TD9 : Modules de longueur finie

A x A
A/(x)A
Z[X]/(12, X5−6) Z
Z[X]
A M A M1M2A
M
`(M1+M2) + `(M1∩M2) = `(M1) + `(M2).
A M A f
M f
f:A→B M B
`A(M)`B(M)M A B
f `A(M) = `B(M)
f
f:A→Bm
A B A
eB/A := `B(B/mB).

M A
`B(M⊗AB)≤eB/A`A(M).
B A
A M A a ∈A
M/aM M[a]A
eA(a, M) = `A(M/aM)−`A(M[a]).
M eA(a, M)
I A IM = 0 eA(a, M) =
eA/I (a, M)a a A/I
0→M→N→P→0A
eA(a, M)eA(a, N)eA(a, P )
eA(a, N) = eA(a, M) + eA(a, P ).
A M A
A M (m)
m∈MA(M) (M)
M
(M) = ∅M= 0
S A S−1A(S−1M)
S−1p p ∈A(M)
p∩S=∅
A
[
p∈(A)
p.
A(A)
(A)
M A
0 = M0⊂M1⊂... ⊂Mr=M

M i Mi/Mi+1
A/pipiA
(M)
A M A M
A M M
M
A I A
A M A
IM =M M
AmN A M
M=N+mM N M
A M A
M
Z
A M A
X=A
d:X→N
p7→ dimk(p)Mp/pMp
.
p∈X n =d(p)α:k(p)n→Mp/pMp
k(p)
f∈Arp
A[f−1]n//
γ
An
p
//
β
k(p)n
α
M[f−1]//Mp
//Mp/pMp
α β :An
p→MpAp

f /∈pβ γ :
A[f−1]n→M[f−1]A[f−1]
n I
A{p∈X|d(p)≥n}
A I
I A V (I)
A I X
V(I)X
d X