Intro1

X X X
Λ ΛΛX−mod Λ
X F ΛΛX−mod
MΛMXX
V/X 7→ M F X F
MXΛM
F
π:W→X π∗(F)
F F
ΛΛX−mod
KΛΛX−mod K
Db
c(X , Λ),
ΛΛX−mod
f:Y→X
(f∗,Rf∗)
f:Y→X
(f!, f!)f!=f∗f f!=f∗f

X
X¯ FpΛp
H∗(X , ΛX)
Λ
π:X→Spec(¯ Fp)
Rπ∗(ΛX)
¯
Fp
Λ
Hn(X , ΛX) = Rnπ∗(ΛX).
l X Ql
Hn(X , Ql) := lim
←−
n>0
Hn(X , Z/ln.Z)⊗Q.
l X
Q⊗Zlim
←−
n>0
Rπ∗(Z/lnZ)X.
l
Rπ∗(Z/lZ)X
RΓ(X, Zl) = lim
←−
n>0
Rπ∗(Z/lnZ)X
RΓ(X, Ql) = lim
←−
n>0
Rπ∗(Z/lnZ)X⊗ZQ
X
X Y ¯ Fp
⊕p+q=nHp(X , Ql)⊗QlHq(Y , Ql)→Hn((Xׯ
FpY),Ql)
Λ
D(X , Λ)
Λp

Λ =
Z/ln.Z
Zp0
//
q0
Y
q
Xp
//Spec(¯
Fp)
h=p◦q0
Rh∗(ΛZ) = Rq∗Rp0
∗(q0∗(ΛX)⊗L Λp0∗(ΛY)) (1)
=Rq∗Rp0
∗q0∗(ΛX)⊗L ΛΛY
(2)
=Rq∗q∗Rp∗(ΛX)⊗L
ΛΛY(1)
=Rp∗(ΛX)⊗L
ΛRq∗(ΛY)
l
RΓ(Z, Zl)'RΓ(X, Zl)ˆ ⊗L
ΛRΓ(Y, Zl)
RΓ(X, Zl)⊗L
ΛRΓ(Y, Zl)→RΓ(X, Zl)ˆ
⊗L
ΛRΓ(Y, Zl)
RΓ(Z, Ql)'RΓ(X, Ql)ˆ ⊗L
ΛRΓ(Y, Ql)
l
X=A1
¯
Fp,Gm
Hi(X , Ql)'(Qli= 0 (i= 1, X =Gm),
0
X¯ Fpn φ :X→
X
deg([φ].[∆X]) =
2n
X
i=0
tr(φ∗, Hi(X , Ql))
CH0(X×X)
X×X

FpA
FA:A→A, f 7→ fp
A X =
Spec(A)FX
A
YFY//
f
Y
f
XFX
//X
f
f
FpX
X
FX:X→X.
FX
X
FX
X/Fp
FFpFX
FX(1X, F ]
X)F] X
OX(U)→ OX(U), f 7→ fp.
XFpK p
FXK X
FX∗:X(K)→X(K)
F∗
KK
FKFX∗FX
FpX FX
f:Y→XFp
Y(p/X):= F−1
X(Y).

Y(p)X
FY/X
Y
FY /X!!
FY
!!
f
!!
Y(p)Y×XFX//
Y×Xf
Y
f
XFX
//X.
FY/X
Y/X
Y/X F Y/X
f:An
X→X
Fp(An
X)(p)An XAn X→(An X)(p)
titp
ipn
XFpY X
Y/X
FY/X
Y/X n FY/X
pn
X n
F X
F∗:H∗(X , Ql)→H∗(X , Ql).
Fr
Nr=
2n
X
i=0
tr((Fr)∗, Hi(X , Ql))
NrFrX
X(Fpr)
dF = 0 ΓF∆XΓF×X×X
∆XFr