Support of a module sheaf

Let $(X,{\mathcal{O}}_X)$ be a ringed space and let $\mathcal{F}$ be an ${\mathcal{O}}_X$-module (Modules on a ringed space). The support of $\mathcal{F}$ is the set of points at which its stalk is nonzero, $\mathrm{Supp}(\mathcal{F}):=\{x\in X:{\mathcal{F}}_x\mathrm{\ne }0\}\subseteq X,$ where ${\mathcal{F}}_x$ denotes the stalk of $\mathcal{F}$ at $x$ (The stalk of a presheaf at a point).

The support is thus a subset of $X$ determined by the stalks of $\mathcal{F}$; it is not determined by the zero set of one chosen global or local section. The definition is invariant under isomorphism: $\mathcal{F}\cong \mathcal{G}$ implies $\mathrm{Supp}(\mathcal{F})=\mathrm{Supp}(\mathcal{G})$. One has $\mathrm{Supp}(0)=\mathrm{\varnothing }$, since the zero module has zero stalks, and, whenever all structure stalks are nonzero (in particular on a scheme), $\mathrm{Supp}({\mathcal{O}}_X)=X$. For an open subscheme $U\subseteq X$ the support of the restriction is $\mathrm{Supp}(\mathcal{F}{\mathrm{\mid }}_U)=\mathrm{Supp}(\mathcal{F})\cap U,$ because the stalk of the restriction at $x\in U$ is ${\mathcal{F}}_x$.