Anneaux factoriels, anneaux noethériens (TD7)

k k[X, Y ]XnY n ≥1
k
k[T2, T 3]k[T]
k k[X, Y ]/(X2−Y3)k[T2, T 3]
k k[X, Y ]/(X2−Y)k[T]
k k[X, Y ]/(XY −1) k[T]
x P ∈Z[X]
P(x) = 0 Q Q C Z
Z Q
Z
A={P∈Q[X]|P(0) ∈Z}
A
A
A
A A
A
x, y A
u, v ∈A ux +vy = 1
I A d ∈A
I
A
A
I A
PiP1P2· · · PnI⊆I
I⊆Pi1≤i≤nI
A P A P

A f :M0→M g :P→M
h:P→M0g=f◦h
A A
A A
A
A
R1R2R=
R1⊕R2R1R2
R1R2R
ΛR→R2π M Λ
f:R→Rf(x+ 2π) = −f(x)x∈R
Λ
Λ⊕Λ∼ −→ M⊕M
(f, g)7→ fcos x
2+gsin x 2, f sin x 2−gcos x 2 .
MΛ
A X M A MX
A X M
M n m M
Am→An→M→0.
A M A M
M AX⊗AM→MX
A
AXA
AX⊗AM→MX
A MXX A
M