Lemma 81.10.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $i : Z \to X$ be a closed immersion of finite presentation. Let $Q \in D_\mathit{QCoh}(\mathcal{O}_ X)$ be supported on $|Z|$. Let $\overline{x}$ be a geometric point of $X$ and let $I_{\overline{x}} \subset \mathcal{O}_{X, \overline{x}}$ be the stalk of the ideal sheaf of $Z$. Then the cohomology modules $H^ n(Q_{\overline{x}})$ are $I_{\overline{x}}$-power torsion (see More on Algebra, Definition 15.88.1).
Proof. Choose an affine scheme $U$ and an étale morphism $U \to X$ such that $\overline{x}$ lifts to a geometric point $\overline{u}$ of $U$. Then we can replace $X$ by $U$, $Z$ by $U \times _ X Z$, $Q$ by the restriction $Q|_ U$, and $\overline{x}$ by $\overline{u}$. Thus we may assume that $X = \mathop{\mathrm{Spec}}(A)$ is affine. Let $I \subset A$ be the ideal defining $Z$. Since $i : Z \to X$ is of finite presentation, the ideal $I = (f_1, \ldots , f_ r)$ is finitely generated. The object $Q$ comes from a complex of $A$-modules $M^\bullet $, see Derived Categories of Spaces, Lemma 75.4.2 and Derived Categories of Schemes, Lemma 36.3.5. Since the cohomology sheaves of $Q$ are supported on $Z$ we see that the localization $M^\bullet _ f$ is acyclic for each $f \in I$. Take $x \in H^ p(M^\bullet )$. By the above we can find $n_ i$ such that $f_ i^{n_ i} x = 0$ in $H^ p(M^\bullet )$ for each $i$. Then with $n = \sum n_ i$ we see that $I^ n$ annihilates $x$. Thus $H^ p(M^\bullet )$ is $I$-power torsion. Since the ring map $A \to \mathcal{O}_{X, \overline{x}}$ is flat and since $I_{\overline{x}} = I\mathcal{O}_{X, \overline{x}}$ we conclude. $\square$
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