Lemma 10.9.13. Localization respects quotients, i.e. if $N$ is a submodule of $M$, then $S^{-1}(M/N)\simeq (S^{-1}M)/(S^{-1}N)$.

Proof. From the exact sequence

$0 \longrightarrow N \longrightarrow M \longrightarrow M/N \longrightarrow 0$

we have

$0 \longrightarrow S^{-1}N \longrightarrow S^{-1}M \longrightarrow S^{-1}(M/N) \longrightarrow 0$

The corollary then follows. $\square$

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