The Stacks project

Lemma 13.13.6. The functors $\text{gr}^ p, \text{gr}, (\text{forget }F)$ induce canonical exact functors

\[ \text{gr}^ p, \text{gr}, (\text{forget }F): DF(\mathcal{A}) \longrightarrow D(\mathcal{A}) \]

which commute with the localization functors.

Proof. This follows from the universal property of localization, see Lemma 13.5.7, provided we can show that a filtered quasi-isomorphism is turned into a quasi-isomorphism by each of the functors $\text{gr}^ p, \text{gr}, (\text{forget }F)$. This is true by definition for the first two. For the last one the statement we have to do a little bit of work. Let $f : K^\bullet \to L^\bullet $ be a filtered quasi-isomorphism in $K(\text{Fil}^ f(\mathcal{A}))$. Choose a distinguished triangle $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ which contains $f$. Then $M^\bullet $ is filtered acyclic, see Lemma 13.13.4. Hence by the corresponding lemma for $K(\mathcal{A})$ it suffices to show that a filtered acyclic complex is an acyclic complex if we forget the filtration. This follows from Homology, Lemma 12.19.15. $\square$

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