Introduction to Commutative Algebra December 20, 2019 About This Document This document was typeset by Jason McCullough. It is based on course notes from a course taught by Professor S.P. Dutta at the University of Illinois texed by Jason McCullough and Bart Snapp. The notes have been redone in 2019 for Math 619 at Iowa State taught by Jason McCullough. Special thanks to: Michael Dewar, Jordan Disch, Dan Lior, Christian McRoberts, Elizabeth Sprangel, and Patrick Szuta, for making many comments and corrections con- cerning these notes. Please report corrections, suggestions, gripes, complaints, and criticisms to:
[email protected] Contents 0 Background 1 0.1 OperationsonIdeals ......................... 1 0.2 ChainConditions........................... 4 0.3 FlatModules ............................. 8 0.4 Localization.............................. 9 1 Primary Decomposition 13 1.1 PrimaryandCoprimaryModules . 13 1.2 ThePrimaryDecompositionTheorem . 15 1.2.1 Primary Decomposition and Localization . 18 1.2.2 AssociatedPrimes ...................... 21 1.3 ArbitraryModules .......................... 25 2 Filtrations and Completions 28 2.1 Limits ................................. 28 2.1.1 Direct Limits . 28 2.1.2 InverseLimits......................... 30 2.2 Filtrations and Completions . 32 2.2.1 Topology and Algebraic Structures . 32 2.2.2 Filtered Rings and Modules . 32 2.2.3 The Topology Corresponding to a Filtration . 33 2.2.4 GradedRingsandModules . 36 2.3 Adic Completions and Local Rings . 39 2.4 Faithfully Flat Modules . 44 3 Dimension Theory 52 3.1 TheGradedCase........................... 52 3.1.1 The Hilbert Polynomial . 54 3.2 TheHilbert-SamuelPolynomial . 59 3.3 TheTopologicalApproach. 66 3.3.1 Basic Definitions . 66 3.3.2 The Zariski Topology and the Prime Spectrum . 68 3.4 Systems of Parametersand the Dimension Theorem .