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 |
Gröbner Bases | 321 |
Homological Algebra | 617 |
A Sketch of Local Cohomology | 691 |
Category Theory | 697 |
Limits and Colimits | 705 |
Where Next? | 719 |
| 757 | |
| 775 | |
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Common terms and phrases
affine algebra annihilator apply associated assume basis called Chapter closed codimension coefficients Cohen-Macaulay complex compute condition consider construction contains Corollary corresponding curve defined definition depth derivation differential dimension direct domain elements equal equivalent exact sequence example Exercise extension fact factor field FIGURE finitely flat follows free module free resolution function functor geometric give given graded Hilbert homogeneous homomorphism idea induced injective integral intersection isomorphism Lemma length linear localization matrix maximal ideal minimal module monomial multiplication natural Noetherian nonzero nonzerodivisor normal Note particular polynomial polynomial ring prime prime ideal principal projective Proof Proposition prove R-module rank reduced regular sequence relation respect result satisfied sense Show simple space statement submodule subset suffices Suppose Theorem theory unique variables variety vector write
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Bibliographic information
| Title | Commutative Algebra: With a View Toward Algebraic Geometry Graduate Texts in Mathematics, ISSN 0072-5285 |
| Authors | David Eisenbud, Professor David Eisenbud |
| Edition | illustrated, reprint |
| Publisher | Springer Science & Business Media, 1995 |
| ISBN | 0387942696, 9780387942698 |
| Length | 785 pages |
| Subjects | › › Mathematics / Algebra / General Mathematics / Geometry / Algebraic |
| Export Citation | BiBTeX EndNote RefMan |