XVI. FONCTIONS DE DIMENSION

X
X
S X ⊂Y
Sdim(Y/X)
X Z⊂Y S X Y
x X
dim(Y/X) = dim(OY,x).
0S
X⊂Y⊂Z S
dim(Z/X) = dim(Z/Y ) + dim(Y/X).
S S
S n ≥0An
S
S
n≥0An
SS
0

S
b
S S
Sb S
s∈S{s}
S
S
S
A1
S
S s ∈S S0
S0→ {s}s0S0dim(OS0, s0) = dim({s})
≤2
≤1
S= Spec(A)P
A L Frac(A/P )B A/P L
B
A/P
B
dim(B) = dim(A/P )
X x X ¯x
x X(x)Xh
(x)b
X(x)X x
X(¯x)X¯x x y X ¯x¯y
x y
SGA 4 ¯x ¯y
X X(¯x)→X(¯y)
SGA 4 ¯x ¯y
X¯x→X(¯y)

r∈N¯x ¯y
r¯x X(¯y)r
y x ¯x ¯y
1
y x y ∈ {x}y
x1
y x x
y f :X(¯y)→X(y)
y f x0∈X(¯y)
dim {x0}= 1
y x y ∈ {x}
y x
x0∈X(¯y)dim {x0}= 1 x=f(x0)
y x0
{x}(¯y)=f−1({x}).
x00 ∈ {x}(¯y)x0f
f(x00) = x x00 =x0x X
x01{x}(¯y)
f f(x0) = x
X
x X X x
y x c :b X(y)→X(y)
y x
c−1(x) 1
X={x}X(¯y)→Xh
(y)
X¯y
1X y
1Xh
(y)
Xh
(y)
1b X(y)1
c1
b
X(y)x X
X
X
X
x y X
{x}=X X(y)1
X(y)
X(¯y)1
X(¯y)−→ Xh
(y)

Xh
(y)1
b
X(y)−→ Xh
(y)
b
X(y)1
1X(y)
X δ :X→Z
x y X
δ(y) = δ(x)−1.
δ X
Xred
Ui →X X δ X δ ◦i
U
X X
X
XZ
X δ :X→
Zδ(x) = −dim(OX,x)X
X δ(y) = δ(x)−1
x y X
dim(OX,y ) = dim(OX,x) + dim(O{y},x) = dim(OX,x)+1.
X δ(x) = −dim(OX,x)
X
X
X1, ..., XnX
Xix x
Xi0≤i≤nFi
XiZ
δi∈ Fi0x1≤i, j ≤n δi−δj
Xi∩Xj0x Fi,j X
Xi∩Xjx U X
Fi,j δiU

S
δ X ⊂Y⊂Z S x y
z x y z
δ(z)−δ(y) = dim(Z/Y ),
δ(y)−δ(x) = dim(Y/X),
δ(z)−δ(x) = dim(Z/X).
dim(Z/X) = dim(Z/Y ) + dim(Y/X)
S
δ Shδ
δhShShδhSh
ShS
X δ :X→Z
δ(x) = −dim OX,x
x X ¯x x
X(¯x)
δ0:X(¯x)→Zy
h:X(¯x)→X
δ(h(y)) = δ0(y)y∈X(¯x)
δ(h(y)) = δ(x)+1 y∈X(¯x)1
X Y
p:Y→X δ
Y p1
p2Y×XY→Y
p?
1δ−p? 2δ:Y×XY−→ Z
1 [δ] H1
Cech(Y→X, Z)
H1
Cech(Y→X, Z)→H1(X, Z).
[δ]ZX
U X [δ]|U
H1(U, Z)U