Mathematics Lecture Notes

Theorem3.7Five Lemma

In an abelian category, given a commutative diagram with exact rows:

\begin{array}{ccccccccc} A_1 & \to & A_2 & \to & A_3 & \to & A_4 & \to & A_5 \\ \downarrow\scriptstyle{\alpha_1} & & \downarrow\scriptstyle{\alpha_2} & & \downarrow\scriptstyle{\alpha_3} & & \downarrow\scriptstyle{\alpha_4} & & \downarrow\scriptstyle{\alpha_5} \\ B_1 & \to & B_2 & \to & B_3 & \to & B_4 & \to & B_5 \end{array}

If $\alpha_1$ is epic and $\alpha_2, \alpha_4$ are monic, then $\alpha_3$ is monic (Four Lemma, left half).
If $\alpha_5$ is monic and $\alpha_2, \alpha_4$ are epic, then $\alpha_3$ is epic (Four Lemma, right half).
If $\alpha_1, \alpha_2, \alpha_4, \alpha_5$ are isomorphisms, then $\alpha_3$ is an isomorphism.

Theorem3.8Short Five Lemma

Given a commutative diagram with exact rows:

\begin{array}{ccccccccc} 0 & \to & A & \to & B & \to & C & \to & 0 \\ & & \downarrow\scriptstyle{\alpha} & & \downarrow\scriptstyle{\beta} & & \downarrow\scriptstyle{\gamma} & & \\ 0 & \to & A' & \to & B' & \to & C' & \to & 0 \end{array}

If $\alpha$ and $\gamma$ are monic, then $\beta$ is monic.
If $\alpha$ and $\gamma$ are epic, then $\beta$ is epic.
If $\alpha$ and $\gamma$ are isomorphisms, then $\beta$ is an isomorphism.

ExampleFive Lemma in R-Mod

In $R\text{-}\mathbf{Mod}$ , the Five Lemma is proved by element chasing. To show $\alpha_3$ is monic: take $x \in \ker \alpha_3$ , chase it left to show it is in the image of $A_2 \to A_3$ , then use $\alpha_2$ being monic to show $x = 0$ .

ExampleComparison of long exact sequences

If two long exact sequences arise from a morphism of short exact sequences of complexes, the Five Lemma applied to the long exact sequences in cohomology shows that if the maps on most cohomology groups are isomorphisms, the remaining map is also an isomorphism.

ExampleQuasi-isomorphism detection

A chain map $f : A_\bullet \to B_\bullet$ is a quasi-isomorphism iff $H^n(f)$ is an isomorphism for all $n$ . The Five Lemma applied to the long exact sequence of the mapping cone shows this is equivalent to $H^n(\mathrm{Cone}(f)) = 0$ for all $n$ .

ExampleExcision in homology

The excision theorem in algebraic topology states that $H_n(X, A) \cong H_n(X \setminus U, A \setminus U)$ under certain conditions. The proof uses the Five Lemma applied to the long exact sequences of the pairs $(X, A)$ and $(X \setminus U, A \setminus U)$ .

ExampleKunneth theorem

The Kunneth theorem relates $H_n(X \times Y)$ to $H_i(X) \otimes H_j(Y)$ . The Five Lemma is used to show that the Kunneth map is an isomorphism when one of the homology groups is free.

ExampleComparison of resolutions

The comparison theorem states that any two projective resolutions of a module are chain homotopy equivalent. The Five Lemma (or induction using the lifting property) establishes the isomorphism on homology.

ExampleFive Lemma for sheaf cohomology

Given a morphism $f : X \to Y$ of spaces and compatible short exact sequences of sheaves, the Five Lemma applied to the resulting long exact sequences in sheaf cohomology relates the higher direct images $R^n f_*$ .

ExampleShort Five Lemma for group extensions

For a commutative diagram of group extensions where the outer maps are isomorphisms, the Short Five Lemma (which holds in $\mathbf{Grp}$ as well) gives that the middle map is an isomorphism. This is used in the classification of group extensions.

ExampleFive Lemma and universal coefficients

The universal coefficient theorem for cohomology involves a short exact sequence $0 \to \mathrm{Ext}^1(H_{n-1}(X), G) \to H^n(X; G) \to \mathrm{Hom}(H_n(X), G) \to 0$ . The Five Lemma applied to morphisms between these sequences shows that maps inducing isomorphisms on homology also induce isomorphisms on cohomology.

ExampleFive Lemma in derived categories

In the context of distinguished triangles in a triangulated category, the analogue of the Five Lemma states: if $f : X \to Y$ induces isomorphisms on all but one vertex of a morphism of distinguished triangles, it induces an isomorphism on the remaining vertex.

ExampleFive Lemma and spectral sequence comparison

Given a morphism of spectral sequences $E_r^{p,q} \to {E'}_r^{p,q}$ , if the map is an isomorphism on the $E_2$ page, then by iterated application of the Five Lemma, it is an isomorphism on all subsequent pages and on the abutment.

ExampleWhitehead theorem via Five Lemma

The Whitehead theorem states that a weak homotopy equivalence between CW-complexes is a homotopy equivalence. The proof uses the Five Lemma on the long exact sequence of homotopy groups of mapping cylinder and mapping cone.

ProofProof (element chase in R-Mod)

We prove $\alpha_3$ is monic assuming $\alpha_1$ epic and $\alpha_2, \alpha_4$ monic.

Let $x_3 \in A_3$ with $\alpha_3(x_3) = 0$ . We need to show $x_3 = 0$ .

Map $x_3$ to $A_4$ : let $x_4 = d(x_3)$ where $d : A_3 \to A_4$ . Then $\alpha_4(x_4) = \alpha_4(d(x_3)) = d'(\alpha_3(x_3)) = d'(0) = 0$ . Since $\alpha_4$ is monic, $x_4 = 0$ .

By exactness of the top row at $A_3$ , $x_3 = d(x_2)$ for some $x_2 \in A_2$ . Then $d'(\alpha_2(x_2)) = \alpha_3(d(x_2)) = \alpha_3(x_3) = 0$ . By exactness of the bottom row at $B_2$ , $\alpha_2(x_2) = d'(b_1)$ for some $b_1 \in B_1$ .

Since $\alpha_1$ is epic, $b_1 = \alpha_1(x_1)$ for some $x_1 \in A_1$ . Then $\alpha_2(x_2) = d'(\alpha_1(x_1)) = \alpha_2(d(x_1))$ . Since $\alpha_2$ is monic, $x_2 = d(x_1)$ . Thus $x_3 = d(x_2) = d(d(x_1)) = 0$ by $d^2 = 0$ (i.e., by exactness at $A_2$ : $x_3 = d(d(x_1)) = 0$ ).

The proof that $\alpha_3$ is epic under the dual hypotheses is the dual argument.

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RemarkValidity in general abelian categories

The element-chasing proof works in $R\text{-}\mathbf{Mod}$ . For a general abelian category, we invoke the Freyd-Mitchell embedding theorem to reduce to a module category. Alternatively, one can give a purely categorical proof using the universal properties of kernels and cokernels.

Math Notes

Five Lemma

Statement

Special Cases

Examples

Proof of the Five Lemma