 157
4. Graded prime ideals . 160
§ 2. General results on filtered rings and modules. 162
1. Filtered rings and modules. 162
2. The order function. 165
3. The graded module associated with a filtered module. ... 165
4. Homomorphisms compatible with filtrations . 169
5. The topology defined by a filtration . 170
6. Complete filtered modules . 173
7. Linear compactness properties of complete filtered modules 176
8. The lift of homomorphisms of associated graded modules 177
9. The lift of families of elements of an associated graded
module. 179
10. Application: examples of Noetherian rings. 183
11. Complete m-adic rings and inverse limits. 185
12. The Hausdorff completion of a filtered module. 187
13. The Hausdorff completion of a semi-local ring. 192
§3. rn-adic topologies on Noetherian rings. 195
1. Good filtrations . 195
2. m-adic topologies on Noetherian rings . 199
3. Zariski rings . 201
4. The Hausdorff completion of a Noetherian ring. 202
5. The completion of a Zariski ring. 206
§ 4. Lifting in complete rings. 209
1. Strongly relatively prime polynomials. 209
2. Restricted formal power series. 212
3. Hensel’s Lemma. 215
4. Composition of systems of formal power series. 218
5. Systems of equations in complete rings. 220
6. Application to decompositions of rings . 225
§ 5. Flatness properties of filtered modules. 226
1. Ideally Hausdorff modules . 226
2. Statement of the flatness criterion . 227
3. Proof of the flatness criterion . 228
4. Applications . 230
Exercises for § 1 . 232
Exercises for § 2. 233
Exercises for § 3. 245
Exercises for § 4. 255
Exercises for § 5. 259
Chapter IV. Associated Prime Ideals and Primary Decomposition 261
§ 1. Prime ideals associated with a module. 261
1. Definition of associated prime ideals. 261
2. Localization of associated prime ideals. 263
3. Relations with the support. 265
4. The case of finitely generated modules over a Noetherian
ring. 265
§ 2. Primary decomposition. 267
1. Primary submodules . 267
2. The existence of a primary decomposition. 270
3. Uniqueness properties in the primary decomposition .... 270
4. The localization of a primary decomposition. 272
5. Rings and modules of finite length . 274
6. Primary decomposition and extension of scalars . 279
§ 3. Primary decomposition in graded modules. 283
1. Prime ideals associated with a graded module. 283
2. Primary submodules corresponding to graded prime ideals 284
3. Primary decomposition in graded modules. 285
Exercises for § 1 . 286
Exercises for § 2. 290
Exercises for § 3. 301
Chapter V. Integers. 303
§ 1. Notion of an integral element . 303
1. Integral elements over a ring. 303
2. The integral closure of a ring. Integrally closed domains 308
3. Examples of integrally closed domains . 309
4. Completely integrally closed domains. 312
5. The integral closure of a ring of fractions. 314
6. Norms and traces of integers . 316
7. Extension of scalars in an integrally closed algebra. 318
8. Integers over a graded ring . 320
9. Application : invariants of a group of automorphisms of an
algebra. 323
§ 2. The lift of prime ideals. 325
1. The first existence theorem. 325
2. Decomposition group and inertia group. 330
3. Decomposition and inertia for integrally closed domains. . 337
4. The second existence theorem . 343
§ 3. Finitely generated algebras over a field. 344
1. The normalization lemma. 344
2. The integral closure of a finitely generated algebra over a
field. 348
3. The Nullstellensatz. 349
4. Jacobson rings. 351
Exercises for §1 . 355
Exercises for § 2. 362
Exercises for § 3. 370
Chapter VI. Valuations. 375
§ 1. Valuation rings . 375
1. The relation of domination between local rings. 375
2. Valuation rings. 376
3. Characterization of integral elements . 378
4. Examples of valuation rings . 379
§2. Places. 381
1. The notion of morphism for laws of composition not every¬
where defined. 381
2. Places. 381
3. Places and valuation rings . 383
4. Extension of places. 384
5. Characterization of integral elements by means of places. . 385
§ 3. Valuations. 385
1. Valuations on a ring . 385
