Diagram lemmas
We can use the The Salamander Lemma (and the various mural maps) to quickly prove many of the named "diagram lemmas" one encounters in homological algebra (and elsewhere). This is just a sampling.
The Four Lemma
Consider a commutative diagram (in some abelian category) of the form below, where the rows are exact,
Then:
In particular, if
By the given assumptions, we can extend the given diagram to a double complex below, in which all columns are exact and the middle two rows are exact:
For the first statement it is enough to show that
(Alternatively, one can show this directly from the definition of the donor.)
Then the long zig-zag of extramural isomorphisms (highlighted in blue) shows that this is isomorphic to
The second statement is proven analogously.
The Five Lemma
Consider a commutative diagram in
Then:
- If
and are epimorphisms and is a monomorphism, then is an epimorphism. - If
and are monomorphisms and is an epimorphism, then is a monomorphism.
Note that as an immediate corollary, if
To see how this result follows from the Four Lemma, let's look at the proof of the first statement. Consider the following part of the diagram:
The Four Lemma applies, since the rows are exact,
An analogous argument proves the second statement.
The Snake Lemma
Here's a famous result:
If both rows are exact in a commutative diagram of the form
then there is a morphism
The morphism
To prove this lemma, complete the given diagram to a double complex:
By assumption, all of the columns are exact, the rows are exact at the
We first prove exactness at
As for the connecting morphism
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