Introduction à la Géométrie Algébrique. Le langage des schémas

f:X→S
S X
An
SS
f:X→S S
Sif−1(Si)i
f:X→S S OSA(X) = f∗OX
AOSSSpec(A)
{OS}Spec //{S}
A
oo
f:X→S S A(X)
S Si= Spec(Ri)
Xi=f−1(Si)fi:Xi→Sif f∗OX
A(X)|Si=fi,∗OXiA
OSf:X= Spec(A)→S
OSU⊂S f−1(U) = Spec(A(U))
f:X→S f
S X

f:X→S
S Sjf−1(Sj)j
f:X→S
f(X)
f(X)
⇒ ⇒ f(X)∩U
U U S f(X)∩U
U S U S f
X Xi= Spec(Ai)f
X1q ··· q Xn→X→S X1q ··· q Xn
A1× ··· × AnX
S= Spec(R)X= Spec(A)s= [p]
T:=f(X)D(g)∩T6=∅g∈R g 6∈ p D(g)∩T
f Xg=D(g1A) = Spec(Ag)Ag16= 0
AgApA R rp Ap= lim
−→g6∈pAg
16= 0 ApSpec(Ap)
Spec(Ap)→Spec(A)→Spec(R)p0p s = [p]
s0= [p0]∈T T s ∈T
f:X→S f
U= Spec(R)S V = Spec(A)f−1(U)
R→A A R f

f:X→S
S Sj= Spec(Rj)X
Xi,j = Spec(Ai,j)f(Xi,j)⊂SjAi,j Rj
f:X→S
U= Spec(R)S V =f−1(U)V= Spec(A)
R→A A R
A R
R R A R
f:X→S R
f
R A
R→κSpec(A⊗Rκ)x1, . . . , xr
A R A ⊗Rκ κ
A⊗Rκ
An
k
f:X→S
S0→S f0:X0=X×SS0→S0
f:X→S

R→A p ⊂R
I⊂A I ∩R⊂p q ⊂A q ∩R=p
R→A x ∈S
R R
f:X→S Y ⊂X T =f(Y)
T S S
U= Spec(R)X f−1(U) = Spec(A)
f Y =V(I)I f(Y) = V(I∩R)
p⊂R p =q∩R
q⊂A I I ∩R
f f0:X0→S0f S0→S
f
f:X→S
U= Spec(R)S V =f−1(U)V= Spec(A)
R→A A R 
A
x K x ∈A x−1∈A
A
A
A
A > 0
A1

k[X](X)k[[X]] kZ(p)ZpA⊂B
B A mB∩A=mA
mAB⊂mBA →B
Spec(B)→Spec(A)
A0K0K/K0
A
K A0
A K A ⊂B(K
K A B =A
x∈B K = Frac(A)x=r/s
r, s ∈A A rA ⊂sA
sA ⊂rA t ∈A r =st x =r/s =t∈A
t∈A s =rt t 6∈ mA1 = rt/s =tx ∈mAB⊂mB
A t ∈A×x= 1/t ∈A
S
S f :X→S
f
A K
Spec(K)// X
f
Spec(A)//
∃!
;;
S,
Spec(A)→X
Spec(K)→Spec(A)A⊂K
X η
K:=OX,η =κ(η)OX(U)
OX,x K K
X