Lemma 57.12.4. Let $X$ be a Noetherian scheme. Let $x \in X$ be a closed point and denote $\mathcal{O}_ x$ the skyscraper sheaf at $x$ with value $\kappa (x)$. Let $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Let $b \in \mathbf{Z}$. The following are equivalent

$H^ i(K)_ x = 0$ for all $i > b$ and

$\mathop{\mathrm{Hom}}\nolimits _ X(K, \mathcal{O}_ x[-i]) = 0$ for all $i > b$.

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