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 BAKBtot
B
A y2=
x2(x+ 1) (yux)(y+ux) = 0 u=1 + xB
A Ahs
B C = (A[X]/(P))qPA[X]
q C B
xC0exB
SR
R A B B AK Btot
exSBeyB
exeySAC= (A[X]/P )qAex
x=u/v u, v A[X]/P v 6∈ q vx =u
RA[X]eu=evexSBuSC
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
n1
X π :
X0X X0=X1q···qXrX
X0X
F0=πF
Epq
2=Hp(X, RqπF0)Hp+q(X0, F 0)xXe
X
e
X0X0e
X0
e
XixXi
(RqπF0)x=Hq(e
Xi, F 0)=0 q1 cd( e
Xi)=0
Hp(X, πF0) = Hp(X0, F 0)
0FπF0G0Gn1
Hm(X, F ) = Hm(X, πF0)
Hp(X0, F 0)
X0X
Epq
1=Hq(X0
p, F 0
p)Hp+q(X, F )X0
p
(p+1) X0X F 0
pF X0
p
X
p1
q > 2(np)Hq(X0
p, F 0
p)=0
Hm(X0, F 0) = Hm(X, F )X0
p(p+ 1)
X
np X0
p
X00
X00
p
(p+ 1)
F gF g :ηX
η F ggF η
0KFggFC0K C
n1I= im(FggF)
Hm(X, F )'Hm(X, I)'
Hm(X, ggF)
RjgFnj j
xX Y X d > nj(RjgF)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(RjgF)x=
iHj(Ki, F |Ki)d>nj
j > n d
k0Kidegtr(Ki/k0)nd i
YAd
kx η Ad
k
XAd
k
k0=k(η)sep X0=X×Ad
kSpec(k0)
k(η) = OAd
kOX,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 X0X0nd k0
Ohs
X0,x0=Ohs
X,x Kik0nd
Hi(X, RjgF) = 0 j1
i > 2(nj)Hp(X, RqgF)Hp+q(k(X), F )
Hm(X, gF) = Hm(k(X), F )m > 2n
cd`(X)dim(X)
X k
cd`(X)dim(X).
XY
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 ) = xX0H0
x(X, F )X0
RpΓc(X, F ) = xX0Hp
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
0H0
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/Kn'Z/nZ
H1
x(X, Z/nZ) = Z/nZ
π:US
j:U X π :XS π =πj
Uj//
π
B
B
B
B
B
B
B
BX
π
S
S
S
US
π:US π =πj
p Rp
cπF:=
Rpπ(j!F)
F X Rpπ(j!F)
Uj0
X0π0
S X0
U X ×SX0X0X g :X0X
X0
g
π0
~~
U
j0==
|
|
|
|
|
|
|
|j//
π
!!
C
C
C
C
C
C
C
CX
π
S
π0
=πg
Epq
2=RpπRqg(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
cπδ
Rp
cπF
F Rp
cπF
U0π0
Uπ
S π π0ππ0
Epq
2=Rp
cπ(Rq
cπ0
F)Rp+q
c(ππ0)F.
Rp
cπ
R0
cπ
π=πj
δ Rp
cπ
j!
j!Rpπ
1 / 6 100%
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