Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
Commutative algebra - Department of Mathematical Sciences - old ...
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46 3. EXACT SEQUENCES OF MODULES<br />
Pro<strong>of</strong>. The commutative diagram<br />
0<br />
0<br />
<br />
N ∩ L<br />
<br />
N ⊕ L<br />
<br />
N + L<br />
<br />
x↦→(x,x) <br />
(x,y)↦→x−y <br />
<br />
M <br />
M ⊕ M <br />
M<br />
where the rows are short exact sequences, gives by 3.2.4 a five term long exact<br />
sequence<br />
0<br />
<br />
M/N ∩ L<br />
<br />
M/N ⊕ M/L<br />
<br />
M/N + L<br />
3.2.8. Proposition (five lemma). Given a commutative diagram <strong>of</strong> homomorphisms<br />
M1<br />
u1<br />
<br />
M ′ 1<br />
<br />
M2<br />
u2<br />
<br />
<br />
M ′<br />
2<br />
<br />
M3<br />
u3<br />
<br />
<br />
M ′<br />
3<br />
<br />
M4<br />
u4<br />
<br />
<br />
M ′<br />
4<br />
<br />
M5<br />
u5<br />
<br />
<br />
M ′<br />
5<br />
where the rows are exact sequences. If u1 is surjective, u2, u4 are isomorphism<br />
and u5 is injective, then u3 is an isomorphism.<br />
Pro<strong>of</strong>. Let fi : Mi → Mi+1, f ′ i : M ′ i → M ′ i+1 . Split the given diagram in three as<br />
follows<br />
0<br />
0<br />
0<br />
0<br />
0<br />
M1<br />
u1<br />
<br />
<br />
Ker f ′<br />
2<br />
<br />
Cok f2<br />
u ′′<br />
3<br />
<br />
<br />
Ker f ′<br />
4<br />
<br />
Im f2<br />
u ′ 3<br />
<br />
<br />
Im f ′<br />
2<br />
<br />
M2<br />
u2<br />
<br />
<br />
M ′<br />
2<br />
<br />
M4<br />
u4<br />
<br />
<br />
M ′<br />
4<br />
<br />
M3<br />
u3<br />
<br />
<br />
M ′<br />
3<br />
<br />
Cok f1<br />
u ′ 3<br />
<br />
<br />
Im f ′<br />
2<br />
<br />
Im f4<br />
u5<br />
<br />
<br />
M ′<br />
5<br />
<br />
Cok f2<br />
u ′′<br />
3<br />
<br />
<br />
Cok f ′<br />
2<br />
Note that Cok f1 Im f2 and Cok f ′ 2 Ker f ′ 4 . Now use 2.3.3 and the snake<br />
lemma to conclude that Ker u3 = 0 and Cok u3 = 0 and u3 is therefore an isomorphism.<br />
3.2.9. Proposition (windmill lemma). Given homomorphism M f<br />
There is induced an eight term long exact sequence<br />
0<br />
<br />
Ker f<br />
<br />
Ker g ◦ f<br />
<br />
<br />
Cok f<br />
f<br />
g<br />
<br />
Ker g<br />
<br />
<br />
Cok g ◦ f<br />
<br />
0<br />
<br />
0<br />
<br />
0<br />
<br />
0<br />
<br />
0<br />
<br />
0<br />
<br />
0<br />
<br />
Cok g<br />
<br />
0<br />
g<br />
<br />
N<br />
<br />
0<br />
<br />
L .
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