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five lemma

Idea

The five lemma is one of the basic lemmas of homological algebra, useful for example in the construction of the connecting homomorphism in the homology long exact sequence.

Statement

Let $𝒜$ be an abelian category. Consider a commutative diagram in $𝒜$ of the form

$$\begin{array}{ccccccccc}{A}_1& \to & A_2& \to & A_3& \to & A_4& \to & A_5\\ \downarrow f_1& & \downarrow f_2& & \downarrow f_3& & \downarrow f_4& & \downarrow f_5\\ B_1& \to & B_2& \to & B_3& \to & B_4& \to & B_5\end{array}$$\array{ A_1 & \to & A_2 & \to & A_3 & \to & A_4 &\to & A_5\\ \downarrow f_1 &&\downarrow f_2 &&\downarrow f_3 &&\downarrow f_4 &&\downarrow f_5 \\ B_1 & \to & B_2 & \to & B_3 & \to & B_4 &\to & B_5 }

where the top and bottom rows are exact sequences. For simplicity we denote all the differentials in both exact sequences by $d$.

Proposition (the lemma on five homomorphisms or the five lemma)

1. sharp five lemma (essentially the weak four lemma)

   1. If $f_2$ and $f_4$ are epi and $f_5$ is mono, then $f_3$ is epi.

   2. If $f_2$ and $f_4$ are mono and $f_1$ is epi, then $f_3$ is mono.

2. (weak) five lemma (conjunction of the two statements above)

   If $f_2$ and $f_4$ are isos, $f_1$ is epi, and $f_5$ is mono, then $f_3$ is iso.

Remark

on terminology

The weak four lemma is another terminology (cf. MacLane, Homology) for the same as 1.1 and 1.2 except that in 1.1 $f_1$ is not required to exist, and in 1.2 $f_5$ is not required to exist (see four lemma), where the dropped requirements are inessential as not used in the proof.

Proof

The four lemma follows immediately from the salamander lemma, as discussed at salamander lemma - impliciations - four lemma. Here is direct proof.

By the Freyd-Mitchell embedding theorem we can always assume that the abelian category is $R$Mod (though this requires the category to be small, one can always take a smaller abelian subcategory containing the morphism in the diagram which is small). Then we can do the diagram chasing using elements in that setup. We prove only 1) as 2) is dual.

Suppose $b\in B_3$. Since $f_4$ is epi, one can choose an element $a_4\in A_4$ such that $f_4(a_4)=d(b)$. Now $0=d^{2}b=d{f}_4(a_4)=f_{5}d(a_4)$. Since $f_5$ is a monomorphism that means that $d{a}_4=0$ as well. By the exactness of the upper row, that means there is $a_3\in A_3$ such that $d{a}_3=a_4$, hence also $d{f}_3(a_3)=f_{4}d(a_3)=f_4(a_4)=db$. We would like that $f_3(a_3)$ be equal to $b$ but this is not so, we just see that $d(b-f_3(a_3))=0$ and hence by exactness of the lower row there is $b\prime \in B_2$ such that $db\prime =b-f_3(a_3)$. Since $f_2$ is also epi, there is $a_2\in A_2$ such that $f_2(a_2)=b\prime$. Now $d{a}_2+a_3\in A_3$ is such that

$$f_3(d{a}_2+a_3)=d{f}_2(a_2)+f_3(a_3)=db\prime +f_3(a_3)=b-f_3(a_3)+f_3(a_3)=b$$f_3 (d a_2 + a_3) = d f_2(a_2) + f_3(a_3) = d b' + f_3(a_3) = b - f_3(a_3) + f_3(a_3) = b

demonstrating that $b$ is in the image of $f_3$.

Hence $f_3$ is an epimorphism.

Remark

The five lemma also holds in the category Grp of groups, by essentially the same diagram-chasing proof.

Remark

One can avoid appealing to the Freyd-Mitchell embedding theorem if one works with generalized elements or uses the device of interpreting regular logic in the given abelian category. The former requires a bit of manual reformulation, while the latter is almost automatic, as the element-based proof given above only uses (constructive) regular reasoning.

Immediate consequences

Short five lemma

Corollary

(short five lemma)

Let $A\to B\to C$ and $A\to \tilde{B}\to C$ be two exact sequences. If a homomorphism $f:B\to \tilde{B}$ makes the diagram

$$\begin{array}{ccc}& & B\\ & \nearrow & & \searrow \\ A& & {\downarrow }^f& & C\\ & \searrow & & \nearrow \\ & & \tilde{B}\end{array}$$\array{ && B \\ & \nearrow && \searrow \\ A &&\downarrow^{\mathrlap{f}}&& C \\ & \searrow && \nearrow \\ && \tilde B }

commute, then $f$ is an isomorphism.

Proof

Apply prop. 1 to the diagram

$$\begin{array}{ccccccccc}0& \to & A& \to & B& \to & C& \to & 0\\ {\downarrow }^{=}& & {\downarrow }^{=}& & {\downarrow }^f& & {\downarrow }^{=}& & {\downarrow }^{=}\\ 0& \to & A& \to & \tilde{B}& \to & C& \to & 0\end{array}$$\array{ 0 &\to& A &\to& B &\to& C &\to& 0 \\ \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}} \\ 0 &\to& A &\to& \tilde B &\to& C &\to& 0 }

Short split five lemma

A special case of the five lemma is the short five lemma where the objects $A_1,B_1,A_5,B_5$ above are all zero objects. It may hold in more general setups, sometimes with additional assumptions.

The short split five lemma is a statement usually stated in the setup of semiabelian categories:

Corollary

(short split five lemma)

Given a commutative diagram

$$\begin{array}{ccccc}L& \stackrel{l}{\to }& H& \stackrel{q}{\to }& C\\ {}^u\downarrow & & {\downarrow }^w& & {\downarrow }^v\\ K& \underset{k}{\to }& G& \underset{p}{\to }& B\end{array}$$\array{L & \overset{l}{\to} & H & \overset{q}{\to} & C\\ ^u\downarrow && \downarrow^w && \downarrow^v \\ K & \underset{k}{\to} & G& \underset{p}{\to} & B}

where $p$ and $q$ are split epimorphisms and $l$ and $k$ are their kernels, if $u$ and $v$ are isomorphisms then so is $w$.

Remark

The short five lemma holds in the category of abelian topological groups, even though that category is not semi-abelian. For a proof, see this paper by Borceux and Clementino.

Related concepts

• connecting homomorphism

• salamander lemma

• 3x3 lemma, snake lemma

• horseshoe lemma

References

Early references of the 5-lemma

• (lemma (5,9) in) D. A. Buchsbaum, Exact categories and duality, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 (JSTOR)
• (prop.1.1, page 5) Henri Cartan, Samuel Eilenberg, Homological algebra, Princeton Univ. Press 1956
• (lemma 3.3 in chapter I) S. MacLane, Homology, Springer 1963, 1975

Modern

• (exercise 1.3.3 in) Charles Weibel, An Introduction to Homological Algebra
• Nicolas Bourbaki, Homological algebra (Algebra ch. X) 1980

In nonabelian context

• Francis Borceux, Dominique Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004

The short 5-lemma also appears in various topological algebra contexts; see for example

• Francis Borceux, Maria Manuel Clementino, Topological semi-abelian categories, Adv. Math. 190 (2005), 425-453 (web)