Commutative Algebra · An Introduction by Hoffman, J. William -- Read -- Imperial Library of Trantor

Index

Cover Page Halftitle Title Page Copyright Contents Preface 1. Introduction 2. Rings and Ideals

2.1 Rings and Ideals 2.2 Localization of a Ring 2.3 Ideals in a Polynomial Ring 2.4 Gröbner Basis of an Ideal 2.5 Elimination and Extension 2.6 Implicitization 2.7 Schemes 2.8 Gröbner Basis Applications

2.8.1 Solving Systems of Equations 2.8.2 Orthogonal Projection 2.8.3 Poncelet’s Algebraic Correspondence

3. Modules

3.1 Modules 3.2 Exact Sequences and Commutative Diagrams 3.3 Projective and Injective Modules 3.4 Tensor Product of Modules 3.5 Flatness 3.6 Localization 3.7 Local Property 3.8 Associated Primes 3.9 Primary Decomposition of Modules 3.10 Modules of Finite Length 3.11 Krull Dimension of a Ring 3.12 Dimension of Modules 3.13 Rank of Modules 3.14 Computational Applications

3.14.1 Tensor Products 3.14.2 Primary Decomposition 3.14.3 Krull Dimension 3.14.4 Generators of Syzygy Modules

4. Graded and Local Rings and Modules

4.1 Graded Rings and Modules 4.2 Graded Localization 4.3 Graded Associated Primes 4.4 Filtration 4.5 System of Parameters 4.6 Regular and Quasi-regular M-Sequence 4.7 Proj of a Graded Ring 4.8 Graded Free Resolution 4.9 Dimension and Multiplicity 4.10 Applications

4.10.1 Compute the Free Resolution 4.10.2 Implicitization via Syzygies 4.10.3 Compute the Rees Algebra 4.10.4 Resultants

5. Homological Method

5.1 Complexes 5.2 Complex of Tor 5.3 Koszul Complex 5.4 Regular Sequences 5.5 Regular Rings 5.6 Complex of Ext 5.7 Exactness Criteria for Complexes 5.8 Local Cohomology 5.9 Applications

5.9.1 Castelnuovo-Mumford Regularity 5.9.2 The MacRae’s Invariant 5.9.3 Approximation Complex

Bibliography Index