The Stacks project

Lemma 48.33.3. Let $X$ be a proper scheme over a field $k$. Let $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with $H^ i(K) = 0$ for $i < 0$. Set $\mathcal{F} = H^0(K)$. Let $Z \subset X$ be closed with complement $U = X \setminus U$. Then

\[ H^0_ c(U, K|_ U) \subset H^0(X, \mathcal{F}) \]

is given by those global sections of $\mathcal{F}$ which vanish in an open neighbourhood of $Z$.

Proof. Consider the map $H^0_ c(U, K|_ U) \to H^0_ X(X, K) = H^0(X, K) = H^0(X, \mathcal{F})$ of Remark 48.32.7. To study this we represent $K$ by a bounded complex $\mathcal{F}^\bullet $ with $\mathcal{F}^ i = 0$ for $i < 0$. Then we have by definition

\[ H^0_ c(U, K|_ U) = \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{I}^ n\mathcal{F}^\bullet ) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Ker}}( H^0(X, \mathcal{I}^ n\mathcal{F}^0) \to H^0(X, \mathcal{I}^ n\mathcal{F}^1)) \]

By Artin-Rees (Cohomology of Schemes, Lemma 30.10.3) this is the same as $\mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{I}^ n\mathcal{F})$. Thus the arrow $H^0_ c(U, K|_ U) \to H^0(X, \mathcal{F})$ is injective and the image consists of those global sections of $\mathcal{F}$ which are contained in the subsheaf $\mathcal{I}^ n\mathcal{F}$ for any $n$. The characterization of these as the sections which vanish in a neighbourhood of $Z$ comes from Krull's intersection theorem (Algebra, Lemma 10.51.4) by looking at stalks of $\mathcal{F}$. See discussion in Algebra, Remark 10.51.6 for the case of functions. $\square$

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