Commutative algebra - Department of Mathematical Sciences - old ...

3.2. THE SNAKE LEMMA 43
3.1.16. Example. Let Zi is the family of modules each a copy of Z indexed by the
natural numbers. Then the short exact sequence
0
i Zi
i Zi
is not split exact.
The element f = (1, 2, 22 , . . . , 2n , . . . ) +
fk = (0, . . . , 0, 2 n−k , . . . ) +
But in
i Zi/ i Zi
0
i Zi is divisible by 2 k for all k. If
i Zi for n ≥ k, then 2 k fk = f in
i Zi the only element divisible with all 2 k is 0, so no section exists.
3.1.17. Exercise. (1) Show that the sequence
0
Z
is short exact, but not split exact.
(2) Show that the sequence
0
Q
n
Z
Z
is exact, but not split exact for n = 0, 1.
(3) Show that the sequence
is exact, but not split exact.
(4) Show that the sequence
is split exact.
0
0
Z/(2) 1↦→2
Z/(4)
Z/(2) 1↦→3
Z/(6)
Q/Z
Z/(n)
3.2. The snake lemma
Z/(2)
Z/(3)
3.2.1. Example. Given a commutative diagram of homomorphisms
M
u
M ′
there is induced a commutative diagram
0
0
Ker f
u
Ker f ′
M
u
M ′
where the rows are exact sequences.
The diagram splits into two diagrams
0
0
Ker f
u
Ker f ′
f
f ′
f
f ′
M
u
M ′
N
v
N ′
N
0
0
Cok f
0
0
v
v
N ′ Cok f ′
f
Im f
f
v
′
Im f ′
0
0
i
0
0
Zi/
i Zi.