The Stacks project

Definition 13.13.2. Let $\mathcal{A}$ be an abelian category.

1. Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of $K(\text{Fil}^ f(\mathcal{A}))$. We say that $\alpha$ is a filtered quasi-isomorphism if the morphism $\text{gr}(\alpha )$ is a quasi-isomorphism.

2. Let $K^\bullet$ be an object of $K(\text{Fil}^ f(\mathcal{A}))$. We say that $K^\bullet$ is filtered acyclic if the complex $\text{gr}(K^\bullet )$ is acyclic.