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Algebraic Geometry

Part I: Schemes. With Examples and Exercises

  • Textbook
  • © 2010

Overview

Authors:
  1. Ulrich Görtz

    1. Institute of Experimental Mathematics, University Duisburg-Essen, Essen, Germany

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  2. Torsten Wedhorn

    1. University of Paderborn, Department of Mathematics, Paderborn, Germany

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    You can also search for this author in PubMed   Google Scholar

  • Der Wegbegleiter in das Feld der modernen
  • algebraischen Geometrie im Bachelor/Master Studium

Part of the book series: Advanced Lectures in Mathematics (ALM)

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Table of contents (17 chapters)

  1. Front Matter

    Pages I-VII
    Download chapter PDF

  2. Introduction

    • Ulrich Görtz, Torsten Wedhorn
    Pages 1-6

  3. Prevarieties

    • Ulrich Görtz, Torsten Wedhorn
    Pages 7-39

  4. Spectrum of a Ring

    • Ulrich Görtz, Torsten Wedhorn
    Pages 40-65

  5. Schemes

    • Ulrich Görtz, Torsten Wedhorn
    Pages 66-92

  6. Fiber products

    • Ulrich Görtz, Torsten Wedhorn
    Pages 93-117

  7. Schemes over fields

    • Ulrich Görtz, Torsten Wedhorn
    Pages 118-144

  8. Local Properties of Schemes

    • Ulrich Görtz, Torsten Wedhorn
    Pages 145-168

  9. Quasi-coherent modules

    • Ulrich Görtz, Torsten Wedhorn
    Pages 169-204

  10. Representable Functors

    • Ulrich Görtz, Torsten Wedhorn
    Pages 205-225

  11. Separated morphisms

    • Ulrich Görtz, Torsten Wedhorn
    Pages 226-240

  12. Finiteness Conditions

    • Ulrich Görtz, Torsten Wedhorn
    Pages 241-285

  13. Vector bundles

    • Ulrich Görtz, Torsten Wedhorn
    Pages 286-319

  14. Affine and proper morphisms

    • Ulrich Görtz, Torsten Wedhorn
    Pages 320-365

  15. Projective morphisms

    • Ulrich Görtz, Torsten Wedhorn
    Pages 366-422

  16. Flat morphisms and dimension

    • Ulrich Görtz, Torsten Wedhorn
    Pages 423-484

  17. One-dimensional schemes

    • Ulrich Görtz, Torsten Wedhorn
    Pages 485-502

  18. Examples

    • Ulrich Görtz, Torsten Wedhorn
    Pages 503-540

  19. Back Matter

    Pages 541-615
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Keywords

About this book

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 .

Authors and Affiliations

  • Institute of Experimental Mathematics, University Duisburg-Essen, Essen, Germany

    Ulrich Görtz

  • University of Paderborn, Department of Mathematics, Paderborn, Germany

    Torsten Wedhorn

About the authors

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.

Bibliographic Information

  • Book Title: Algebraic Geometry

  • Book Subtitle: Part I: Schemes. With Examples and Exercises

  • Authors: Ulrich Görtz, Torsten Wedhorn

  • Series Title: Advanced Lectures in Mathematics

  • DOI: https://doi.org/10.1007/978-3-8348-9722-0

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH, Wiesbaden 2010

  • eBook ISBN: 978-3-8348-9722-0Published: 06 August 2010

  • Series ISSN: 0932-7134

  • Series E-ISSN: 2512-7039

  • Edition Number: 1

  • Number of Pages: IV, 615

  • Topics: Algebraic Geometry, Algebra

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