Let $\mathcal A$ be an abelian category.

Show that $$\mathrm{Im}(\bigoplus_{1\leqslant i\leqslant n}\ \varphi_i)\simeq \bigoplus_{1\leqslant i\leqslant n}\mathrm{Im}\varphi_i$$ There are two conclusions that I have proved:$$\mathrm{Ker}(\prod_{1\leqslant i\leqslant n}\ \varphi_i)\simeq \prod_{1\leqslant i\leqslant n}\mathrm{Ker}\varphi_i$$$$\mathrm{Coker}(\bigoplus_{1\leqslant i\leqslant n}\ \varphi_i)\simeq \bigoplus_{1\leqslant i\leqslant n}\mathrm{Coker}\varphi_i$$If $$\mathrm{Ker}(\bigoplus_{1\leqslant i\leqslant n}\ \varphi_i)\simeq \bigoplus_{1\leqslant i\leqslant n}\mathrm{Ker}\varphi_i$$ then by the definition of $\mathrm{Im} \varphi$,it's easy to prove the first isomorphism.But I can't prove it.

  • $\begingroup$ In an abelian category, finite product and finite direct sum are canonically isomorphic. $\endgroup$ – Roland Jul 19 '16 at 13:27

I assume that your definition of abelian category includes additive and I define additive categories as follows: An Ab-category $\mathscr C$ is additive if there is a zero object $0 \in \mathscr C$ and products $X \times Y$ exist for every pair $X, Y \in \mathscr C$.

Denote products $X \times Y$ by $X \oplus Y$. Now prove the following theorem: In an additive category $\mathscr C$, there exist unique morphisms $$ X \xrightarrow{i_1} X \oplus Y \xleftarrow{i_2} Y $$ such that $p_1 i_1 = 1_{X}$, $p_2 i_2 = 1_{Y}$, $p_1i_2 = 0$, $p_2i_1 = 0$ and $i_1p_1 + i_2p_2 = 1_{X \oplus Y}$, where $$X \xleftarrow{p_1} X \oplus Y \xrightarrow{p_2} Y$$ are the projection maps of the product $X \oplus Y$.

Then we get the following corollary: Suppose maps $i_1, i_2$ are constructed as above. Then $(X \oplus Y, i_1, i_2)$ is the coproduct of $X$ and $Y$. Thus, sums and products are the same in an additive category.

Finally, prove that if $g$ and $h$ are isomorphisms then $\operatorname{Ker} f \cong \operatorname{Ker}hfg$ and similarly for cokernels.

Now your result is easy: $$\begin{align}\operatorname{Im} \left( \bigoplus_{1\le i\le n} \varphi_i \right) &\cong \operatorname{Ker} \left(\bigoplus_{1\le i\le n} \operatorname{Coker} \varphi_i \right) \\ &\cong \operatorname{Ker} \left(\prod_{1\le i\le n} \operatorname{Coker} \varphi_i \right) \\ &\cong \prod_{1\le i\le n} \operatorname{Ker}\operatorname{Coker} \varphi_i \\ &\cong \bigoplus_{1\le i\le n} \operatorname{Ker}\operatorname{Coker} \varphi_i \\ &\cong \bigoplus_{1\le i\le n} \operatorname{Im} \varphi_i.\end{align}$$


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