Lemma 15.95.9. Let $A$ be a ring and let $f, g \in A$ be nonzerodivisors. Let $M^\bullet$ be a complex of $A$-modules such that $fg$ is a nonzerodivisor on all $M^ i$. Then $\eta _ f\eta _ gM^\bullet = \eta _{fg}M^\bullet$.

Proof. The statement means that in degree $i$ we obtain the same submodule of the localization $M^ i_{fg} = (M^ i_ g)_ f$. We omit the details. $\square$

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