The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 17.5.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $U \subset X$ open.

  1. The support of $s \in \mathcal{F}(U)$ is closed in $U$.

  2. The support of $fs$ is contained in the intersections of the supports of $f \in \mathcal{O}_ X(U)$ and $s \in \mathcal{F}(U)$.

  3. The support of $s + s'$ is contained in the union of the supports of $s, s' \in \mathcal{F}(U)$.

  4. The support of $\mathcal{F}$ is the union of the supports of all local sections of $\mathcal{F}$.

  5. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of $\mathcal{O}_ X$-modules, then the support of $\varphi (s)$ is contained in the support of $s \in \mathcal{F}(U)$.

Proof. This is true because if $s_ x = 0$, then $s$ is zero in an open neighbourhood of $x$ by definition of stalks. Similarly for $f$. Details omitted. $\square$


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