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Nine Lemma

NineLemma

A diagram lemma also known as 3×3 lemma. According to its most general statement, the commutative diagram illustrated above with exact rows and columns can be completed by two morphisms

A-->^alphaA^' A^'-->^(alpha^')A^('')

without losing commutativity.

Moreover, the short exact sequence

0-->A-->^alphaA^'-->^(alpha^')A^('')-->0

is exact.

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

Mitchell, B. Theory of Categories. New York: Academic Press, pp. 20-21, 1965.Mac Lane, S. Homology. Berlin: Springer-Verlag, pp. 49-50, 1967.

Referenced on Wolfram|Alpha

Nine Lemma

Cite this as:

Barile, Margherita. "Nine Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NineLemma.html

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