nLab
four lemma

Idea

The four lemma is one of the basic diagram chasing lemmas in homological algebra. It follows directly from the salamander lemma. It directly implies the five lemma.

Statement

Let $𝒜$ be an abelian category.

Proposition

Consider a commuting diagram in $𝒜$ of the form

$$\begin{array}{ccccccc}& \to & & \stackrel{\xi }{\to }& & \to & \\ {\downarrow }^{\tau }& & {\downarrow }^f& & {\downarrow }^g& & {\downarrow }^{\nu }\\ & \to & & \stackrel{\eta }{\to }& & \to & \end{array}$$\array{ &\to& &\stackrel{\xi}{\to}& &\to& \\ \downarrow^{\mathrlap{\tau}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{\nu}} \\ &\to& &\stackrel{\eta}{\to}& &\to& }

where

1. the rows are exact sequences,

2. $\tau$ is an epimorphism,

3. $\nu$ is a monomorphism.

Then

1. $\xi (\ker (f))=\ker (g)$ (the image under $\xi$ of the kernel of $f$ is the kernel of $g$)

2. $\mathrm{im}(f)={\eta }^{-1}(\mathrm{im}(g))$ (the preimage under $\eta$ of the image of $g$ is the image of $f$)

(the “strong four lemma”) and hence in particular

1. if $g$ is an epimorphism then so is $f$;

2. if $f$ is a monomorphism then so is $g$

(the “weak four lemma”).

A direct proof from the salamander lemma is spelled out at salamander lemma – implications – four lemma.

References

The strong/weak four lemma appears as lemma 3.2, 3.3 in chapter I and then with proof in lemma 3.1 of chapter XII of

• Saunders MacLane, Homology (1967) reprinted as Classics in Mathematics, Springer (1995)