Snake Lemma

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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 $i = 1, 2, 3$ respectively $i = 1, 2$:

• $\ker \phi_i$ is the kernel of $\phi_i$
• $\operatorname{coker} \phi_i$ is the cokernel of $\phi_i$
• $\iota_i$ is the inclusion mapping
• $\pi_i$ is the quotient epimorphism
• $\tilde \alpha_i = \alpha_i {\restriction \ker \phi_i}$, the restriction of $\alpha_i$ to $\ker \phi_i$
• $\bar \beta_i$ is defined by:

$\forall n_i + \Img {\phi_i} \in \operatorname {coker} \phi_i : \map {\bar \beta_i} {n_i + \Img {\phi_i} } = \map {\beta_i} {n_i} + \Img {\phi_{i + 1} }$

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

$\begin{xy}\xymatrix@L+2mu@+1em{ \ker \phi_1 \ar[r]_*{\tilde\alpha_1} & \ker \phi_2 \ar[r]_*{\tilde\alpha_2} & \ker\phi_3 \ar[r]_*{\delta} & \operatorname{coker}\phi_1 \ar[r]_*{\bar\beta_1} & \operatorname{coker}\phi_2 \ar[r]_*{\bar\beta_2} & \operatorname{coker}\phi_3 }\end{xy}$

Proof

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