Nine lemma

In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well.

The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).

There are two variants of nine lemma: sharp nine lemma and symmetric nine lemma (see Lemmas 3.3, 3.4 in Chapter XII of [1]).