Dimension cohomologique et images directes supérieures à support

2 dim(X)
dim(X)
x X OX,x
OX,x =Ohs
X,x
X   X
cd`(X)n m > n Hm(X, F )=0
 F X X
cd`(X)=0
cd`(X)≤2 dim(X)
K/k n
cd`(K)≤cd`(k) + n
A K B
Btot B⊗AK→Btot
B
A y2=
x2(x+ 1) (y−ux)(y+ux) = 0 u=√1 + x∈B
A Ahs
B C = (A[X]/(P))qP∈A[X]
q C →B
x∈C0ex∈B
SR
R A →B B ⊗AK Btot
ex∈SBey∈B
exey∈SAC= (A[X]/P )qAex

x=u/v u, v ∈A[X]/P v 6∈ q vx =u
R∈A[X]eu=evex∈SBu∈SC
R(X) = XS(X) + a a SA
ey vS(vx)Bexey=vxS(vx) = −a
X k
cd`(X)≤2 dim(X).
cd = cd`
n= dim(X) F m > 2n
Hm(X, F ) = 0 n= 0 cd(XqY) = max(cd(X),cd(Y)) cd(X) =
cd(Xred)X K k
cd(X) = cd(k)=0
n≥1
X π :
X0→X X0=X1q···qXrX
X0X
F0=π∗F
Epq
2=Hp(X, Rqπ∗F0)⇒Hp+q(X0, F 0)x∈Xe
X
e
X0X0e X0
e
Xix∈Xi
(Rqπ∗F0)x=⊕Hq(e Xi, F 0)=0 q≥1 cd( e Xi)=0
Hp(X, π∗F0) = Hp(X0, F 0)
0→F→π∗F0→G→0G≤n−1
Hm(X, F ) = Hm(X, π∗F0)
Hp(X0, F 0)
X0→X
Epq
1=Hq(X0
p, F 0 p)⇒Hp+q(X, F )X0 p
(p+1) X0X F 0
pF X0 p
X
p≥1
q > 2(n−p)Hq(X0
p, F 0 p)=0
Hm(X0, F 0) = Hm(X, F )X0
p(p+ 1)
X
≤n−p X0
p
X00
•X00 p
(p+ 1)
F g∗F g :η→X
η F →g∗g∗F η
0→K→F→g∗g∗F→C→0K C
≤n−1I= im(F→g∗g∗F)
Hm(X, F )'Hm(X, I)'
Hm(X, g∗g∗F)

Rjg∗F≤n−j j
x∈X Y ⊂X d > n−j(Rjg∗F)x= 0
X X, Y
e
X= Spec(OX,x)e
Xηg
Ohs
X,x ⊗OX,x k(X) = Q(Ohs X,x) = K1× ··· × KrKi
x(Rjg∗F)x=
⊕iHj(Ki, F |Ki)d>n−j
j > n −d
k0⊂Kidegtr(Ki/k0)≤n−d i
Y→Ad
kx η Ad k
X→Ad
k
k0=k(η)sep X0=X×Ad
kSpec(k0)
k(η) = OAd
k,η →OX,x Ohs
X,x k0
u: Spec(Ohs
X,x)→Y0
Spec(Ohs
X,x)
''
%%
u
$$Y0//
X0//
Spec(k0)
Y//X//Ad
k.
k(x)/k(η)k(x)sep/k0x0
u X0X0n−d k0
Ohs
X0,x0=Ohs
X,x Kik0≤n−d
Hi(X, Rjg∗F) = 0 j≥1
i > 2(n−j)Hp(X, Rqg∗F)⇒Hp+q(k(X), F )
Hm(X, g∗F) = Hm(k(X), F )m > 2n
cd`(X)≤dim(X)
X k
cd`(X)≤dim(X).
X→Y
Y
k
X Y
Y X

X k Z X k
F X
H0
Z(X, F ) = ker H0(X, F )→H0(X\Z, F )
Z
Γc(X, F ) = [
Z
H0
Z(X, F )
Γc(X, −)
X k k
Γc(X, F ) = ⊕x∈X0H0
x(X, F )X0
RpΓc(X, F ) = ⊕x∈X0Hp
x(X, F )X
R1ΓcR2Γc
k X k z ∈X n
k
H0
x(X, Z/nZ) = 0
H1
x(X, Z/nZ) = 0
H2
x(X, Z/nZ)'Z/nZ
M=Z/nZe X X x
k Hp
x(X, M) = Hp(e
X, M)
p X e X
K U =X\x= Spec(K)
x →X← U
0−→ H0
x(X, M)−→ H0(X, M)−→ H0(U, M)−→ H1 x(X, M)−→ H1(X, M)−→
−→ H1(U, M)→H2
x(X, M)−→ H2(X, M)−→ . . .
Xcd(X)=0 H1(X, M) = H2(X, M)=0
M H0(X, M) = H0(U, M) = M
H0
x(X, M) = H1 x(X, M)=0
H2
x(X, M) = H1(U, M) = K∗/K∗n'Z/nZ
H1
x(X, Z/nZ) = Z/nZ
π:U→S
j:U →X π :X→S π =π◦j
Uj//
π
B
B
B
B
B
B
B
BX
π
S

S
S
U→S
π:U→S π =π◦j
p Rp
cπ∗F:=
Rpπ∗(j!F)
F X Rpπ∗(j!F)
Uj0 →X0π0 →S X0
U X ×SX0X0X g :X0→X
X0
g
π0
~~
U
j0==
|
|
|
|
|
|
|
|j//
π
!!
C
C
C
C
C
C
C
CX
π
S
π0
∗=π∗◦g∗
Epq
2=Rpπ∗Rqg∗(j0
!F)⇒Rp+qπ0
∗(j0
!F).
Rqg∗(j0
!F)
g U
U j0
!F= 0 R0g∗(j0 !F) = j!F Rqg∗(j0 !F)=0
q > 0Rpπj!F=Rpπ0j0
!F
Rp
cπ∗δ
Rp
cπ∗F
F Rp
cπ∗F
U0π0 →Uπ →S π π0π◦π0
Epq
2=Rp
cπ∗(Rq cπ0 ∗F)⇒Rp+q c(ππ0)∗F.
Rp
cπ∗
R0
cπ∗
π=π◦j
δ Rp
cπ∗
j!
j!Rpπ∗