Snake Lemma

Theorem
Let $A$ be a commutative ring with unity.

Let:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

& M_1 \ar[r]_*{\alpha_1} \ar[d]^*{\phi_1} & M_2 \ar[r]_*{\alpha_2} \ar[d]^*{\phi_2} & M_3 \ar[d]^*{\phi_3} \ar[r] & 0 \\ 0 \ar[r] & N_1 \ar[r]_*{\beta_1} & N_2 \ar[r]_*{\beta_2} & N_3 & }\end{xy}$

be a commutative diagram of $A$-modules.

Suppose that the rows are exact.

Then we have a commutative diagram:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

& \ker \phi_1 \ar[r]_*{\tilde\alpha_1} \ar[d]^*{\iota_1} & \ker \phi_2 \ar[r]_*{\tilde\alpha_2} \ar[d]^*{\iota_2} & \ker \phi_3 \ar[d]^*{\iota_3} & \\ & M_1 \ar[r]_*{\alpha_1} \ar[d]^*{\phi_1} & M_2 \ar[r]_*{\alpha_2} \ar[d]^*{\phi_2} & M_3 \ar[d]^*{\phi_3} \ar[r] & 0 \\ 0 \ar[r] & N_1 \ar[r]_*{\beta_1} \ar[d]^*{\pi_1} & N_2 \ar[r]_*{\beta_2} \ar[d]^*{\pi_2} & N_3 \ar[d]^*{\pi_3} \\ & \operatorname{coker} \phi_1 \ar[r]_*{\bar\beta_1} & \operatorname{coker} \phi_2 \ar[r]_*{\bar\beta_2} & \operatorname{coker} \phi_3 & }\end{xy}$

where:
 * For $i=1,2,3$, $\iota_i$ is the inclusion mapping
 * For $i=1,2,3$, $\pi_i$ is the canonical epimorphism
 * For $i = 1,2$, $\tilde\alpha_i = \left({ \alpha_i }\right)_{| \ker \phi_i}$
 * For $i = 1,2$, $\bar\beta_i$ is defined by:
 * $\forall\; n_i + \operatorname{im}\phi_i \in \operatorname{coker}\phi_i : \bar\beta_i\left({ n_i + \operatorname{im}\phi_i }\right) = \beta_i\left({ n_i }\right) + \operatorname{im} \phi_{i+1}$

Moreover there exists a morphism $\delta : \ker\phi_3 \to \operatorname{coker}\phi_1$ such that we have an exact sequence:


 * $\begin{xy}\xymatrix@L+2mu@+1em{

& \ker \phi_1 \ar[r]_*{\tilde\alpha_1} & \ker \phi_2 \ar[r]_*{\tilde\alpha_2} & \ker\phi_3 \ar[r]_*{\delta} & \operatorname{coker}\phi_1 \ar[r]_*{\bar\beta_1} & \operatorname{coker}\phi_2 \ar[r]_*{\bar\beta_2} & \operatorname{coker}\phi_3 }\end{xy}$

which is functorial in the sense that if:


 * $\begin{xy}\xymatrix{

&&& M_1 \ar@{->}[rr] \ar@{->}[dl] \ar@{->}[dd]|!{[d];[d]}\hole && M_2 % \ar@{->}[rr] \ar@{->}[dl] \ar@{->}[dd]|!{[d];[d]}\hole && M_3 \ar@{->}[dl] \ar@{->}[dd] \ar@{->}[rr] && 0 \\ && M_1' \ar@{->}[rr] \ar@{->}[dd] && M_2' \ar@{->}[rr] \ar@{->}[dd] && M_3' \ar@{->}[dd] \ar@{->}[rr]|!{[r];[r]}\hole && 0 \\ & 0 \ar@{->}[rr]|!{[r];[r]}\hole && N_1 \ar@{->}[rr]|!{[r];[r]}\hole \ar@{->}[dl]_{F’} && N_2 \ar@{->}[rr]|!{[r];[r]}\hole \ar@{->}[dl] && N_3 \ar@{->}[dl]_{F’’} \\ % 0 \ar@{->}[rr] && N_1' \ar@{->}^(.65){e’}[rr] && N_2' \ar@{->}[rr] && N_3' }\end{xy}$

is a commutative diagram.

}