It has been estimated that, at the present stage of our knowledge, one could give a 200 semester course on commutative algebra and algebraic geometry without ever repeating himself. So any introduction to this subject must be highly selective. I first want to indicate what point of view guided the selection of material for this book. This introduction arose from lectures for students who had taken a basic course in algebra and could therefore be presumed to have a knowledge of linear algebra, ring and field theory, and Galois theory. The present text shouldn't require much more. In the lectures and in this text I have undertaken with the fewest possible auxiliary means to lead up to some recent results of commutative algebra and algebraic geometry concerning the representation of algebraic varieties as in tersections of the least possible number of hypersurfaces and- a closely related problem-with the most economical generation of ideals in Noetherian rings. The question of the equations needed to describe an algebraic variety was addressed by Kronecker in 1882. In the 1940s it was chiefly Perron who was interested in this question; his discussions with Severi made the problem known and contributed to sharpening the rei event concepts. Thanks to the general progress of commutative algebra many beautiful results in this circle of questions have been obtained, mainly after the solution of Serre's problem on projective modules. Because of their relatively elementary character they are especially suitable for an introduction to commutative algebra.
Rezensionen / Stimmen
"An excellent text for a student who wants to learn the basic facts from commutative algebra and algebraic geometry... Many examples and exercises complete the text." -REVUE ROUMANIE DE MATHEMATIQUES PURES ET APPLIQUEES "An excellent introduction to the subject... The center of gravity lies in commutative algebra, and this book can serve as a preparation for more advanced topics. The presentation is very clear and the theory is accompanied by numerous interesting exercises...This is a highly recommended text for students and lecturers." -MATHEMATICA "Outrageous as it may sound this is the first really introductory book written about the new algebraic geometry of the sixties. At last something that we can give our students without cautionary words, and where we ourselves can learn basic concepts that cut across the party lines of mathematics." -Gian-Carlo Rota, formerly MIT
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Research
Editions-Typ
Produkt-Hinweis
Illustrationen
black & white illustrations
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Dicke: 15 mm
Gewicht
ISBN-13
978-0-8176-3065-2 (9780817630652)
DOI
10.1007/978-1-4612-5290-0
Schweitzer Klassifikation
I. Algebraic varieties.- §1. Affine algebraic varieties.- §2. The Hilbert Basis Theorem. Decomposition of a variety into irreducible components.- §3. Hilbert's Nullstellensatz.- §4. The spectrum of a ring.- §5. Projective varieties and the homogeneous spectrum.- References.- II. Dimension.- §1. The Krull dimension of topological spaces and rings.- §2. Prime ideal chains and integral ring extensions.- §3. The dimension of affine algebras and affine algebraic varieties.- §4. The dimension of projective varieties.- References.- III. Regular and rational functions on algebraic varieties Localization.- §1. Some properties of the Zariski topology.- §2. The sheaf of regular functions on an algebraic variety.- §3. Rings and modules of fractions. Examples.- §4. Properties of rings and modules of fractions.- §5. The fiber sum and fiber product of modules. Gluing modules.- References.- IV. The local-global principle in commutative algebra.- §1. The passage from local to global.- §2. The generation of modules and ideals.- §3. Projective modules.- References.- V. On the number of equations needed to describe an algebraic variety.- §1. Any variety in n-dimensional space is the intersection of n hypersurfaces.- §2. Rings and modules of finite length.- §3. Krull's Principal Ideal Theorem. Dimension of the intersection of two varieties.- §4. Applications of the Principal Ideal Theorem in Noetherian rings.- §5. The graded ring and the conormal module of an ideal.- References.- VI. Regular and singular points of algebraic varieties.- §1. Regular points of algebraic varieties. Regular local rings.- §2. The zero divisiors of a ring or module. Primary decomposition.- §3. Regular sequences. Cohen-Macaulay modules and rings.- §4. A connectedness theorem for set-theoretic complete intersections in projective space.- References.- VII. Projective Resolutions.- §1. The projective dimension of modules.- §2. Homological characterizations of regular rings and local complete intersections.- §3. Modules of projective dimension ? 1.- §4. Algebraic curves in A3 that are locally complete intersections can be represented as the intersection of two algebraic surfaces.- References.- A. Textbooks.- B. Research papers.- List of symbols.
