Lemma 57.12.3. Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. 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 $D_{perf}(\mathcal{O}_ X)$.

If $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{O}_ x, K) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^{i - d_ x}(K)|_ U = 0$ where $d_ x = \dim (\mathcal{O}_{X, x})$.

If $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{O}_ x, K[i]) = 0$ for all $i \in \mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$.

If $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \mathcal{O}_ x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^ i(K^\vee )|_ U = 0$.

If $\mathop{\mathrm{Hom}}\nolimits _ X(K, \mathcal{O}_ x[i]) = 0$ for all $i \in \mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$.

If $H^ i(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^ i(K)|_ U = 0$.

If $H^ i(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ x) = 0$ for $i \in \mathbf{Z}$ then $K$ is zero in an open neighbourhood of $x$.

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