Algebraic Geometry: Part I: Schemes. With Examples and Exercises (Google eBook)

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Springer Science & Business Media, Aug 9, 2010 - Mathematics - 615 pages
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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 .
  

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Contents

Introduction
1
1 Prevarieties
7
2 Spectrum of a Ring
40
3 Schemes
66
4 Fiber products
93
5 Schemes over fields
118
6 Local Properties of Schemes
145
7 Quasicoherent modules
169
14 Flat morphisms and dimension
423
15 Onedimensional schemes
485
16 Examples
503
A The language of categories
541
B Commutative Algebra
547
C Permanence for properties of morphisms of schemes
573
D Relations between properties of morphisms of schemes
576
E Constructible and open properties
578

8 Representable Functors
205
9 Separated morphisms
226
10 Finiteness Conditions
241
11 Vector bundles
286
12 Affine and proper morphisms
320
13 Projective morphisms
366
Bibliography
583
Detailed List of Contents
588
Index of Symbols
598
Index
602
Copyright

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About the author (2010)

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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