Lemma 57.11.3 (0G03)—The Stacks project

Lemma 57.11.3. 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 $x$.

Proof. Observe that $H^ i(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ x)$ is equal to $K_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \kappa (x)$. Hence part (5) follows from More on Algebra, Lemma 15.76.4. Part (6) follows from part (5). Part (1) follows from part (5), Lemma 57.11.2, and the fact that the Matlis dual of $\kappa (x)$ is $\kappa (x)$. Part (2) follows from part (1). Part (3) follows from part (5) and the fact that $\mathop{\mathrm{Ext}}\nolimits ^ i(K, \mathcal{O}_ x) = H^ i(X, K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ x)$ by Cohomology, Lemma 20.50.5. Part (4) follows from part (3) and the fact that $K \cong (K^\vee )^\vee $ by the lemma just cited. $\square$

The Stacks project

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