XVII. THÉORÈME DE LEFSCHETZ AFFINE

X δ F
X
δ(F) = sup δ(x), x ∈X|F¯x6= 0.
f:Y→X
X δXY
δYδXn X
FZ/nZY
δY(Rqf?(F)) ≤δX(F)−q .
S f :Y→X
S X
δXY δYδX
n S FZ/nZY
δY(Rqf?(F)) ≤δX(F)−q .
X
f:Y→X X
X δX(x) = dim({x})
Y δYδX
n X FZ/nZY
Hq(Y, F) = 0 q > δY(F)
δX(Rqf∗F)≤δY(F)−q
⇐⇒ Rqf∗(F)¯x= 0 δX(x)> δY(F)−q∀¯x→x∈X
⇐⇒ Hq(Y(¯x),F)=0 q > δY(F)−δX(x)≥δY(¯x)(F)∀¯x→x∈X.

Y(¯x)X
SGA 4
X d U
X n X F
Z/nZUHq(U, F) = 0 q > d
F=Z/nZX
V//
Y
U//X
X Y X Y U V
X Y RΓ(U, Z/nZ)→
RΓ(V, Z/nZ)U V
U →X V →Y
U•→X• V•→Y•U•V•
U X 0
X I
X U0
X0I X
Hq(U, Z/nZ)∼ −→ Hq(U0,Z/nZ).
X d
U X n X F
Z/nZUHq(U, F) = 0 q > d
F=Z/nZ
Z/nZU, X F
SGA 4 Ff?j!Z/nZ
W0j//U0f//U
j U0f
X0X U0X X0X
dimX≥dimX0X U0X0
X0X Y 0=X0\W0
W0X0V0=U0\W0Y0dimY0<dimX0
j!(Z/nZ)W0−→ (Z/nZ)U0−→ (Z/nZ)V0
U0
V0Hq(U0, j!Z/nZ) = 0 q > dimX0
Hq(U, f?j!Z/nZ)=Hq(U0, j!Z/nZ),

X
b
X X b X X
X U X b Ub X
b
X U
b
U
//U
b
X//X
Hq(b U, Z/nZ) = Hq(U, Z/nZ)q≥0n X
V// Y
U//X
X Y Y →X
U V X Y
RΓ(U, Z/nZ)→RΓ(V, Z/nZ)
X Z
j:U=X\Z →X{Zi}i∈In
X
R1j?(Z/nZ)∼ −→ M
i∈I
(Z/nZ)Zi(−1)
Rqj?(Z/nZ)∼ −→
q
^R1j?(Z/nZ)
Vj0
//
Y
g
T
oo
Uj//X Z
oo
X Y U V
Z T
¯y→Y f1,· · · , fr
Z g(¯y)Z →X T →Y¯y
g∗(f1),· · · , g∗(fr)T¯y
fiZ →X T →Y
Y

Z →X T →Y
g?Rj?(Z/nZ)∼ −→ Rj0
?(Z/nZ)
X Y g
RΓ(U, Z/nZ)∼ −→ RΓ(V, Z/nZ).
U V 1
gH1(U, Z/nZ)∼ →H1(V, Z/nZ)
Z T g
Vj0
//
Y
g
T
oo
Uj//X Z
oo
X Y x
y g U V X Y
h:x→X h0:y→Y
U•
j•//
εU
X•
εX
Z•
εZ
oo
Uj//X Z
oo
εY:Y•→Y εV:V•→V εT:T•→T εX,x :X•,x →x
εY,y :Y•,y →y h•:X•,x →X•h0
•:Y•,y →Y•
h h0g•:Y•→X•g
i
XiYi
ZiTi
(Zi→Xi) (Ti→Yi)Yi
n X
RΓ(U, Z/nZ)∼ −→ RΓ(V, Z/nZ)
RΓ(U, Z/nZ)∼ −→ h∗Rj?(Z/nZ)
∼
−→ R(εX,x)?h?
•R(j•)?(Z/nZ) ( et )
∼
−→ R(εY,y)?(h0
•)?g? •R(j•)?(Z/nZ)
∼
−→ R(εY,y)?(h0
•)?R(j0 •)?(Z/nZ) ( )
∼
−→ (h0)∗Rj0
?(Z/nZ) ( et )
∼
−→ RΓ(V, Z/nZ)
h∗Rj?Z/nZ→R(εX,x)?h?
•R(j•)?Z/nZ

Y•g?
•R(j•)?Z/nZ→
R(j0
•)?Z/nZ(h0 •)?g? •R(j•)?Z/nZ→
(h0
•)?R(j0 •)?Z/nZ
εX:X•→X N ∈N
X•≤N=sqN(X•)X•≤NG∈D+
c(X, Z/nZ)
r∈Nτ≤rGr
F•Z/nZX•τ≤rR(εX)?F•
X•≤r+1
q≥0fq:Xq→X
q Aq,. Rfq,? Fq≥0
R(εX)?(F•)
Aq+1,p //Aq+1,p+1
Aq,p //
OO
Aq,p+1
OO
p q ≥0
Xq+1 →Xqτ≤qR(εX)?F•
Aq,p p+q≤r
X•≤r+1
U•≤N//
εU
X•≤N
εX
Z•≤N
εZ
oo
U//X Z
oo
εY:Y•≤N→Y εV:V•≤N→V
εT:T•≤N→T
i≤N
XiYi
ZiTi
(Zi→Xi) (Ti→Yi)Yi
n X
Hq(U, Z/nZ)−→ Hq(V, Z/nZ)
q < N
X•= cosqNX•≤N
Y•Y