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

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