feuille 1

1 1
A A
A A[X]
A A
A A nil(A)√0
A
nil(A) = 0 A Ar´ed =A/ nil(A)
X= Spec(A)Ar´ed
X X
A X = Spec(A)
Y
Y=Y1∪ ··· ∪ YnYiYi6⊂ Yj
i6=j YiY
Y
k Y ⊂A3
ky2=xz z2=y3
Y
f=y/x Y x 6= 0 f
Y
X
X
X Y ⊂XY
A X = Spec(A)
X
A A1×A2A16= 0 A26= 0
A e ∈A e2=e e = 0 e= 1
⇒ ⇒ ⇒
Y⊂A3
kx2−yz = 0 xz −x= 0 Y
O(Y)Y

A S ⊂A1∈S
x, y ∈S⇒xy ∈S S−1A ASf:A→S−1A
g:A→B g(s)
s∈Seg:S−1A→B g =egf
S−1A A S S
A→S−1A
S−1A A A S−1A
Spec(S−1A) = {p∈Spec(A), p ∩S=∅}
A I ⊂A M A M =IM
a∈A a ≡1 mod I aM = 0
A m M A M =mM M = 0
k=A/m M =mM M ⊗Ak= 0
A M A
d: Spec(A)→Np⊂A d(p) = dimk(p)(M⊗Ak(p))
p∈Spec(A)n=d(p)
U=D(f)⊂Spec(A)p d|U≤n
ϕ:k(p)n→Mp⊗Apk(p)
f6∈ p App
Mp=M⊗AAp
α, β, γ, δ
ϕ
Apψ: (Ap)n→Mp
A[1
f]n
α
χ//M[1
f]
γ
(Ap)n
β
ψ//Mp
δ
k(p)nϕ//M⊗k(p)
f ψ χ:A[1
f]n→M[1 f]
f χ
k A1k[X1, . . . , Xn]
f∈k[X1, . . . , Xn]An
kA2=
k[[x]] A3
Zp
A1A2A3
A K k
A M
M⊗AK= 0 M⊗AK= 0
M
n∈N{p∈Spec(A) ; d(p)≥n}