Five lemma

The five lemma is a standard diagram-chase tool in $R$-modules and, more generally, in any abelian category . It strengthens the injectivity/surjectivity conclusions of the four lemma and is often used alongside the snake lemma .

Theorem (Five lemma)

Consider a commutative diagram with exact rows

Assume:

• $f_1, f_2, f_4, f_5$ are isomorphisms,
• $f_2$ is an epimorphism and $f_4$ is a monomorphism (in module terms: $f_2$ is surjective and $f_4$ is injective).

Then $f_3$ is an isomorphism.

(There are common variants with slightly different hypotheses; a frequent special case is when the sequences begin and end with $0$, so surjectivity/injectivity assumptions are automatic.)

See also five lemma corollary for standard “short exact sequence” consequences.

Examples

1. Two-out-of-three for quasi-isomorphisms in a short exact sequence of complexes.
   Suppose

   $$0\to C'_\bullet \to C_\bullet \to C''_\bullet \to 0 \quad\text{and}\quad 0\to D'_\bullet \to D_\bullet \to D''_\bullet \to 0$$

   are short exact sequences of chain complexes , and we have a morphism between them inducing a commutative diagram of long exact sequences in homology .
   If the induced maps $H_n(C'_\bullet)\to H_n(D'_\bullet)$ and $H_n(C''_\bullet)\to H_n(D''_\bullet)$ are isomorphisms for all $n$, then the induced maps $H_n(C_\bullet)\to H_n(D_\bullet)$ are isomorphisms for all $n$, by applying the five lemma degree-by-degree to the long exact homology sequences.

2. Comparing Tor groups via a map of short exact sequences.
   Given a morphism between short exact sequences

   $$\begin{array}{ccccccccc} 0 &\to& A' &\to& A &\to& A'' &\to& 0\\ && \downarrow && \downarrow && \downarrow &&\\ 0 &\to& B' &\to& B &\to& B'' &\to& 0 \end{array}$$

   and a fixed module $M$, naturality of the long exact Tor sequence gives a morphism of long exact sequences

   $$\cdots \to \operatorname{Tor}_n(A'',M)\to \operatorname{Tor}_{n-1}(A',M)\to \cdots$$

   If the induced maps $\operatorname{Tor}_n(A',M)\to \operatorname{Tor}_n(B',M)$ and $\operatorname{Tor}_n(A'',M)\to \operatorname{Tor}_n(B'',M)$ are isomorphisms for all relevant neighboring terms (for instance, if $A'\to B'$ and $A''\to B''$ are isomorphisms), then the five lemma implies $\operatorname{Tor}_n(A,M)\to \operatorname{Tor}_n(B,M)$ is an isomorphism as well.

3. Comparing Ext groups (same pattern).
   With the same setup, applying the long exact Ext sequence and using the five lemma shows that if the induced maps on the surrounding $\operatorname{Hom}$ and $\operatorname{Ext}$ terms are isomorphisms, then the middle $\operatorname{Ext}$-map is an isomorphism. This is a standard way to propagate isomorphisms through long exact sequences.