Site cristallin et cristaux de Dieudonné Table des matières 1

p
p
p
X k
k X
H∗(X, Ω∗)X X
X X →Y
(Ω∗
Y)∧
/X Y

X
X
U⊂X U →T
p > 0
U →T
k p > 0X  6=p
 Hi
`(X) = Hi(X, Q`)X
XW(k)W(k)→C
 Hi
`(XC) = Hi(XC,Q`)
Hi
sing(XC) = Hi(XC,Z)Hi
`(X)'Hi `(XC)
Hi
`(XC)'Hi
sing(XC)⊗Q`
X Hi
sing(XC)
Q`
6=p p
p Hi
sing(XC)X
=p
p
U →T
XH∗(X, (Ω∗
Y)∧
/X ))
k p > 0
X →Y
A B
B
P(x+h) = P(x) + P0(x)h+P00(x)h[2] +· · · +P(n)(x)h[n], x ∈B, h .
Qh[n]=hn/n!
B

{x7→ xn/n!}n>1
I⊂A I
γn:I→I n >1
γ1(x) = x γ0(x)=1
γn(λx) = λnγn(x)
γm(x)γn(x) = ((m, n)) γm+n(x)
γn(x+y) = Pn
i=0 γi(x)γn−i(y)
γm(γn(x)) = Cm,nγmn(x)
Cm,n := (mn)!
m! (n!)mmn m n
(A, I, γ)A I ⊂A
γ= (γn)n>1I
ϕ: (A, I, γ)→(B, J, δ)
ϕ:A→B ϕ(I)⊂J δn◦ϕ=ϕ◦γnn
U →T U X
(A, I, γ)B A γ
B γ IB (A, I, γ)→(B, IB, γ)
A7→ (A, 0,0)
(A, I, γ)→(B, IB, γ)
γ
γ
IB = 0
I
A→B
IB P16i6rbixibi∈B xi∈I
γ
γ(Xbixi) = X
n1+···+nr=n
bn1
1. . . bnr
rγn1(x1). . . γnr(xr).
γ
IB = 0 γn(0) = 0
I= (x)γn(bx) = bnγn(x)bx =b0x(b−b0)x= 0
bnγn(x)−(b0)nγn(x) = (bn−(b0)n)γn(x) = (bn−1+· · · + (b0)n−1)(b−b0)γn(x)=0

γn(x)∈I= (x)
B
A z ∈IB
z=Pxibiz=Px0
i0b0 i0c1, . . . , cs
A B ai,j, a0
i0,j ∈A bi=Pai,jcj
b0
i0=Pa0 i0,jcjyj=Pxiai,j =Px0 i0a0 i0,j j γ
z=P16i6rxibiIB
γn(Xxibi) = X
e1+···+er=n
be1
1. . . ber
rγe1(x1). . . γer(xr).
z z =Px0
i0b0 i0
γn(z)
X
e1+···+er=n
be1
1. . . ber
rγe1(x1). . . γer(xr) = X
d1+···+ds=n
cd1
1. . . cds
tγd1(y1). . . γds(ys).
Z[ai,j]cd1
1. . . cds
tγe1(x1). . . γer(xr)
Q[x1, . . . , xr, c1, . . . , cs, ai,j]
(x1, . . . , xr)
(A, I, γ) (B, J, δ)A→B
δ γ
γ J +IB (A, I, γ)→(B, J +IB, γ) (B, J, δ)→(B, J +IB, γ)
Σ (I, γ)SΣ
p S γ S
I
Σ = Spec(Z(p)) Σ = Spec(W(k)) I= (p)
CRIS(S/Σ,I, γ)
(U, T, i, δ)U S T Σp
i:U →T δ
OTi γ
(U0, T 0, i0, δ0)→(U, T, i, δ) (u, v)S
u:U0→UΣv:T0→T

v◦i0=i◦u
U0i0
//
u
T0
v
Ui//T
pOTU →T
(U, T, i, δ)T
Ti(u, v) CRIS(S/Σ,I, γ)
(u, v)
v i = 1 i= 2
i= 3 i= 3
CRIS(S/Σ,I, γ)
T1
Tiτ
CRIS(S/Σ,I, γ)τ(S/Σ,I, γ)CRIS,τ
(u, v):(U0, T 0, i0, δ0)→(U, T, i, δ)Ti
(U0, T 0, i0, δ0)×(U,T,i,δ)(U1, T1, i1, δ1)
CRIS(S/Σ,I, γ)U0×UU1→T0×TT1
CRIS(S/Σ,I, γ)τ
F(U,T,δ)τ T (U, T, δ)
ρ(u,v):v−1F(U,T,δ)−→ F(U0,T 0,δ0)(u, v):(U0, T 0, δ0)→(U, T, δ)
v−1
ρ(id,id) = id
ρ(uu0,vv0)=ρ(u0,v0)◦(v0)−1ρ(u,v)
ρ(u,v)(u, v)∈Ti
(S/Σ,I, γ)τOS/Σ
Γ((U, T, δ),OS/Σ) = Γ(T, OT).
IS/Σ
Γ((U, T, δ),IS/Σ) = ker(Γ(T, OT)→Γ(U, OU)).
O×
S/ΣΓ((U, T, δ),O× S/Σ) = Γ(T, OT)×