nLab 3x3 lemma

Idea

The $3 \times 3$-lemma or nine lemma is one of the basic diagram chasing lemmas in homological algebra.

Statement

Lemma

Let

$$0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$$

be a short exact sequence of chain complexes. Then if two of the three complexes $A_\bullet, B_\bullet, C_\bullet$ are exact, so is the remaining third.

Lemma

Let

$$\array{ && 0 && 0 && 0 && \\ && \downarrow && \downarrow && \downarrow && \\ 0 &\to& A' &\to& B' &\to& C' &\to& 0 \\ && \downarrow && \downarrow && \downarrow && \\ 0 &\to& A &\to& B &\to& C &\to& 0 \\ && \downarrow && \downarrow && \downarrow && \\ 0 &\to& A'' &\to& B'' &\to& C'' &\to& 0 \\ && \downarrow && \downarrow && \downarrow && \\ && 0 && 0 && 0 && }$$

be a commuting diagram in some abelian category such that each of the three columns is an exact sequence. Then

1. If the two bottom rows are exact, then so is the top.

2. If the top two rows are exact, then so is the bottom.

3. If the top and bottom rows are exact and $A \to C$ is the zero morphism, then also the middle row is exact.

A proof by way of the salamander lemma is spelled out in detail at Salamander lemma - Implications - 3x3 lemma.

Related concepts

• salamander lemma

• snake lemma, 5-lemma

• horseshoe lemma

References

In abelian categories

An early appearance of the $3 \times 3$-lemma is as lemma (5.5) in

• D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)

In

• Charles Weibel, An introduction to homological algebra

it appears as exercise 1.3.2.

The sharp $3 \times 3$-lemma appears as lemma 2 in

• Temple Fay, Keith Hardie, Peter Hilton, The two-square lemma, Publicacions Matemàtiques, Vol 33 (1989) (pdf)

Also lemma 3.2-3.4 of

• Saunders MacLane, Homology, Grundlehren der math. Wissenshaften vol 114, Springer (1995)

In non-abelian categories

Discussion of generalization to non-abelian categories is in

• Marino Gran, Diana Rodelo, Goursat categories and the $3 \times 3$-lemma, Applied Categorical Structures, Vol. 20, No 3, 2012, 229-238. (journal, pdf slides)

• Marino Gran, Zurab Janelidze and Diana Rodelo, $3 \times 3$ lemma for star-exact sequences, Homology, Homotopy and Applications, Vol. 14 (2012), No. 2, pp.1-22. (journal)

• Dominique Bourn, $3 \times 3$-lemma and protomodularity, Journal of Algebra, Volume 236, Number 2, 15 February 2001 , pp. 778-795(18)

• Dominique Bourn, The denormalized $3 \times 3$ lemma, Journal of Pure and Applied Algebra, Volume 177, Issue 2, 24 January 2003, Pages 113-129, doi:10.1016/S0022-4049(02)00143-3