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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