Connecting Homomorphism
The homomorphism S which, according to the snake lemma, permits construction of an exact sequence
| Ker(alpha)-->Ker(beta)-->Ker(gamma)-->^Scoker(alpha)-->coker(beta)-->coker(gamma) |
(1)
|
from the above commutative diagram with exact rows. The homomorphism S is defined by
| S(c)=a^'+Im(alpha) |
(2)
|
for all c in Ker(gamma), Im denotes the image, and a^' is obtained through the following construction, based on diagram chasing.
1. Exploit the surjectivity of g to find b in B such that c=g(b).
2. Since 0=gamma(c)=gamma(g(b))=g^'(beta(b)) because of the commutativity of the right square, beta(b) belongs to Ker(g^'), which is equal to Im(f^') due to the exactness of the lower row at B^'. This allows us to find a^' in A^' such that beta(b)=f^'(a^').
While the elements b and a^' are not uniquely determined, the coset a^'+Im(alpha) is, as can be proven by using more diagram chasing. In particular, if b^_ and a^_^' are other elements fulfilling the requirements of steps (1) and (2), then c=g(b^_) and beta(b^_)=f^'(a^_^'), and
| 0=c-c=g(b)-g(b^_)=g(b-b^_), |
(3)
|
hence b-b^_ in Ker(g)=Im(f) because of the exactness of the upper row at B. Let a in A be such that
| b-b^_=f(a). |
(4)
|
Then
| f^'(a^'-a^_^')=f^'(a^')-f^'(a^_^')=beta(b)-beta(b^_)=beta(b-b^_)=beta(f(a))=f^'(alpha(a)), |
(5)
|
because the left square is commutative. Since f^' is injective, it follows that
| a^'-a^_^'=alpha(a) in Im(alpha), |
(6)
|
and so
| a^'+Im(alpha)=a^_^'+Im(alpha). |
(7)
|
See also
Cokernel, Commutative Diagram, Diagram Chasing, Exact Sequence, Group Kernel, Snake LemmaThis entry contributed by Margherita Barile
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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.Referenced on Wolfram|Alpha
Connecting HomomorphismCite this as:
Barile, Margherita. "Connecting Homomorphism." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ConnectingHomomorphism.html