Four lemma

The four lemma is a pair of related statements about a commutative diagram of exact sequences. It is weaker than the five lemma (it gives injectivity or surjectivity, not necessarily an isomorphism), and it is often proved by diagram chasing using ideas similar to the snake lemma .

In categorical language, “injective/surjective” correspond to monomorphisms and epimorphisms .

Theorem (Four lemma: injective and surjective forms)

Let

be a commutative diagram of $R$-modules with exact rows.

(1) Injective form

If $f_1$ is surjective and $f_2$ and $f_4$ are injective, then $f_3$ is injective.

(2) Surjective form

If $f_2$ and $f_4$ are surjective and $f_5$ is injective, then $f_3$ is surjective.

Combining (1) and (2) (under the usual extra hypotheses that make the “remaining” injective/surjective conditions automatic) yields the five lemma .

Examples

1. Injectivity in the middle when the ends are $0$.
   In many applications (e.g. long exact sequences), one works with a 5-term exact piece

   $$0\to A_2\to A_3\to A_4\to 0$$

   and similarly for the $B_i$. Then $f_1:0\to 0$ is automatically surjective and $f_5:0\to 0$ is automatically injective.
   The injective form reduces to: if $f_2$ and $f_4$ are injective, then $f_3$ is injective.
   The surjective form reduces to: if $f_2$ and $f_4$ are surjective, then $f_3$ is surjective.

2. Propagation of injectivity/surjectivity through long exact Tor or Ext sequences.
   Naturality gives morphisms between long exact sequences such as the long exact Tor sequence or long exact Ext sequence .
   Applying the four lemma to a 5-term window inside these long exact sequences is a standard way to conclude:

   • a map on a “middle” $\operatorname{Tor}_n$ or $\operatorname{Ext}^n$ group is injective, provided the adjacent maps satisfy the injective-form hypotheses;
   • or surjective, provided the adjacent maps satisfy the surjective-form hypotheses.

3. Homology of chain complexes: injectivity/surjectivity at one degree.
   Given a morphism between long exact homology sequences arising from a short exact sequence of chain complexes , the four lemma lets one conclude that the induced map

   $$H_n(C_\bullet)\to H_n(D_\bullet)$$

   is injective (or surjective) from corresponding injectivity/surjectivity information at neighboring degrees—often a step in proving the full isomorphism statement later via the five lemma .