## Commutative Algebra: With a View Toward Algebraic GeometryCommutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text. |

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### Contents

Introduction | 1 |

Elementary Definitions | 11 |

Roots of Commutative Algebra | 21 |

Localization | 57 |

Associated Primes and Primary Decomposition | 87 |

Integral Dependence and the Nullstellensatz | 117 |

Filtrations and the ArtinRees Lemma | 147 |

Flat Families | 157 |

Modules of Differentials | 385 |

Homological Methods | 421 |

Depth Codimension and CohenMacaulay Rings | 451 |

Homological Theory of Regular Local Rings | 473 |

Free Resolutions and Fitting Invariants | 493 |

Duality Canonical Modules and Gorenstein Rings | 523 |

Field Theory | 561 |

Multilinear Algebra | 571 |

Completions and Hensels Lemma | 181 |

Dimension Theory | 213 |

Fundamental Definitions of Dimension Theory | 227 |

The Principal Ideal Theorem and Systems of Parameters | 233 |

Dimension and Codimension One | 251 |

Dimension and HilbertSamuel Polynomials | 275 |

The Dimension of Affine Rings | 285 |

Elimination Theory Generic Freeness and the Dimension | 307 |

Grobner Bases | 321 |

Homological Algebra | 617 |

A Sketch of Local Cohomology | 691 |

Category Theory | 697 |

Limits and delimits | 705 |

Where Next? | 719 |

757 | |

775 | |

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### Common terms and phrases

affine ring algebraic set algebraically closed annihilator associated prime canonical module Chapter closed field codim codimension coefficients Cohen-Macaulay ring colimits commutative algebra compute contains coordinate ring Corollary corresponding curve defined definition degree depth differential dimension direct sum domain elements example Exercise factor fiber finite length finitely generated module flat follows free module free resolution functor geometric given Gorenstein graded module graded ring Grobner basis homology homomorphism induction injective intersection invertible irreducible isomorphism k[xi kernel Koszul complex linear matrix maximal ideal minimal primes monomial ideal monomial order monomorphism morphism multiplication Nakayama's lemma Noetherian ring nonzero nonzerodivisor polynomial ring primary decomposition prime ideal principal ideal projective Proof Proposition prove R-algebra R-module regular local ring regular sequence result short exact sequence spectral sequence submodule subset Suppose syzygies tensor product Theorem theory Torf unique variables vector space write