Commutative Algebra - with a View Toward Algebraic Geometry

Commutative 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.

Sommario

Advice for the Beginner.- Information for the Expert.- Prerequisites.- Sources.- Courses.- Acknowledgements.- 0 Elementary Definitions.- 0.1 Rings and Ideals.- 0.2 Unique Factorization.- 0.3 Modules.- I Basic Constructions.- 1 Roots of Commutative Algebra.- 2 Localization.- 3 Associated Primes and Primary Decomposition.- 4 Integral Dependence and the Nullstellensatz.- 5 Filtrations and the Artin-Rees Lemma.- 6 Flat Families.- 7 Completions and Hensel's Lemma.- II Dimension Theory.- 8 Introduction to Dimension Theory.- 9 Fundamental Definitions of Dimension Theory.- 10 The Principal Ideal Theorem and Systems of Parameters.- 11 Dimension and Codimension One.- 12 Dimension and Hilbert-Samuel Polynomials.- 13 The Dimension of Affine Rings.- 14 Elimination Theory, Generic Freeness, and the Dimension of Fibers.- 15Gröbner Bases.- 16 Modules of Differentials.- III Homological Methods.- 17 Regular Sequences and the Koszul Complex.- 18 Depth, Codimension, and Cohen-Macaulay Rings.- 19 Homological Theory of Regular Local Rings.- 20 Free Resolutions and Fitting Invariants.- 21 Duality, Canonical Modules, and Gorenstein Rings.- Appendix 1 Field Theory.- A1.1 Transcendence Degree.- A1.2 Separability.- A1.3.1 Exercises.- Appendix 2 Multilinear Algebra.- A2.1 Introduction.- A2.2 Tensor Product.- A2.3 Symmetric and Exterior Algebras.- A2.3.1 Bases.- A2.3.2 Exercises.- A2.4 Coalgebra Structures and Divided Powers.- A2.5 Schur Functors.- A2.5.1 Exercises.- A2.6 Complexes Constructed by Multilinear Algebra.- A2.6.1 Strands of the Koszul Comple.- A2.6.2 Exercises.- Appendix 3 Homological Algebra.- A3.1 Introduction.- I: Resolutions and Derived Functors.- A3.2 Free and Projective Modules.- A3.3 Free and Projective Resolutions.- A3.4 Injective Modules and Resolutions.- A3.4.1 Exercises.- Injective Envelopes.- Injective Modules over Noetherian Rings.- A3.5 Basic Constructions with Complexes.- A3.5.1 Notation and Definitions.- A3.6 Maps and Homotopies of Complexes.- A3.7 Exact Sequences ofComplexes.- A3.7.1 Exercises.- A3.8 The Long Exact Sequence in Homology.- A3.8.1 Exercises.- Diagrams and Syzygies.- A3.9 Derived Functors.- A3.9.1 Exercise on Derived Functors.- A3.10 Tor.- A3.10.1 Exercises: Tor.- A3.1l Ext.- A3.11.1 Exercises: Ext.- A3.11.2 Local Cohomology.- II: From Mapping Cones to Spectral Sequences.- A3.12 The Mapping Cone and Double Complexe.- A3.12.1 Exercises: Mapping Cones and Double Complexes.- A3.13 Spectral Sequences.- A3.13.1 Mapping Cones Revisited.- A3.13.2 Exact Couples.- A3.13.3 Filtered Differential Modules and Complexes.- A3.13.4 The Spectral Sequence of a Double Complex.- A3.13.5 Exact Sequence of Terms of Low Degree.- A3.13.6 Exercises on Spectral Sequences.- A3.14 Derived Categories.- A3.14.1 Step One: The Homotopy Category of Complexes.- A3.14.2 Step Two: The Derived Category.- A3.14.3 Exercises on the Derived Category.- Appendix 4 A Sketch of Local Cohomology.- A4.1 Local Cohomology and Global Cohomology.- A4.2 Local Duality.- A4.3 Depth andDimensio.- Appendix 5 Category Theory.- A5.1 Categories, Functors, and Natural Transformations.- A5.2 Adjoint Functors.- A5.2.1 Uniqueness.- A5.2.2 Some Examples.- A5.2.3 Another Characterization of Adjoints.- A5.2.4 Adjoints and Limits.- A5.3 Representable Functors and Yoneda's Lemma.- Appendix 6 Limits and Colimits.- A6.1 Colimits in the Category of Modules.- A6.2 Flat Modules as Colimits of Free Modules.- A6.3 Colimits in the Category of Commutative Algebras.- A6.4 Exercises.- Appendix 7 Where Next?.- References.- Index of Notation.

Info autore

The author taught at Brandeis University for twenty-seven years, with sabbatical time spent in Paris, Bonn, and Berkeley, and became Director of the Mathematical Sciences Research Institute in Berkeley in the Summer of 1997. At the same time he joined the faculty of UC Berkeley as Professor of Mathematics. In 2003 he became President of the American Mathematical Society. He currently serves on several editorial boards (Annals of Mathematics, Bulletin du Société Mathématique de France, Springer-Verlag's book series Algorithms and Computation in Mathematics).

Riassunto

Commutative 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.

Testo aggiuntivo

D. Eisenbud
Commutative Algebra with a View Toward Algebraic Geometry
"This text has personality—Those familiar with Eisenbud"s own research will recognize its traces in his choice of topics and manner of approach. The book conveys infectious enthusiasm and the conviction that research in the field is active and yet accessible."—MATHEMATICAL REVIEWS

Relazione

D. Eisenbud
Commutative Algebra with a View Toward Algebraic Geometry
"This text has personality-Those familiar with Eisenbud"s own research will recognize its traces in his choice of topics and manner of approach. The book conveys infectious enthusiasm and the conviction that research in the field is active and yet accessible."-MATHEMATICAL REVIEWS

Dettagli sul prodotto

• Autori: David Eisenbud
• Editore: Springer, Berlin
• Lingue: Inglese
• Formato: Copertina rigida
• Pubblicazione: 01.01.1995
• EAN: 9780387942681
• ISBN: 978-0-387-94268-1
• Pagine: 800
• Dimensioni: 155 mm x 47 mm x 235 mm
• Illustrazioni: XVI, 800 p.
• Serie: Graduate Texts in Mathematics
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Aritmetica, algebra; Dimension, B, Field, Mathematics and Statistics, Algebraic Geometry, Linear Algebra, Commutative algebra, Homological algebra, cohomology, polynomial, colimit