Local algebra /
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| Author / Creator: | Serre, Jean Pierre, 1926- |
|---|---|
| Uniform title: | Algèbre locale, multiplicités. English |
| Imprint: | Berlin ; New York : Springer, c2000. |
| Description: | xiii, 128 p. ; 25 cm. |
| Language: | English |
| Series: | Springer monographs in mathematics, 1439-7382 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4315195 |
Table of Contents:
- Preface
- Contents
- Introduction
- 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 - \mathfr {{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[X 1 , ..., X n ]
- 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-Poincare 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
- Bibliography
- Index
- Index of Notation
