Algebraic Geometry

1. Front Matter

   Pages I-VII

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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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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 .

• Algebraic Geometry
• Geometry
• Schemes
• Vector bundles
• algebra
• morphisms

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

  Ulrich Görtz

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

  Torsten Wedhorn

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.

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