The Stacks project

Lemma 13.13.3. Let $\mathcal{A}$ be an abelian category.

1. The functor $K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \text{Gr}(\mathcal{A})$, $K^\bullet \longmapsto H^0(\text{gr}(K^\bullet ))$ is homological.

2. The functor $K(\text{Fil}^ f(\mathcal{A})) \rightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0(\text{gr}^ p(K^\bullet ))$ is homological.

3. The functor $K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0((\text{forget }F)K^\bullet )$ is homological.