long exact sequence of homology

Let be complexes that fit into a short exact sequence of complexes

There exists a long exact sequence in homology

Proof

Most of the maps are exact by functoriality of homology, the only ones we need to construct are the connecting homomorphsims .

Consider

Let satisfy so that . By exactness of the jth row, there exists such that . Then

that is, . By exactness of the row, there exists a unique such that . Note that

So we may define

Now we must check that this is well-defined: First, we check that the construction is independent of the choice of . Say also satisfies . Then

that is,

for some . Then

that is . Then , so it is independent of the choice of .

This gives a well defined map

Now we must check that is in the kernel.

Say for some . By exactness of the row, there exists such that

Then

that is, we may take . The unique which satisfies

is . Therefore, this descends to the map

This map makes the above diagram exact. #todo Prove the exactness.