# Induced Exact Sequences in Abelian Categories

Let $$\mathcal{A}$$ be an abelian category. Show that for every $$f:A\to B$$ the following sequences are exact:

$$0\to \text{ker}(f)\xrightarrow{i} A\xrightarrow{\pi}\text{coim}(f)\to 0$$ $$0\to \text{im}(f)\xrightarrow{j} B\xrightarrow{\rho}\text{coker}(f)\to 0$$

I've used the universal properties of $$\text{ker, im, coim}$$ and $$\text{coker}$$ to prove that $$i, j$$ are monomorphisms and $$\pi,\rho$$ are epimorphisms.

But I have no idea how to prove that $$\text{ker}(\pi)=\text{im}(i)$$ and $$\text{ker}(\rho)=\text{im}(j)$$. In explicit cases, like in the categories of $$A$$-modules or groups, we obviously have that $$\text{im}(i)=\text{ker}(f)=\text{ker}(\pi)$$, for example. But in general, I don't even know where to begin.

What is the idea?

## 1 Answer

I define $\text{coim}(f) := \text{coker}(\ker(f) \to A) = \text{coker}(i),$ so the upper sequence is clearly exact in the middle and right places. The left you already took care of. Similarly, $\text{im}(f) = \ker(\rho),$ so you're done unless you're using different definitions!