Indexaﬃne coordinate ring, 20
aﬃne morphism, 172aﬃne n-space, 31
aﬃne scheme, 108aﬃne variety, 30
algebraic set, 8, 12
associated point of coherent OX -module, 207

generic point [aﬃne version], 94
generic point [prescheme version], 114
generic stalk, 29
geometric fibre of a morphism, 155geometric point of a preschems over k0 , 162
global section of a presheaf, 27
going-down theorem of Cohen-Seidenberg, 179
going-up theorem of Cohen-Seidenberg, 4Grothendieck, 212
Grothendieck’s existence problem, 160

bijective morphism, 31
birational correspondence, 128
birational morphism of varieties, 73
blowing-up [aﬃne space version], 220
blowing-up [scheme version], 224
branch locus of a finite morphism, 303

Hensel’s lemma, 247
Hilbert’s Nullstellensatz, 8Hironaka, 186
homological method, 192
homomorphism of OX -modules, 193

Chevalley, 72
Chow’s lemma, 85
closed immersion, 146closed subprescheme, 145
coherent OX -module, 206
complete non-singular model of a curve, 284
complete variety, 77
complex variety, 80
component of an algebraic set, 11conic, 16conjugation map, 131
connectedness theorem of Zariski, 183
constructible subset of a variety, 71
cotangent bundle, 236
cotangent space, 235

invertible OX -module, 208
irreducible algebraic set, 10
irreducible topological space, 36
Krull dimension of a local ring, 63
k0 -topology, 136
K-valued point of a prescheme over S, 158
local criterion for varieties, 55
local criterion for variety [proof], 165
locally free OX -module, 208
map of presheaves, 24
maximal atlas, 209morphism of irreducibe algebraic sets, 18
morphism of preschemes, 108
morphism of prevarieties, 41

dimension of a variety, 56
dimension [topological characterization], 61
dimension (upper semi-continuity of —), 72
discrete valuation ring, 102
dominating morphism of varieties, 67

Nagata, 271
Nakayama’s lemma, 79, 212
noetherian decomposition theorem, 10
noetherian scheme, 205
noetherian topological space, 22
Noether’s normalization lemma, 3, 176
Noether’s normalization lemma (geometric form of—), 59
Noether’s normalization lemma over R, 180
non-singular closed point, 232
non-singular in codimension one, 273
non-singular point (not necessarily closed), 236
non-singular variety, 236
normal point of a variety, 272
normal variety, 272
normalization of variety, 277
Nullstellensatz (Hilbert’s —), 8
Nullstellensatz (weak —), 6

elimination theory, 76embedded component of a scheme, 208
embedding dimension, 236
equality of two morphisms at a point, 163étale morphism, 243
exceptional divisor, 221
exceptional subscheme, 224
factorial variety, 259, 289
fibre of a morphism, 153
fibre product, 116
field of definition, 131
finite morphism, 172
finite morphism of aﬃne varieties, 57
finite type (morphism of —), 168
finite type (prescheme of — over R), 122
flat morphism, 297
flat (OX -module — over OY ), 297
flat R-module, 295
function field of a prevariety, 37
functor of points, 155order ordZ (f ) of vanishing of f along Z, 273
power series methods, 192
prescheme, 108
1

prescheme of finite type over R, 122
prescheme over R, 116
presheaf, 24
prevariety, 123prevariety [definition], 35
prevariety over k0 , 143
principal ideal theorem (Krull’s —), 57
principal valuation ring, 102, 198, 214
product of two prevarieties, 47
projective methods, 192
projective variety, 40
projectivized tangent cone, 225
proper morphism, 168
pure codimension, 61
pure dimension, 61

valuative criterion for completeness, 80
valuative criterion for properness, 171variety, 52, 164
vector bundle with atlas, 209
Zariski topology of a projective algebraic set, 23
Zariski topology of an aﬃne algebraic set, 21
Zariski topology of an aﬃne scheme, 93
Zariski’s main theorem [connectedness form], 289
Zariski’s main theorem [Grothendieck’s form], 289
Zariski’s main theorem [motivation], 286
Zariski’s main theorem [original form], 288
Zariski’s main theorem [power series form], 289
Zariski’s main theorem [toplogical form], 288
zeroth cohomology of a presheaf, 27

quasi-coherent OX -ideal, 150
quasi-coherent OX -module, 195
ramification point of a finite morphism, 303
rational closed point over k0 , 136
rational function on a prevariety, 37
reduced prescheme, 123
relative diﬀerential [definition], 203
relative diﬀerential (sheaf ΩX/Y of —s), 199
R-morphism, 116
scheme, 164section of a presheaf, 27
separable, 142, 143, 201, 202, 245, 246, 256
separably algebraic closed, 142, 143
separating transcendence basis, 201
sheaf, 25
sheafication of a presheaf, 26smooth morphism, 304
space curve, 66
specialization of a closed subset of Pn (k), 182
specialization of x ∈ Pn (k), 177
Spec(R), 93
stalk, 26
strong topology, 81
structure sheaf of an aﬃne scheme, 100
structure sheaf of an irreducible algebraic set, 28
structure theorem of integrally closed noetherianrings, 272
tangent cone, 215
tangent cone [final definition], 216
tangent cone [provisional definition], 215
tangent space [final definition], 231
tangent space [provisional definition], 228
torsion part, 199trivialization, 212
twisted cubic, 14
UFD property at non-singular points, 259
UFD property at singular points, 269
uniformazing parameters [definition], 256
uniformazing parameters [motivation], 255
universally torsion-free, 297
upper semi-continuity of dimension, 722