Algebraic Geometry I

624 Pages·2012·3.3 MB·English

Ulrich Görtz | Torsten Wedhorn
Algebraic Geometry I

Ulrich Görtz | Torsten Wedhorn
Algebraic
Geometry I
Schemes
With Examples and Exercises
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Prof. Dr. Ulrich Görtz
Institute of Experimental Mathematics
University Duisburg-Essen
Ellernstraße 29
45326 Essen
[email protected]
Prof. Dr. Torsten Wedhorn
Department of Mathematics
University of Paderborn
Warburger Straße 100
33098 Paderborn
[email protected]
1st Edition 2010
All rights reserved
© Vieweg+Teubner Verlag|Springer Fachmedien Wiesbaden GmbH 2010
Editorial Office: Ulrike Schmickler-Hirzebruch
Vieweg+Teubner Verlag is a brand of Springer Fachmedien.
Springer Fachmedien is part of Springer Science+Business Media..
www.viewegteubner.de
No part of this publication may be reproduced, stored in a retrieval system or
transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the copyright holder.
Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication
are part of the law for trade-mark protection and may not be used free in any form or by any means
even if this is not specifically marked.
Cover design:KünkelLopka Medienentwicklung, Heidelberg
Printed on acid-free paper
Printed in Germany
ISBN 978-3-8348-0676-5
V
Contents
Introduction 1
1 Prevarieties 7
Affine algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Affine algebraic sets as spaces with functions . . . . . . . . . . . . . . . . . . 17
Prevarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Spectrum of a Ring 40
Spectrum of a ring as a topological space . . . . . . . . . . . . . . . . . . . . 41
Excursion: Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Spectrum of a ring as a locally ringed space . . . . . . . . . . . . . . . . . . . 57
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Schemes 66
Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Examples of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Basic properties of schemes and morphisms of schemes . . . . . . . . . . . . . 74
Prevarieties as Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Subschemes and Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Fiber products 93
Schemes as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Fiber products of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Base change, Fibers of a morphism . . . . . . . . . . . . . . . . . . . . . . . . 105
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Schemes over fields 118
Schemes over a field which is not algebraically closed . . . . . . . . . . . . . . 118
Dimension of schemes over a field . . . . . . . . . . . . . . . . . . . . . . . . . 120
Schemes over fields and extensions of the base field . . . . . . . . . . . . . . . 133
Intersections of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 Local Properties of Schemes 145
The tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Regular schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Normal schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
VI Contents
7 Quasi-coherent modules 169
Excursion: O -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
X
Quasi-coherent modules on a scheme . . . . . . . . . . . . . . . . . . . . . . . 181
Properties of quasi-coherent modules . . . . . . . . . . . . . . . . . . . . . . . 189
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8 Representable Functors 205
Representable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
The example of the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . 209
Brauer-Severi schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9 Separated morphisms 226
Diagonal of scheme morphisms and separated morphisms . . . . . . . . . . . 226
Rational maps and function fields. . . . . . . . . . . . . . . . . . . . . . . . . 232
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10 Finiteness Conditions 241
Finiteness conditions (noetherian case) . . . . . . . . . . . . . . . . . . . . . . 242
Finiteness conditions in the non-noetherian case . . . . . . . . . . . . . . . . 249
Schemes over inductive limits of rings . . . . . . . . . . . . . . . . . . . . . . 258
Constructible properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11 Vector bundles 286
Vector bundles and locally free modules . . . . . . . . . . . . . . . . . . . . . 286
Flattening stratification for modules . . . . . . . . . . . . . . . . . . . . . . . 296
Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Vector bundles on P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
12 Affine and proper morphisms 320
Affine morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
Finite and quasi-finite morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 324
Serre’s and Chevalley’s criteria to be affine . . . . . . . . . . . . . . . . . . . 334
Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Proper morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Zariski’s main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
13 Projective morphisms 366
Projective spectrum of a graded algebra . . . . . . . . . . . . . . . . . . . . . 367
Embeddings into projective space . . . . . . . . . . . . . . . . . . . . . . . . . 384
Blowing-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
VII
14 Flat morphisms and dimension 423
Flat morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Properties of flat morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Faithfully flat descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Dimension and fibers of morphisms . . . . . . . . . . . . . . . . . . . . . . . . 463
Dimension and regularity conditions . . . . . . . . . . . . . . . . . . . . . . . 473
Hilbert schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
15 One-dimensional schemes 485
Morphisms into and from one-dimensional schemes . . . . . . . . . . . . . . . 485
Valuative criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
Curves over fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Divisors on curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
16 Examples 503
Determinantal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Cubic surfaces and a Hilbert modular surface . . . . . . . . . . . . . . . . . . 520
Cyclic quotient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
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
Bibliography 583
Detailed List of Contents 588
Index of Symbols 598
Index 602

1
Introduction
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 coefficients in a field k.
i 1 n
ThesetofsolutionsisasubsetV(f ,...,f )ofkn.Polynomialequationsareomnipresent
1 r
inandoutsidemathematics,andhavebeenstudiedsinceantiquity.Thefocusofalgebraic
geometry is studying the geometric structure of their solution sets.
If the polynomials f are linear, then V(f ,...,f ) is a subvector space of kn. Its
i 1 r
“size” is measured by its dimension and it can be described as the kernel of the linear
map kn →kr, x=(x ,...,x )(cid:4)→(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 first manifestation is the following: If g =g f +...g f
1 1 r r
is a linear combination of the f (with coefficients 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 . On the other hand we may consider polynomials
1 n i
f ∈k[T ,...,T ], view them as polynomial functions kn →k, and define the ideal a(cid:2) of
1 n
thosef suchthatf vanishesonV(f ,...,f ).Itisoneofthefirstmainresults(Hilbert’s
1 r
Nullstellensatz) that if k is algebraically closed, then a(cid:2) is closely related to a, more
precisely
a(cid:2) =rad(a):={f ∈k[T ,...,T ] ; ∃m>0:fm ∈a}.
1 n
The quotient k[T ,...,T ]/a(cid:2) may be considered as thek -algebra of polynomial func-
1 n
tions on the affine variety V(f ,...,f ). We obtain a close relation between ideals a of
1 r
k[T ,...,T ] (or, equivalently, quotient algebras of k[T ,...,T ]) and affine varieties in
1 n 1 n
kn –atleastifk isalgebraicallyclosed.Fornotalgebraicallyclosedfieldsk thisapproach
is too naive.
Besides this algebraic description, one can endow the sets V(f ,...,f ) with a “geo-
1 r
metricstructure”.Theonlyreasonabletopologywhichcanbedefinedpurelyinalgebraic
terms, i.e., without appealing to analytic notions as convergence, is the Zariski topology
which is the coarsest topology (on kn, say) such that all zero sets of polynomials are
closed. Not surprisingly, it is very coarse and therefore is not sufficient to determine
the “geometric structure” of the spaces in question. The right way to remedy this, is to
considereachspacetogetherwiththeentiretyoffunctionsonthespace.Similarlyasadif-
ferentiable(orholomorphic)manifoldisdeterminedbyitstopologicalstructuretogether
with the entirety of differentiable (or holomorphic) functions on all its open subsets, we
obtain a satisfactory notion of algebraic geometric objects, “affine varieties over k”, by
considering closed subsets of kn together with the entirety of functions on them, which
in this case means all functions defined by rational polynomials.