Corrigé du TD8

C=Z[X]/(2X)
Z/2Z⊗ZZ/3Z
Z/nZ⊗ZZ/mZ
Q/Z⊗ZQ/Z
Q⊗ZQ/Z
Q⊗ZQ Q
Q⊗QQ Q
M⊗AA/I A A I
A M A
A[X1, ..., Xn]⊗AA[X1, ..., Xm]A A
A[X1, ..., Xn]/I ⊗AB B A
B A I A[X1, ..., Xn]
Z[i]⊗ZR
Q(√2) ⊗QR Q
Q(3
√2) ⊗QR Q
(1,3
√2,3 √4) Q
Q(i)⊗QQ(i)Q
A M A (Ni)i∈IA
M⊗A(LiNi)∼
=Li(M⊗ANi)

f:M×(M
i
Ni)→M
i
(M⊗ANi)
m∈M n ∈Ni(m, n)m⊗n∈M⊗ANi
f:M⊗A(LiNi)→Li(M⊗ANi)
Ni→LiNiM⊗A
Ni→M⊗A(LiNi)i
g:Li(M⊗ANi)→M⊗A(LiNi)
f g
M⊗A(QiNi)∼
=Qi(M⊗ANi)
A=ZM=QI=NNi=Z/2iZ
i∈IQi(M⊗ANi) = 0 n
(1,1,1, ...)QiNi1⊗n
M⊗A(QiNi) (Z\ {0})−1(QiNi)n
1
(Z\ {0})−1(QiNi)s∈Z\ {0}sn = 0
QiNin
1(Z\ {0})−1(QiNi) 1 ⊗n
M⊗A(QiNi)
J(Pj)j∈J(Qj)j∈J
A
(lim
−→
j
Pj)⊗A(lim
−→
j
Qj)∼
=lim
−→
j
(Pj⊗AQj)?
fk:Pk→lim
−→jPjgk:Qk→lim −→jQj
Pk×Qk→(lim
−→
j
Pj)⊗A(lim
−→
j
Qj)
(p, q)7→ fk(p)⊗gk(q)
Pk⊗AQk→(lim
−→
j
Pj)⊗A(lim
−→
j
Qj).
φ: lim
−→
j
(Pj⊗AQj)→(lim
−→
j
Pj)⊗A(lim
−→
j
Qj).
j∈J
Pj⊕Qj→Pj⊗AQj,(p, q)7→ p⊗q.

Pj⊗AQj→lim
−→j(Pj⊗AQj)
lim
−→
j
(Pj⊕Qj)→lim
−→
j
(Pj⊗AQj).
lim
−→j(Pj⊕Qj) (lim −→jPj)×(lim −→jQj)
(lim
−→
j
Pj)×(lim
−→
j
Qj)→lim
−→
j
(Pj⊗AQj).
ψ: (lim
−→
j
Pj)⊗A(lim
−→
j
Qj)→lim
−→
j
(Pj⊗AQj).
φ ψ
A=Z[X]I= (2, X)
2⊗X−X⊗2I⊗AI
2⊗X−X⊗2
X I ⊗AI
A I ⊗AI2⊗X−X⊗2
A/I
A M, N, P, Q A
A
M→N→P→0.
M⊗AQ→N⊗AQ→P⊗AQ→0.
−⊗AQ:A− → A−
M→N M ⊗AQ→
N⊗AQ
A Q −⊗AQ
0→M→N→P→0
0→M⊗AQ→N⊗AQ→P⊗AQ→0.

A A
A
A A
B A M A M ⊗AB
B
B A M B M
A
M A M
I A I ⊗AM→M
I→A
A M A M
A M A M
M
A=C[X, Y ]m= (X, Y )m
A
B=C[T2, T 3]n= (T2, T 3)
nB
A S A M A
S−1A S−1M M ⊗AS−1A
S−1A A
A K M A
M K
M⊗AK M
0→M→N→P→0
A N =M+P
K= (A\ {0})−1A
K A A
0→M→N→P→0
K
0→M⊗AK→N⊗AK→P⊗AK→0.
N=M+P

M A
MpAppA
MmAmmA
A
Ap
pA M M
M A
Am
0→M→N→P→0A
P A A
Q
0→M⊗AQ→N⊗AQ→P⊗AQ→0
M A M =
mM M = 0
M A
M M A