Lemma 57.10.5. Let $X$ be a regular Noetherian scheme of dimension $d < \infty$. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ and $a \in \mathbf{Z}$. If $H^ i(K) = 0$ for $a < i < a + d$, then $K = \tau _{\leq a}K \oplus \tau _{\geq a + d}K$.

Proof. We have $\tau _{\leq a}K = \tau _{\leq a + d - 1}K$ by the assumed vanishing of cohomology sheaves. By Derived Categories, Remark 13.12.4 we have a distinguished triangle

$\tau _{\leq a}K \to K \to \tau _{\geq a + d}K \xrightarrow {\delta } (\tau _{\leq a}K)[1]$

By Derived Categories, Lemma 13.4.11 it suffices to show that the morphism $\delta$ is zero. This follows from Lemma 57.10.4. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).