The Stacks project

Lemma 13.13.4. Let $\mathcal{A}$ be an abelian category. The full subcategory $\text{FAc}(\mathcal{A})$ of $K(\text{Fil}^ f(\mathcal{A}))$ consisting of filtered acyclic complexes is a strictly full saturated triangulated subcategory of $K(\text{Fil}^ f(\mathcal{A}))$. The corresponding saturated multiplicative system (see Lemma 13.6.10) of $K(\text{Fil}^ f(\mathcal{A}))$ is the set $\text{FQis}(\mathcal{A})$ of filtered quasi-isomorphisms. In particular, the kernel of the localization functor

\[ Q : K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \text{FQis}(\mathcal{A})^{-1}K(\text{Fil}^ f(\mathcal{A})) \]

is $\text{FAc}(\mathcal{A})$ and the functor $H^0 \circ \text{gr}$ factors through $Q$.

Proof. We know that $H^0 \circ \text{gr}$ is a homological functor by Lemma 13.13.3. Thus this lemma is a special case of Lemma 13.6.11. $\square$

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