Eléments de géométrie algébrique

C[E]
ECnC[E]'C[X1, ..., Xn]
E
E
V={z= (z1, ..., zn)∈E, Pi(z) = 0 ∀i∈I},
(Pi)iC[E]F⊂C[E]
V(F)F
(Pi)iI⊂C[E]
V((Pi)i)V(I)
I⊂C[E]I
√I={P∈C[E],∃n∈N∗, P n∈I}.
IC[E]P∈C[E]P|V(I)≡0
P∈√I
V(I)
V

C[E]
EC[E]
radicaux(C[E]) →var(E)
I7→ V(I).
V
V7→ I(V),
I(V) = {P∈C[E], P|V≡0}
I(V(I)) = sqrt(I)
Y⊂E
VI(V)
P Q ∈I(V)V⊂V(P Q) = V(P)∪V(Q)
V= (V∩V(P)) ∪(V∩V(Q)) ,
V V
V=V∩V(P)V⊂V(P)P∈I(V)
IV(I) = V1∪V2I=I(V1)∩I(V2)
I=I(V1)I=I(V2)V(I) = V1V2
C[E]
EC[E]
nil(C[E]) →var(E)
C[E]/I 7→ V(I).
C[E]