# nLab four lemma

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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

$\begin{array}{ccccccc}& \to & & \stackrel{\xi }{\to }& & \to & \\ {↓}^{\tau }& & {↓}^{f}& & {↓}^{g}& & {↓}^{\nu }\\ & \to & & \stackrel{\eta }{\to }& & \to & \end{array}$\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. $\xi \left(\mathrm{ker}\left(f\right)\right)=\mathrm{ker}\left(g\right)$ (the image under $\xi$ of the kernel of $f$ is the kernel of $g$)

2. $\mathrm{im}\left(f\right)={\eta }^{-1}\left(\mathrm{im}\left(g\right)\right)$ (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)

Revised on September 24, 2012 23:46:27 by Urs Schreiber (82.169.65.155)