The Stacks project

Exercise 111.30.8. Let $\mathcal{A}$ be an abelian category. Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of bounded below complexes of $\text{Fil}^ f(\mathcal{A})$. (Note the superscript $f$.) Show that the following are equivalent:

1. $\alpha$ is a filtered quasi-isomorphism,

2. for each $p \in \mathbf{Z}$ the map $\alpha : F^ pK^\bullet \to F^ pL^\bullet$ is a quasi-isomorphism,

3. for each $p \in \mathbf{Z}$ the map $\alpha : K^\bullet /F^ pK^\bullet \to L^\bullet /F^ pL^\bullet$ is a quasi-isomorphism, and

4. the cone of $\alpha$ (see Derived Categories, Definition 13.9.1) is a filtered acyclic complex.

Moreover, show that if $\alpha$ is a filtered quasi-isomorphism then $\alpha$ is also a usual quasi-isomorphism.