1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ In an abelian category, finite product and finite direct sum are canonically isomorphic. $\endgroup$ – Roland Jul 19 '16 at 13:27
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.