Direct-Sum

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Joined: Jan 12, 2011

by iarnab_kundu, Dec 10, 2018, 2:14 PM

Proposition- Let $\mathcal{C}$ be an abelian category. Let $M_1,M_2$ be two objects. Then $F$ with four maps $i_1:M_2\to F, p_1:F\to M_1$ and $i_2:M_2\to F, p_2:F\to M_2$ is the direct sum of $M_1$ and $M_2$ if and only if we have that $i_1p_1+i_2p_2=\text{id}_F$ . Where the equation is viewed in $\text{Hom}(F,F)$ .
Proof- We have a morphism $(i_1+i_2):M_1\oplus M_2\to F$ and $(p_1,p_2):F\to M_1\oplus M_2$ . Then $(i_1+i_2)\circ (p_1,p_2)=(\text{id}_{M_1},\text{id}_{M_2})$ . Also $(p_1,p_2)\circ (i_1+i_2)=p_1\circ i_1+p_2\circ i_2=\text{id}_F$ .

This post has been edited 4 times. Last edited by iarnab_kundu, Dec 10, 2018, 2:25 PM

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Cracking the Nut Open

Direct-Sum

by iarnab_kundu, Dec 10, 2018, 2:14 PM

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