Lemma 13.13.8. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet \in K(\text{Fil}^ f(\mathcal{A}))$.

If $H^ n(\text{gr}(K^\bullet )) = 0$ for all $n < a$, then there exists a filtered quasi-isomorphism $K^\bullet \to L^\bullet $ with $L^ n = 0$ for all $n < a$.

If $H^ n(\text{gr}(K^\bullet )) = 0$ for all $n > b$, then there exists a filtered quasi-isomorphism $M^\bullet \to K^\bullet $ with $M^ n = 0$ for all $n > b$.

If $H^ n(\text{gr}(K^\bullet )) = 0$ for all $|n| \gg 0$, then there exists a commutative diagram of morphisms of complexes

\[ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u] } \]where all the arrows are filtered quasi-isomorphisms, $L^\bullet $ bounded below, $M^\bullet $ bounded above, and $N^\bullet $ a bounded complex.

## Comments (1)

Comment #9777 by ZL on