Lemma 12.8.4. Let $\mathcal{A}$ be an abelian category.

1. If $S$ is a left multiplicative system, then the category $S^{-1}\mathcal{A}$ has cokernels and the functor $Q : \mathcal{A} \to S^{-1}\mathcal{A}$ commutes with them.

2. If $S$ is a right multiplicative system, then the category $S^{-1}\mathcal{A}$ has kernels and the functor $Q : \mathcal{A} \to S^{-1}\mathcal{A}$ commutes with them.

3. If $S$ is a multiplicative system, then the category $S^{-1}\mathcal{A}$ is abelian and the functor $Q : \mathcal{A} \to S^{-1}\mathcal{A}$ is exact.

Proof. Assume $S$ is a left multiplicative system. Let $a : X \to Y$ be a morphism of $S^{-1}\mathcal{A}$. Then $a = s^{-1}f$ for some $s : Y \to Y'$ in $S$ and $f : X \to Y'$. Since $Q(s)$ is an isomorphism we see that the existence of $\mathop{\mathrm{Coker}}(a : X \to Y)$ is equivalent to the existence of $\mathop{\mathrm{Coker}}(Q(f) : X \to Y')$. Since $\mathop{\mathrm{Coker}}(Q(f))$ is the coequalizer of $0$ and $Q(f)$ we see that $\mathop{\mathrm{Coker}}(Q(f))$ is represented by $Q(\mathop{\mathrm{Coker}}(f))$ by Categories, Lemma 4.27.9. This proves (1).

Part (2) is dual to part (1).

If $S$ is a multiplicative system, then $S$ is both a left and a right multiplicative system. Thus we see that $S^{-1}\mathcal{A}$ has kernels and cokernels and $Q$ commutes with kernels and cokernels. To finish the proof of (3) we have to show that $\mathop{\mathrm{Coim}}= \mathop{\mathrm{Im}}$ in $S^{-1}\mathcal{A}$. Again using that any arrow in $S^{-1}\mathcal{A}$ is isomorphic to an arrow $Q(f)$ we see that the result follows from the result for $\mathcal{A}$. $\square$

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