Local algebra

The present book is an English translation of Algebre Locale - Multiplicites published by Springer-Verlag as no. 11 of the Lecture Notes series. The original text was based on a set of lectures, given at the College de France in 1957-1958, and written up by Pierre Gabriel. Its aim was to give a short account of Commutative Algebra, with emphasis on the following topics: a) Modules (as opposed to Rings, which were thought to be the only subject of Commutative Algebra, before the emergence of sheaf theory in the 1950s); b) H omological methods, a la Cartan-Eilenberg; c) Intersection multiplicities, viewed as Euler-Poincare characteristics. The English translation, done with great care by Chee Whye Chin, differs from the original in the following aspects: - The terminology has been brought up to date (e.g. "cohomological dimension" has been replaced by the now customary "depth"). I have rewritten a few proofs and clarified (or so I hope) a few more. - A section on graded algebras has been added (App. III to Chap. IV). - New references have been given, especially to other books on Commu- tive Algebra: Bourbaki (whose Chap. X has now appeared, after a 40-year wait) , Eisenbud, Matsumura, Roberts, .... I hope that these changes will make the text easier to read, without changing its informal "Lecture Notes" character.

Preface	p. v
Contents	p. vii
Introduction	p. xi
I Prime Ideals and Localization	p. 1
1 Notation and definitions	p. 1
2 Nakayama's lemma	p. 1
3 Localization	p. 2
4 Noetherian rings and modules	p. 4
5 Spectrum	p. 4
6 The noetherian case	p. 5
7 Associated prime ideals	p. 6
8 Primary decompositions	p. 10
II Tools	p. 11
A Filtrations and Gradings	p. 11
1 Filtered rings and modules	p. 11
2 Topology defined by a filtration	p. 12
3 Completion of filtered modules	p. 13
4 Graded rings and modules	p. 14
5 Where everything becomes noetherian again - \mathfr {{q}} -adic filtrations	p. 17
B Hilbert-Samuel Polynomials	p. 19
1 Review on integer-valued polynomials	p. 19
2 Polynomial-like functions	p. 21
3 The Hilbert polynomial	p. 21
4 The Samuel polynomial	p. 24
III Dimension Theory	p. 29
A Dimension of Integral Extensions	p. 29
1 Definitions	p. 29
2 Cohen-Seidenberg first theorem	p. 30
3 Cohen-Seidenberg second theorem	p. 32
B Dimension in Noetherian Rings	p. 33
1 Dimension of a module	p. 33
2 The case of noetherian local rings	p. 33
3 Systems of parameters	p. 36
C Normal Rings	p. 37
1 Characterization of normal rings	p. 37
2 Properties of normal rings	p. 38
3 Integral closure	p. 40
D Polynomial Rings	p. 40
1 Dimension of the ring A[X 1 , ..., X n ]	p. 40
2 The normalization lemma	p. 42
3 Applications. I. Dimension in polynomial algebras	p. 44
4 Applications. II. Integral closure of a finitely generated algebra	p. 46
5 Applications. III. Dimension of an intersection in affine space	p. 47
IV Homological Dimension and Depth	p. 51
A The Koszul Complex	p. 51
1 The simple case	p. 51
2 Acyclicity and functorial properties of the Koszul complex	p. 53
3 Filtration of a Koszul complex	p. 56
4 The depth of a module over a noetherian local ring	p. 59
B Cohen-Macaulay Modules	p. 62
1 Definition of Cohen-Macaulay modules	p. 63
2 Several characterizations of Cohen-Macaulay modules	p. 64
3 The support of a Cohen-Macaulay module	p. 66
4 Prime ideals and completion	p. 68
C Homological Dimension and Noetherian Modules	p. 70
1 The homological dimension of a module	p. 70
2 The noetherian case	p. 71
3 The local case	p. 73
D Regular Rings	p. 75
1 Properties and characterizations of regular local rings	p. 75
2 Permanence properties of regular local rings	p. 78
3 Delocalization	p. 80
4 A criterion for normality	p. 82
5 Regularity in ring extensions	p. 83
Appendix I Minimal Resolutions	p. 84
1 Definition of minimal resolutions	p. 84
2 Application	p. 85
3 The case of the Koszul complex	p. 86
Appendix II Positivity of Higher Euler-Poincare Characteristics	p. 88
Appendix III Graded-polynomial Algebras	p. 91
1 Notation	p. 91
2 Graded-polynomial algebras	p. 92
3 A characterization of graded-polynomial algebras	p. 93
4 Ring extensions	p. 93
5 Application: the Shephard-Todd theorem	p. 95
V Multiplicities	p. 99
A Multiplicity of a Module	p. 99
1 The group of cycles of a ring	p. 99
2 Multiplicity of a module	p. 100
B Intersection Multiplicity of Two Modules	p. 101
1 Reduction to the diagonal	p. 101
2 Completed tensor products	p. 102
3 Regular rings of equal characteristic	p. 106
4 Conjectures	p. 107
5 Regular rings of unequal characteristic (unramified case)	p. 108
6 Arbitrary regular rings	p. 110
C Connection with Algebraic Geometry	p. 112
1 Tor-formula	p. 112
2 Cycles on a non-singular affine variety	p. 113
3 Basic formulae	p. 114
4 Proof of theorem 1	p. 116
5 Rationality of intersections	p. 116
6 Direct images	p. 117
7 Pull-backs	p. 117
8 Extensions of intersection theory	p. 119
Bibliography	p. 123
Index	p. 127
Index of Notation	p. 129

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