2. Valuations on a field. 387
3. Translations . 389
4. Examples of valuations . 389
5. Ideals of a valuation ring . 391
6. Discrete valuations. 392
§ 4. The height of a valuation. 393
1. Inclusion of valuation rings of the same field. 393
2. Isolated subgroups of an ordered group . 394
3. Comparison of valuations . 395
4. The height of a valuation. 396
5. Valuations of height 1 .. 397
§ 5. The topology defined by a valuation . 399
1. The topology defined by a valuation. 399
2. Topological vector spaces over a field with a valuation . . 401
3. The completion of a field with a valuation . 402
§ 6. Absolute values . 403
1. Preliminaries on absolute values . 403
2. Ultrametric absolute values. 405
3. Absolute values on Q.. 406
4. Structure of fields with a non-ultrametric absolute value. . 407
§ 7. Approximation theorem. 412
1. The intersection of a finite number of valuation rings ... 412
2. Independent valuations. 413
3. The case of absolute values. 415
§ 8. Extensions of a valuation to an algebraic extension
1. Ramification index. Residue class degree
2. Extension of a valuation and completion
3. The relation 2 e t f 4. Initial ramification index..
5. The relation 2 e t f = n .
6. Valuation rings in an algebraic extension.
7. The extension of absolute values ..
§ 9. Application : locally compact fields.
1. The modulus function on a locally compact field.
2. Existence of representatives.
3. Structure of locally compact fields.
§10. Extensions of a valuation to a transcendental extension.
1. The case of a monogenous transcendental extension
2. The rational rank of commutative group.
3. The case of any transcendental extension.
Exercises for § 1
Exercises for § 2
Exercises for § 3
Exercises for § 4
Exercises for § 5
Exercises for § 6
416
416
418
420
422
423
427
428
431
431
432
433
434
434
437
438
441
444
446
449
454
459
Exercises for § 7. 460
Exercises for § 8. 461
Exercises for § 9. 470
Exercises for §10. 471
Chapter VII. DrvisORS . 475
§1. Krull domains. 475
1. Divisorial ideals of an integral domain. 475
2. The monoid structure on D(A) . 478
3. Krull domains. 480
4. Essential valuations of a Krull domain. 482
5. Approximation for essential valuations. 484
6 . Prime ideals of height 1 in a Krull domain . 485
7. Application: new characterizations of discrete valuation
rings. 487
8 . The integral closure of a Krull domain in a finite extension
of its field of fractions . 487
9. Polynomial rings over a Krull domain . 488
10. D ivisor classes in Krull domains . 489
§ 2. Dedekind domains. 493
1. Definition of Dedekind domains. 493
2. Characterizations of Dedekind domains . 494
3. Decomposition of ideals into products of prime ideals .... 496
4. The approximation theorem for Dedekind domains. 497
5. The Krull-Akizuki Theorem. 499
§ 3. Factorial domains . 502
1. Definition of factorial domains. 502
2. Characterizations of factorial domains. 502
3. Decomposition into extremal elements . 504
4. Rings of fractions of a factorial domain. 505
5. Polynomial rings over a factorial domain. 505
6 . Factorial domains and Zariski rings. 506
7. Preliminaries on automorphisms of rings of formal power
series. 506
8 . The preparation theorem . 507
9. Factoriality of rings of formal power series. 511
§4. Modules over integrally closed Noetherian domains. 512
1. Lattices. 512
2. Duality; reflexive modules. 517
3. Local construction of reflexive modules . 521
4. Pseudo-isomorphisms. 523
5. Divisors attached to torsion modules. 527
6 . Relative invariant of two lattices. 529
7. Divisor classes attached to finitely generated modules_ 531
8 . Properties relative to finite extensions of the ring of scalars 535
9. A reduction theorem . 540
10. Modules over Dedekind domains. 543
Exercises for § 1 . 545
Exercises for § 2. 555
CONTENTS
Exercises for § 3. 563
Exercises for § 4. 571
Historical note (Chapters I to VII). 579
Bibliography . 603
Index of notation. 607
Index of terminology . 610
Table of implications. 621
Table of invariances — 1. 622
Table of invariances — II . 624
Invariances under completion .