CA - Commutative Algebra Keerthi Madapusi Contents Chapter...

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Unformatted text preview: Commutative Algebra Keerthi Madapusi Contents Chapter 1. Graded Rings and Modules I: Basics 7 1. Basic definitions 7 2. *Local Rings 9 3. Finiteness Conditions 11 4. Associated Primes and Primary Decomposition 13 5. The Category of Graded Modules 15 6. Dehomogenization: Preliminaries for Projective Geometry 17 Chapter 2. Graded Rings and Modules II: Filtrations and Hilbert Functions 21 1. Filtered Rings 21 2. Finiteness Conditions: The Artin-Rees Lemma 25 3. The Hilbert-Samuel Polynomial 27 Chapter 3. Flatness 37 1. Basics 37 2. Homological Criterion for Flatness 39 3. Equational Criterion for Flatness 41 4. Local Criterion for Flatness 45 5. The Graded Case 48 6. Faithfully Flat Modules 51 Chapter 4. Integrality: the Cohen-Seidenberg Theorems 55 1. The Cayley-Hamilton Theorem 55 2. Integrality 56 3. Integral Closure and Normality 57 4. Lying Over and Going Up 63 5. Finite Group Actions 65 6. Going Down for Normal Domains 67 7. Valuation Rings and Extensions of Homomorphisms 68 Chapter 5. Completions and Hensels Lemma 71 1. Basics 71 2. Convergence and some Finiteness Results 74 3. The Noetherian Case 77 4. Hensels Lemma and its Consequences 79 5. Lifting of Idempotents: Henselian Rings 83 6. More on Actions by Finite Groups 85 Chapter 6. Dimension Theory I: The Main Theorem 87 1. Krull Dimension and the Hauptidealsatz 87 2. The Main Theorem of Dimension Theory 89 3 4 CONTENTS 3. Regular Local Rings 93 4. Dimension Theory of Graded Modules 94 5. Integral Extensions and the Going Up property 96 6. Dimensions of Fibers 96 7. The Going Down property 98 Chapter 7. Invertible Modules and Divisors 101 1. Locally Free Modules 101 2. Invertible Modules 103 3. Unique Factorization of Ideals 106 4. Cartier and Weil Divisors 108 5. Discrete Valuation Rings and Dedekind Domains 108 6. The Krull-Akizuki Theorem 110 7. Grothendieck Groups 111 Chapter 8. Noether Normalization and its Consequences 115 1. Noether Normalization 115 2. Generic Freeness 116 3. Finiteness of Integral Closure 116 4. Jacobson Rings and the Nullstellensatz 118 5. Dimension Theory for Affine Rings 119 6. Dimension of Fibers 120 Chapter 9. Quasi-finite Algebras and the Main Theorem of Zariski 123 1. Quasi-finite Algebras 123 2. Proof of Zariskis Main Theorem 124 Chapter 10. Regular Sequences and Depth 127 1. Regular Sequences 127 2. Flatness 129 3. Quasiregular Sequences 132 4. Grade and Depth 136 5. Behavior of Depth under Flat Extensions 141 Chapter 11. The Cohen Macaulay Condition 143 1. Basic Definitions and Results 143 2. Characterizations of Cohen-Macaulay Modules 144 Chapter 12. Homological Theory of Regular Rings 145 1. Regular Local Rings 145 2. Characterization of Regular Rings 145 3. Behavior under Flat Extensions 147 4. Stably Free Modules and Factoriality of Regular Local Rings 148 Chapter 13. Formal Smoothness and the Cohen Structure Theorems 151 Chapter 14. Witt Rings 153 1. Cohen Structure Theorem: The Equicharacteristic Case 153 2. The Witt Scheme 156 3. Cohen Structure Theorem: The Unequal Characteristic Case 161 4. Finiteness of Integral Closure 161 CONTENTS...
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This note was uploaded on 10/26/2011 for the course MATH 8020 taught by Professor Clark during the Summer '11 term at University of Georgia Athens.

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CA - Commutative Algebra Keerthi Madapusi Contents Chapter...

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