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
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
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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)