Connecting Homomorphism

The homomorphism which, according to the snake lemma, permits construction of an exact sequence

(1)

from the above commutative diagram with exact rows. The homomorphism is defined by

(2)

for all , denotes the image, and is obtained through the following construction, based on diagram chasing.

1. Exploit the surjectivity of to find such that .

2. Since because of the commutativity of the right square, belongs to , which is equal to due to the exactness of the lower row at . This allows us to find such that .

While the elements and are not uniquely determined, the coset is, as can be proven by using more diagram chasing. In particular, if and are other elements fulfilling the requirements of steps (1) and (2), then and , and

(3)

hence because of the exactness of the upper row at . Let be such that

(4)

Then

(5)

because the left square is commutative. Since is injective, it follows that

(6)

and so

(7)