nLab
four lemma

Context

Diagram chasing lemmas

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

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\mathcal{A} be an abelian category.

Proposition

Consider a commuting diagram in 𝒜\mathcal{A} of the form

ξ τ f g ν η \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. τ\tau is an epimorphism,

  3. ν\nu is a monomorphism.

Then

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

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

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

  1. if gg is an epimorphism then so is ff;

  2. if ff is a monomorphism then so is gg

(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)