Milne JS - Algebraic Geometry.pdf

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Algebraic Geometry
J.S. Milne
Version 5.20
September 14, 2009
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These notes are an introduction to the theory of algebraic varieties. In contrast to most
such accounts they study abstract algebraic varieties, and not just subvarieties of affine and
projective space. This approach leads more naturally into scheme theory.
BibTeXinformation
@misc{milneAG,
author={Milne,JamesS.},
title={AlgebraicGeometry(v5.20)},
year={2009},
note={Availableatwww.jmilne.org/math/},
pages={239+vi}
}
v2.01 (August 24, 1996). First version on the web.
v3.01 (June 13, 1998).
v4.00 (October 30, 2003). Fixed errors; many minor revisions; added exercises; added two
sections/chapters; 206 pages.
v5.00 (February 20, 2005). Heavily revised; most numbering changed; 227 pages.
v5.10 (March 19, 2008). Minor fixes; TeX style changed, so page numbers changed; 241
pages.
v5.20 (September 14, 2009). Minor corrections; revised Chapters 1, 11, 16; 245 pages.
Available at www.jmilne.org/math/
Please send comments and corrections to me at the address on my web page.
The photograph is of Lake Sylvan, New Zealand.
Copyright c 1996, 1998, 2003, 2005, 2008, 2009 J.S. Milne.
Single paper copies for noncommercial personal use may be made without explicit permis-
sion from the copyright holder.
Contents
Contents
iii
Notations vi ; Prerequisites vi ; References vi ; Acknowledgements vi
Introduction
1
1
Preliminaries
4
Rings and algebras 4 ; Ideals 4 ; Noetherian rings 6 ; Unique factorization 8 ; Polynomial
rings 10 ; Integrality 11 ; Direct limits (summary) 14 ; Rings of fractions 15 ; Tensor Prod-
ucts 18 ; Categories and functors 21 ; Algorithms for polynomials 23 ; Exercises 29
2
Algebraic Sets
30
Definition of an algebraic set 30 ; The Hilbert basis theorem 31 ; The Zariski topology 32 ;
The Hilbert Nullstellensatz 33 ; The correspondence between algebraic sets and ideals 34 ;
Finding the radical of an ideal 37 ; The Zariski topology on an algebraic set 37 ; The coor-
dinate ring of an algebraic set 38 ; Irreducible algebraic sets 39 ; Dimension 41 ; Exercises
44
3
Affine Algebraic Varieties
45
Ringed spaces 45 ; The ringed space structure on an algebraic set 47 ; Morphisms of ringed
spaces 50 ; Affine algebraic varieties 50 ; The category of affine algebraic varieties 51 ;
Explicit description of morphisms of affine varieties 53 ; Subvarieties 55 ; Properties of the
regular map defined by specm() 56 ; Affine space without coordinates 57 ; Exercises 58
4
Algebraic Varieties
60
Algebraic prevarieties 60 ; Regular maps 61 ; Algebraic varieties 62 ; Maps from varieties to
affine varieties 63 ; Subvarieties 64 ; Prevarieties obtained by patching 65 ; Products of vari-
eties 66 ; The separation axiom revisited 70 ; Fibred products 73 ; Dimension 74 ; Birational
equivalence 75 ; Dominant maps 76 ; Algebraic varieties as a functors 76 ; Exercises 78
5
Local Study
80
Tangent spaces to plane curves 80 ; Tangent cones to plane curves 82 ; The local ring at a
point on a curve 82 ; Tangent spaces of subvarieties of A
m 83 ; The differential of a regular
map 85 ; Etale maps 86 ; Intrinsic definition of the tangent space 88 ; Nonsingular points 91 ;
Nonsingularity and regularity 92 ; Nonsingularity and normality 93 ; Etale neighbourhoods
94 ; Smooth maps 96 ; Dual numbers and derivations 97 ; Tangent cones 100 ; Exercises 102
6
Projective Varieties
103
n 103 ; The Zariski topology on P
n 106 ; Closed subsets of A
n and
Algebraic subsets of P
iii
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n 107 ; The hyperplane at infinity 107 ; P
n is an algebraic variety 108 ; The homogeneous
P
n 110 ; Regular functions on a projective variety 111 ;
Morphisms from projective varieties 112 ; Examples of regular maps of projective vari-
eties 113 ; Projective space without coordinates 118 ; Grassmann varieties 118 ; Bezout’s
theorem 122 ; Hilbert polynomials (sketch) 123 ; Exercises 124
coordinate ring of a subvariety of P
7
Complete varieties
125
Definition and basic properties 125 ; Projective varieties are complete 127 ; Elimination
theory 128 ; The rigidity theorem 130 ; Theorems of Chow 131 ; Nagata’s Embedding The-
orem 132 ; Exercises 132
8
Finite Maps
133
Definition and basic properties 133 ; Noether Normalization Theorem 137 ; Zariski’s main
theorem 138 ; The base change of a finite map 140 ; Proper maps 140 ; Exercises 141
9
Dimension Theory
143
Affine varieties 143 ; Projective varieties 150
10 Regular Maps and Their Fibres
152
Constructible sets 152 ; Orbits of group actions 155 ; The fibres of morphisms 157 ; The
fibres of finite maps 159 ; Flat maps 160 ; Lines on surfaces 161 ; Stein factorization 167 ;
Exercises 167
11 Algebraic spaces; geometry over an arbitrary field
168
Preliminaries 168 ; Affine algebraic spaces 172 ; Affine algebraic varieties. 173 ; Algebraic
spaces; algebraic varieties. 174 ; Local study 179 ; Projective varieties. 180 ; Complete
varieties. 181 ; Normal varieties; Finite maps. 181 ; Dimension theory 181 ; Regular maps
and their fibres 181 ; Algebraic groups 181 ; Exercises 182
12 Divisors and Intersection Theory
183
Divisors 183 ; Intersection theory. 184 ; Exercises 189
13 Coherent Sheaves; Invertible Sheaves
190
Coherent sheaves 190 ; Invertible sheaves. 192 ; Invertible sheaves and divisors. 193 ; Direct
images and inverse images of coherent sheaves. 195 ; Principal bundles 195
14 Differentials (Outline)
196
15 Algebraic Varieties over the Complex Numbers
198
16 Descent Theory
201
Models 201 ; Fixed fields 201 ; Descending subspaces of vector spaces 202 ; Descending
subvarieties and morphisms 204 ; Galois descent of vector spaces 205 ; Descent data 206 ;
Galois descent of varieties 210 ; Weil restriction 211 ; Generic fibres and specialization 212 ;
Rigid descent 212 ; Weil’s descent theorems 215 ; Restatement in terms of group actions
216 ; Faithfully flat descent 218
17 Lefschetz Pencils (Outline)
222
Definition 222
iv
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18 Algebraic Schemes and Algebraic Spaces
225
A
Solutions to the exercises
226
B
Annotated Bibliography
233
Index
236
v
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