The Stacks project

Lemma 57.9.5. 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 57.9.4. $\square$

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