Short exact sequence and pushout

Question

Let $0\to N\stackrel{f}{\to} E\stackrel{g}{\to} M\to 0$ be a short exact sequence of $R$-modules. Let $\varphi:N\to N'$ be a morphism of $R$-modules and let $E'$ be the pushout of $N\to E$ and $\varphi:N\to N'$. Show that there is a short exact sequence $0\to N'\to E'\to M\to 0$ of $R$-modules.

My try:

$0\circ \phi=g\circ f$ so by universal property of pullback $\exists \psi:E'\to M$ such that $\psi\circ \ell=0$.

Now $\psi$ is surjective since $g$ is and $\ell\circ k=g$.

Now it remains to show that $Im \ell=Ker \psi$ and $\ell$ injective. Does anyone have ideas about this?

Answer 1

Hint and answer to one of your questions: You can take zero map $0:N'\to M$. Then $0=0\circ\varphi=g\circ f$, and from universal property of pushout you get map $\psi:E'\to M$ such that $\psi\circ (N'\to E')=0$.

Answer 2

We should use the following approach. It is property of pushout that $l$ is injective because $f$ is injective. It is also property of pushout that $coker (l)$ is isomorphic to $coker (f)=M$

We give some theoretic background.

Consider the square $S$: $\require{AMScd}$ \begin{CD} A_0 @>{\phi_1}>> A_1\\ @V{\phi_2}VV @VV{\psi_1}V\\ A_2 @>>{\psi_2}> A \end{CD} We define $\{\phi_1,\phi_2\}:A_0\to A_1\bigoplus A_2$ with $\{\phi_1,\phi_2\}(a_0)=(\phi_1(a_0),\phi_2(a_0))$.

We define $<\psi_1,\psi_2>: A_1\bigoplus A_2\to A$ with $<\psi_1,\psi_2>(a_1,a_2)=\psi_1(a_1)+\psi_2(a_2)$.

We first observe that square $S$ is commutative iff $<\psi_1,-\psi_2>\circ \{\phi_1,\phi_2\}=0$.

Further, it can be shown that following holds:

(1a) $S$ is a pullback iff $\{\phi_1,\phi_2\}$ is kernel of $<\psi_1,-\psi_2>$;

(1b) $S$ is a pushout iff $<\psi_1,-\psi_2>$ is cokernel of $\{\phi_1,\phi_2\}$.

Now, as a corollary we have:

(2a) if $S$ is a pullback and $<\psi_1,-\psi_2>$ is epic, then $S$ is bicartesian (pullback and pushout);

(2b) if $S$ is a pushout and $\{\phi_1,\phi_2\}$ is monic, then $S$ is bicartesian.

Note that $<\psi_1,-\psi_2>$ is epic if one of $\phi_1,\phi_2$ is epic and that $\{\phi_1,\phi_2\}$ is monic if one of $\phi_1,\phi_2$ is monic.

Then we have following:

If $S$ is a pullback (pushout) then induced map of kernels $ker(\phi_1)\to ker(\psi_2)$ (cokernels $coker(\phi_1)\to coker(\psi_2)$) is isomoprhism.

and

(3a) Let $S$ be a pullback. Then if $\psi_2$ is surjective, $\phi_1$ is surjetive.

(3b) Let $S$ be a pushout. Then if $\phi_1$ is injective, $\psi_2$ is injective.