Snake Lemma


A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequence


This commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map S is called a connecting homomorphism and describes a curve from the end of the upper row (Ker(gamma) subset= C) to the beginning of the lower row (coker(alpha)=A^'/Im(alpha)), which suggested the name given to this lemma.

The snake lemma is explained in the first scene of Claudia Weill's film It is My Turn (1980), starring Jill Clayburgh and Michael Douglas.

See also

Commutative Diagram, Connecting Homomorphism, Diagram Lemma, Exact Sequence

Portions of this entry contributed by Margherita Barile

Portions of this entry contributed by Christopher Stover

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Bourbaki, N. "Le diagramme du serpent." §1.2 in Algèbre, Ch. 10: Algèbre Homologique. Paris, France: Masson, pp. 3-7, 1980.The Internet Movie Database. "Memorable Quotes from It is My Turn.", 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

Snake Lemma

Cite this as:

Barile, Margherita; Stover, Christopher; and Weisstein, Eric W. "Snake Lemma." From MathWorld--A Wolfram Web Resource.

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