Overview

"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."

Full Product Details

Author:   Jean-Pierre Serre ,  C.W. Chin
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2000 ed.
Dimensions:   Width: 15.50cm , Height: 0.90cm , Length: 23.50cm
Weight:   0.454kg
ISBN:  

9783540666417

ISBN 10:   3540666419
Pages:   130
Publication Date:   16 June 2000
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
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Table of Contents

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.

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