Algebraic Geometry

Part I: Schemes. With Examples and Exercises

Ulrich Görtz, Torsten Wedhorn (Autoren)

Buch | Softcover

IV, 615 Seiten

2010 | 2010
Vieweg & Teubner (Verlag)
978-3-8348-0676-5 (ISBN)

Infos

Algebraic geometry has its origin in the study of systems of polynomial equations f (x ,. . . ,x )=0, 1 1 n . . . f (x ,. . . ,x )=0. r 1 n Here the f ? k[X ,. . . ,X ] are polynomials in n variables with coe?cients in a ?eld k. i 1 n n ThesetofsolutionsisasubsetV(f ,. . . ,f)ofk . Polynomialequationsareomnipresent 1 r inandoutsidemathematics,andhavebeenstudiedsinceantiquity. Thefocusofalgebraic geometry is studying the geometric structure of their solution sets. n If the polynomials f are linear, then V(f ,. . . ,f ) is a subvector space of k. Its i 1 r “size” is measured by its dimension and it can be described as the kernel of the linear n r map k ? k , x=(x ,. . . ,x ) ? (f (x),. . . ,f (x)). 1 n 1 r For arbitrary polynomials, V(f ,. . . ,f ) is in general not a subvector space. To study 1 r it, one uses the close connection of geometry and algebra which is a key property of algebraic geometry, and whose ?rst manifestation is the following: If g = g f +. . . g f 1 1 r r is a linear combination of the f (with coe?cients g ? k[T ,. . . ,T ]), then we have i i 1 n V(f ,. . . ,f)= V(g,f ,. . . ,f ). Thus the set of solutions depends only on the ideal 1 r 1 r a? k[T ,. . . ,T ] generated by the f .

Autor

Prof. Dr. Ulrich Görtz, Institut für Experimentelle Mathematik, Universität Duisburg-Essen.Essen. Prof. Dr. Torsten Wedhorn, Institut für Mathematik, Universität Paderborn. Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen. Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn.

Inhalt

Prevarieties - Spectrum of a Ring - Schemes - Fiber products - Schemes over fields - Local Properties of Schemes - Quasi-coherent modules - Representable Functors - Separated morphisms - Finiteness Conditions - Vector bundles - Affine and proper morphisms - Projective morphisms - Flat morphisms and dimension - One-dimensional schemes - Examples

Produkt-Details

• Erscheint lt. Verlag: 25.6.2010
• Reihe/Serie: Advanced Lectures in Mathematics
• Zusatzinfo: IV, 615 p.
• Verlagsort: Wiesbaden
• Sprache: englisch
• Maße: 168 x 240 mm
• Gewicht: 1006 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: Algebra • Algebraic Geometry • Algebraische Geometrie; Handbuch/Lehrbuch • Geometry • Morphisms • schemes • Vector Bundles
• ISBN-10: 3-8348-0676-5 / 3834806765
• ISBN-13: 978-3-8348-0676-5 / 9783834806765
• Zustand: Neuware