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As in the proof of the 5-lemma, we assume without loss of generality that we are working in modules over a ring. In keeping with the notion that the maps between the 's (as well as between the 's and the 's) are cohomology sequences, we denote all vertical maps by . The map
is denoted , and the map
is denoted . We must show that
is surjective;
is injective;
-
;
-
(i.e.
)
is surjective: Choose . Then
, and
, so
. Thus
, so
. Finally,
. But is injective, so
.
is injective: This is clear, since
, and and both 's are injective.
: Suppose
. Then
, so
. But then
, and is injective, so
and . Finally,
. is injective and thus
.
:
. But is injective, so
.
Similar diagram chasing can be used to prove that if the top two rows are exact then so is the bottom row.
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