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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