Milne JS - Algebraic Geometry.pdf
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Algebraic Geometry
J.S. Milne
Version 5.20
September 14, 2009
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
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
18 Algebraic Schemes and Algebraic Spaces
225
A
Solutions to the exercises
226
B
Annotated Bibliography
233
Index
236
v
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