
Algebraic Geometry
Part I: Schemes. With Examples and Exercises
Vieweg+Teubner Verlag
Published on 25. June 2010
Book
Paperback/Softback
IV, 615 pages
978-3-8348-0676-5 (ISBN)
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Description
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 .
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Product info
Book
Series
Edition
2010
Language
English
Place of publication
Wiesbaden
Germany
Publishing group
Vieweg & Teubner
Target group
Professional and scholarly
Studierende der Mathematik an Universitäten ab 5. Semester
Mathematiker an Instituten und Forschungsinstituten
Product notice
Paperback (trade)
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biography
Dimensions
Height: 24 cm
Width: 16.8 cm
Thickness: 31 mm
Weight
1157 gr
ISBN-13
978-3-8348-0676-5 (9783834806765)
DOI
10.1007/978-3-8348-9722-0
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Persons
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.
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.
Content
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