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)

REFERENCES:

Bourbaki, N. "Le diagramme du serpent." §1.2 in Algèbre. Chap. 10, Algèbre Homologique. Paris, France: Masson, 3-7, 1980.

Lang, S. Algebra, rev. 3rd ed. New York: Springer Verlag, pp. 158-159, 2002.

Mac Lane, S. Categories for the Working Mathematician. New York: Springer Verlag, pp. 202-204, 1971.

Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub.,p. 141, 1993.