Includes supplementary material: sn.pub/extras
This is an English translation of the now classic "Algbre Locale - Multiplicits" originally published by Springer as LNM 11. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities. Many modifications to the original French text have been made for this English edition, making the text easier to read, without changing its intended informal character.|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original in the following aspects: - The terminology has been brought up to date (e.g. "cohomological dimension" has been replaced by the now customary "depth").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.
I. Prime Ideals and Localization.- §1. Notation and definitions.- §2. Nakayama´s lemma.- §3. Localization.- §4. Noetherian rings and modules.- §5. Spectrum.- §6. The noetherian case.- §7. Associated prime ideals.- §8. Primary decompositions.- II. Tools.- A: Filtrations and Gradings.- §1. Filtered rings and modules.- §2. Topology defined by a filtration.- §3. Completion of filtered modules.- §4. Graded rings and modules.- §5. Where everything becomes noetherian again — q -adic filtrations.- B: Hilbert-Samuel Polynomials.- §1. Review on integer-valued polynomials.- §2. Polynomial-like functions.- §3. The Hilbert polynomial.- §4. The Samuel polynomial.- III. Dimension Theory.- A: Dimension of Integral Extensions.- §1. Definitions.- §2. Cohen-Seidenberg first theorem.- §3. Cohen-Seidenberg second theorem.- B: Dimension in Noetherian Rings.- §1. Dimension of a module.- §2. The case of noetherian local rings.- §3. Systems of parameters.- C: Normal Rings.- §1. Characterization of normal rings.- §2. Properties of normal rings.- §3. Integral closure.- D: Polynomial Rings.- §1. Dimension of the ring A[X1,..., Xn].- §2. The normalization lemma.- §3. Applications. I. Dimension in polynomial algebras.- §4. Applications. II. Integral closure of a finitely generated algebra.- §5. Applications. III. Dimension of an intersection in affine space.- IV. Homological Dimension and Depth.- A: The Koszul Complex.- §1. The simple case.- §2. Acyclicity and functorial properties of the Koszul complex.- §3. Filtration of a Koszul complex.- §4. The depth of a module over a noetherian local ring.- B: Cohen-Macaulay Modules.- §1. Definition of Cohen-Macaulay modules.- §2. Several characterizations of Cohen-Macaulay modules.- §3. The support of a Cohen-Macaulay module.- §4. Prime ideals and completion.- C: Homological Dimension and Noetherian Modules.- §1. The homological dimension of a module.- §2. The noetherian case.- §3. The local case.- D: Regular Rings.- §1. Properties and characterizations of regular local rings.- §2. Permanence properties of regular local rings.- §3. Delocalization.- §4. A criterion for normality.- §5. Regularity in ring extensions.- Appendix I: Minimal Resolutions.- §1. Definition of minimal resolutions.- §2. Application.- §3. The case of the Koszul complex.- Appendix II: Positivity of Higher Euler-Poincaré Characteristics.- Appendix III: Graded-polynomial Algebras.- §1. Notation.- §2. Graded-polynomial algebras.- §3. A characterization of graded-polynomial algebras.- §4. Ring extensions.- §5. Application: the Shephard-Todd theorem.- V. Multiplicities.- A: Multiplicity of a Module.- §1. The group of cycles of a ring.- §2. Multiplicity of a module.- B: Intersection Multiplicity of Two Modules.- §1. Reduction to the diagonal.- §2. Completed tensor products.- §3. Regular rings of equal characteristic.- §4. Conjectures.- §5. Regular rings of unequal characteristic (unramified case).- §6. Arbitrary regular rings.- C: Connection with Algebraic Geometry.- §1. Tor-formula.- §2. Cycles on a non-singular affine variety.- §3. Basic formulae.- §4. Proof of theorem 1.- §5. Rationality of intersections.- §6. Direct images.- §7. Pull-backs.- §8. Extensions of intersection theory.- Index of Notation.
This is an English translation of the now classic "Algebre Locale - Multiplicites" originally published by Springer as LNM 11, in several editions since 1965. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities ("Tor-formula"). Many modifications to the original French text have been made by the author for this English edition: They make the text easier to read, without changing its intended informal character.
I. Prime Ideals and Localization.-
1. Notation and definitions.-
2. Nakayama's lemma.-
3. Localization.-
4. Noetherian rings and modules.-
5. Spectrum.-
6. The noetherian case.-
7. Associated prime ideals.-
8. Primary decompositions.- II. Tools.- A: Filtrations and Gradings.- B: Hilbert-Samuel Polynomials.- III. Dimension Theory.- A: Dimension of Integral Extensions.- B: Dimension in Noetherian Rings.- C: Normal Rings.- D: Polynomial Rings.- IV. Homological Dimension and Depth.- A: The Koszul Complex.- B: Cohen-Macaulay Modules.- C: Homological Dimension and Noetherian Modules.- D: Regular Rings.- Appendix I: Minimal Resolutions.- Appendix II: Positivity of Higher Euler-Poincaré Characteristics.- Appendix III: Graded-polynomial Algebras.- V. Multiplicities.- A: Multiplicity of a Module.- B: Intersection Multiplicity of Two Modules.- C: Connection with Algebraic Geometry.- Index of Notation.
Professor Jean-Pierre Serre ist ein renommierter französischer Mathematiker am College de France in Paris.
Klappentext
This is an English translation of the now classic "Algbre Locale - Multiplicits" originally published by Springer as LNM 11. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities. Many modifications to the original French text have been made for this English edition, making the text easier to read, without changing its intended informal character.
