Local Algebra / Edition 1

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Overview

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.

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Product Details

  • ISBN-13: 9783642085901
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 8/12/2011
  • Series: Springer Monographs in Mathematics Series
  • Edition description: Softcover reprint of hardcover 1st ed. 2000
  • Edition number: 1
  • Pages: 130
  • Product dimensions: 6.10 (w) x 9.00 (h) x 0.60 (d)

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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