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Commutative algebra (Record no. 8609)

000 -Label
leader 05864nac 22003371u 4500
010 ## - ISBN
ISBN 9780387942698
qualificatif br.
090 ## - Numéro biblio (koha)
Numéro biblioitem (koha) 8609
001 - Numéro de notice
Numéro d'identification notice 8609
101 ## - Langue
langue du document anglais
102 ## - Pays de publication ou de production
pays de publication Etats Unis
100 ## - Données générales de traitement
données générales de traitement 20091130 frey50
200 ## - Titre
titre propre Commutative algebra
type de document Monographie
complément du titre with a view toward algebraic geometry
Auteur David Eisenbud
210 ## - Editeur
lieu de publication New York
nom de l'éditeur Springer
date de publication 1993
215 ## - Description
Importance matérielle 1 vol. (XVI-785 p.)
format 23 cm
225 ## - collection
titre de la collection Graduate texts in mathematics
numérotation du volume 150
lien interne koha 169059
ISSN de la collection ou de la sous-collection 0072-5285
300 ## - Note
note With so many texts on commutative algebra available, one's reaction to this one might be "Why yet another one?'', and "Why is it so fat?'' The answer to the second question answers the first as well—this text has a distinctively different flavor than existing texts, both in coverage and style. Motivation and intuitive explanations appear throughout, there are many worked examples, and both text and problem sets lead up to contemporary research.
Structured historically, the introductory chapter is unusually long, and opens the book with a bang. Theorems are stated, and where possible, even at this early stage, proven. Topics such as geometric invariant theory, Hilbert functions, and free resolutions are discussed. I particularly liked the motivation of graded rings via invariant theory. (Graded rings are singled out for special attention in conjunction with many topics, a good idea since they are used so often in geometric and computational settings.) Strands are traced from their roots to current research, and even if not completely understood give reason for the technical work ahead.
This style continues. In the chapter on associated primes and primary decomposition there is a preview of local cohomology. Symbolic powers are treated with much more than the usual attention: there is an outline of the Zariski-Nagata theorem on functions vanishing to high order, and a thoroughly worked determinantal example. Flatness is motivated by flat families, perhaps an unusual topic for a beginning text, but certainly one that establishes "a view toward algebraic geometry''. Application to deformations appears in the exercises, and further application arises in the context of Gröbner bases.
The second section, dimension theory, also begins with an overview. There is a nice discussion of the geometry of systems of parameters and the Krull principal ideal theorem. The latter is illustrated by a picture, which suggests one reason for the book's length, and at the same time its pedagogical strength. Eisenbud gives pictures that illustrate the geometry of primary decomposition, normalization and gluing, deformation under various term orderings, duality in graded Artinian rings, and others. These serve to build intuition about subtle or complicated concepts. Students sometimes complain that algebra is too formal for intuition; this sort of exposition should encourage them to think about algebraic concepts in new ways. Topics in this section include special features of dimension and codimension one, the Hilbert-Samuel polynomial, Noether normalization and the last (of five!) proofs of the Nullstellensatz, elimination theory, semi-continuity of fiber dimension, differentials, and the Jacobian criterion for regularity.
A chapter devoted to Gröbner bases is novel for a text on commutative algebra, but is appropriate because of the increasing ubiquity of computational problems and methods in the field. Eisenbud describes what can (and what cannot, at least so far) be effectively computed. The chapter concludes with eight computer algebra topics designed to be explorations growing out of classes of examples.
The third major section, homological methods, has a faster pace. The chapters are vignettes of representative methods and applications, mostly in the context of regular, Gorenstein, and Cohen-Macaulay rings. Topics include the use of Serre's conditions in proving primeness of ideals, the acyclicity lemma, a proof of the Hilbert-Burch theorem and several applications, and a discussion of Castelnuovo-Mumford regularity. The text proper concludes with a quick glimpse into linkage theory, but the book continues with seven appendices, 190 pages in length. Especially notable are treatments of co-algebra and divided power structures, multi-linear algebra and Koszul-like complexes, and applications of these to determinantal varieties. Exercises explore applications to rational and elliptic normal curves, and scrolls.
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.
Consistent with his computer-friendly attitude, Eisenbud maintains a list of corrections (MathSciNet)
320 ## - Note
note Bibliogr. p. [745]-762. Index
410 ## - collection
lien interne koha 168282
Editeur Springer
titre Graduate texts in mathematics
numéro de volume 0150
ISSN 0072-5285
676 ## - annee msc
annee msc 2010
686 ## - Classification MSC
Indice 13C15
lien interne koha 161859
Libellé Commutative algebra -- Theory of modules and ideals
Sous-catégorie Dimension theory, depth, related rings (catenary, etc.)
code du système msc
686 ## - Classification MSC
lien interne koha 161839
Indice 13A50
Libellé Commutative algebra -- General commutative ring theory
Sous-catégorie Actions of groups on commutative rings; invariant theory
code du système msc
686 ## - Classification MSC
Indice 13Axx
lien interne koha 161832
Libellé Commutative algebra
Sous-catégorie General commutative ring theory
code du système msc
686 ## - Classification MSC
lien interne koha 161852
Indice 13Cxx
Libellé Commutative algebra
Sous-catégorie Theory of modules and ideals
code du système msc
700 ## - Auteur
code de fonction Auteur
auteur Eisenbud
partie du nom autre que l'élément d'entrée David
koha internal code 173186
dates 1947-
856 ## - accès
URI http://link.springer.com/book/10.1007/978-1-4612-5350-1
note Springerlink
856 ## - accès
URI http://www.ams.org/mathscinet-getitem?mr=1322960
note MathSciNet
856 ## - accès
URI http://zbmath.org/?q=an:0819.13001
note Zentralblatt
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