In the following commutative diagram, with exact rows and columns, is $j_1$ surjective?

Question

I have the following commutative digram of abelian groups and homomorphisms between them. The rows and columns are exact: \begin{array} A & && && & & & & 0 & \ \\ & && & & & && &\downarrow{}\\ & && && A_1 & \stackrel{h_1}{\longrightarrow} & A_2 &\stackrel{p_1}{\longrightarrow} & A_3 & \ \\ & && && \downarrow{i_1} & &\downarrow{i_2}& &\downarrow{i_3}\\ & && && B_1 & \stackrel{h_2}{\longrightarrow} & B_2 &\stackrel{p_2}{\longrightarrow} & B_3 & \ \\ & && && \downarrow{j_1} & &\downarrow{j_2}& &\downarrow{j_3}\\ & 0 && \stackrel{}{\longrightarrow} &&C_1 & \stackrel{h_3}{\longrightarrow} &C_2 & \stackrel{p_3}{\longrightarrow} & C_3 & {\longrightarrow} & 0 & \\ & && && & && &\downarrow{}\\ & && && & & & & 0 & & \\ \end{array}

I need to show that:

$p_1$ and $j_2$ are surjective $\implies $ $p_2$ and $j_1$ are surjective

I've been able to use the four lemma to show that $p_2$ is surjective. But I don't know how to show $j_1$ is also surjective.

Answer 1

A little diagram chase should suffice.

Start with $x\in C_1$. As $h_3(x)\in C_2$ and $j_2$ is surjective by assumption take $y\in B_2$ such that $j_2(y)=h_3(x)$. By commutativity of the diagram and by exactness we have $$(j_3\circ p_2)(y)=(p_3\circ j_2)(y)=(p_3\circ h_3)(x)=0$$ Therefore $p_2(y)\in\ker j_3$ and again by exactness there is $z\in A_3$ such that $i_3(z)=p_2(y)$. As $p_1$ is surjective by assumption take $w\in A_2$ such that $p_1(w)=z.$ Now $$(p_2\circ i_2)(w)=(i_3\circ p_1)(w)=i_3(z)=p_2(y)$$ Let $k=y-i_2(w)$, then $$p_2(k)=p_2(y-i_2(w))=p_2(y)-(p_2\circ i_2)(w)=0$$ By exactness there is $u\in B_1$ with $h_2(u)=k$. Now, observe that $$(h_3\circ j_1)(u)=(j_2\circ h_2)(u)=j_2(k)=j_2(y-i_2(w))=j_2(y)-(j_2\circ i_2)(w)=h_3(x)$$ Finally, as $h_3$ is injective by assumption we obtain $j_1(u)=x$ and $j_1$ is surjective as desired.

I am almost entirely sure that the details are painful to read and I think there is a smarter way maybe involving the Four Lemma too. But I could not resist writing down the whole chase; my apologies.