The nine lemma is a diagram lemma also known as the 
lemma. According to its most
general statement, the commutative diagram
illustrated above whose rows and columns are exact
sequences can be completed by two morphisms
without losing commutativity.
Moreover, the sequence
is a short exact sequence.
The lemma is also true if the roles of the first and the third row are interchanged.
See also
Commutative Diagram,
Exact
Sequence
This entry contributed by Margherita
Barile
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References
Mac Lane, S. Homology. Berlin, Germany: Springer-Verlag, pp. 49-50, 1967.Mitchell, B.
Theory
of Categories. New York: Academic Press, pp. 20-21, 1965.Referenced
on Wolfram|Alpha
Nine Lemma
Cite this as:
Weisstein, Eric W., with contributions by Margherita Barile. "Nine Lemma." From MathWorld--A Wolfram
Resource. https://mathworld.wolfram.com/NineLemma.html
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