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Nine lemma (3×3 lemma)

In a commutative 3×3 diagram in an abelian category, exact columns and two exact rows force the remaining row to be exact.

Let A\mathcal A be an . Consider a commutative 3×33\times 3 diagram

0→A′→A→A′′→0↓↓↓0→B′→B→B′′→0↓↓↓0→C′→C→C′′→0\begin{array}{ccccccccc} 0 &\to& A' &\xrightarrow{}& A &\xrightarrow{}& A'' &\to& 0\\ && \downarrow && \downarrow && \downarrow && \\ 0 &\to& B' &\xrightarrow{}& B &\xrightarrow{}& B'' &\to& 0\\ && \downarrow && \downarrow && \downarrow && \\ 0 &\to& C' &\xrightarrow{}& C &\xrightarrow{}& C'' &\to& 0 \end{array}

If the three columns and the first two rows are , then the third row

0→C′→C→C′′→00\to C' \to C \to C'' \to 0

is exact. Dually, exact rows together with two exact columns force the remaining column to be exact.

Remarks

The lemma can be proved by a diagram chase using the , or by a kernel–cokernel argument in the abelian category.