TD4 : Anneaux noethériens et modules

A
A Ann
A An
n
M Mtor
M Mtor M
A A
Z
0→M1→M2→M3→0A
M1M3M2
α∈C Z[α]
A A A
A
I A A/I A
Ann
AnAn
An
Z
Z

0→M1→M2→M3→0A
M1M3M2(M1) +
(M3)
A={P∈Q[X]|P(0) ∈Z}
A
A
A
A
x
P∈Z[X]P(x)=0 Z
Z
Z
Z
Z
X X
XR
A A[[X]]
A
I A
pip1p2· · · pnI⊆I
I⊆pi1≤i≤nI
A
A up1...pnu∈A×pii
A

A I A a A
I+ (a){x∈A|ax ∈I}A I
A
A M A n ∈N
A u :M→An(u)
A
C×
A M N P A
A i :M→N
A p :N→P p ◦i= 0 N
M P (p)/(i)
A M A u :M→M
A n ∈N(un)∩
(un)=0 u u
M
A M A M[X]A[X]
M A
A f :M→N g :N→P
A
0→(f)→(g◦f)→(g)→(f)→(g◦f)→(g)→0.
A
M1//
f1
M2//
f2
M3//
f3
M4//
f4
M5
f5
N1//N2//N3//N4//N5.

f1, f2, f4, f5
f3
R n f :Rn→Rn
R A f
Rn
f
f
(A)∈R×
f
(A)R
R
M R m1, ..., mn
M f M
A= (aij )∈ Mn(R)f(mi) = Pn
j=1 aij mj1≤i≤n
P(X) = det(XIn−A)P(f) = 0
M R
I R IM =M a ∈I
(1 −a)M= 0
M
m n g :Rm→Rn
R m ≤n
A
M N P A 0→
M→N→P→0N∼
=M⊕P
M N P A
•N∼
=M⊕P
•A0//Mf//Ng//P//0
A s :P→N g ◦s=P
•A0//Mf//Ng//P//0
A t :N→M t ◦f=M

A P
M N A
0→M→N→P→0N∼
=M⊕P
A
A
A A
A=ZA
A
A=B×C B C
B A
A2π
R R
P A R R
f(x+ 2π) = −f(x)x∈R
A P ⊕P∼
=A⊕A P
N A f R
Rn≥0
lim
x→+∞xnf(x) = lim x→−∞ xnf(x)=0
A N P
A M
N P A
0→M→N→P→0N∼
=M⊕P
A M
I A A I →M
A A →M
A A
Z Z n > 0
Z/nZ Z/nZ
A=Z
M
m∈M n ∈N∗
m0∈M m =nm0
R/Z
R/Q