Four Lemma


A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds:

1. If alpha is surjective, and beta and delta are injective, then gamma is injective;

2. If delta is injective, and alpha and gamma are surjective, then beta is surjective.

This lemma is closely related to the five lemma, which is based on a similar diagram obtained by adding a single column.

See also

Commutative Diagram, Diagram Lemma, Exact Sequence, Five Lemma

This entry contributed by Margherita Barile

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Mac Lane, S. Homology. Berlin: Springer-Verlag, p. 364, 1967.

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Four Lemma

Cite this as:

Barile, Margherita. "Four Lemma." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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