Lemma 12.8.3. Let $\mathcal{C}$ be an additive category. Let $S$ be a multiplicative system. Let $X$ be an object of $\mathcal{C}$. The following are equivalent

$Q(X) = 0$ in $S^{-1}\mathcal{C}$,

there exists $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that $0 : X \to Y$ is an element of $S$, and

there exists $Z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ such that $0 : Z \to X$ is an element of $S$.

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Comment #332 by arp on

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