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five lemma

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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

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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\mathcal{A} be an abelian category. Consider a commutative diagram in 𝒜\mathcal{A} of the form

A 1 A 2 A 3 A 4 A 5 f 1 f 2 f 3 f 4 f 5 B 1 B 2 B 3 B 4 B 5\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 dd.

Proposition (the lemma on five homomorphisms or the five lemma)
  1. sharp five lemma (essentially the weak four lemma)

    1. If f 2f_2 and f 4f_4 are epi and f 5f_5 is mono, then f 3f_3 is epi.

    2. If f 2f_2 and f 4f_4 are mono and f 1f_1 is epi, then f 3f_3 is mono.

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

    If f 2f_2 and f 4f_4 are isos, f 1f_1 is epi, and f 5f_5 is mono, then f 3f_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 1f_1 is not required to exist, and in 1.2 f 5f_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 RRMod (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 bB 3b\in B_3. Since f 4f_4 is epi, one can choose an element a 4A 4a_4\in A_4 such that f 4(a 4)=d(b)f_4(a_4) = d(b). Now 0=d 2b=df 4(a 4)=f 5d(a 4)0 = d^2 b = d f_4 (a_4) = f_5 d (a_4). Since f 5f_5 is a monomorphism that means that da 4=0d a_4 = 0 as well. By the exactness of the upper row, that means there is a 3A 3a_3\in A_3 such that da 3=a 4d a_3 = a_4, hence also df 3(a 3)=f 4d(a 3)=f 4(a 4)=dbd f_3 (a_3) = f_4 d (a_3) = f_4(a_4) = d b. We would like that f 3(a 3)f_3(a_3) be equal to bb but this is not so, we just see that d(bf 3(a 3))=0d (b-f_3(a_3)) = 0 and hence by exactness of the lower row there is bB 2b'\in B_2 such that db=bf 3(a 3)d b' = b-f_3(a_3). Since f 2f_2 is also epi, there is a 2A 2a_2\in A_2 such that f 2(a 2)=bf_2(a_2) = b'. Now da 2+a 3A 3d a_2+a_3\in A_3 is such that

f 3(da 2+a 3)=df 2(a 2)+f 3(a 3)=db+f 3(a 3)=bf 3(a 3)+f 3(a 3)=bf_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 bb is in the image of f 3f_3.

Hence f 3f_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 ABCA \to B \to C and AB˜CA \to \tilde B \to C be two exact sequences. If a homomorphism f:BB˜f \colon B \to \tilde B makes the diagram

B A f C B˜ \array{ && B \\ & \nearrow && \searrow \\ A &&\downarrow^{\mathrlap{f}}&& C \\ & \searrow && \nearrow \\ && \tilde B }

commute, then ff is an isomorphism.

Proof

Apply prop. 1 to the diagram

0 A B C 0 = = f = = 0 A B˜ C 0 \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 5A_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

L l H q C u w v K k G p B\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 pp and qq are split epimorphisms and ll and kk are their kernels, if uu and vv are isomorphisms then so is ww.

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.

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

In nonabelian context

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)

Revised on September 30, 2012 21:00:35 by Urs Schreiber (89.204.137.60)