Support of a section of a sheaf

Let $X$ be a topological space, let $\mathcal{F}$ be a presheaf of $k$-vector spaces (or of abelian groups) on $X$ (A presheaf on a topological space), let $U\subseteq X$ be open and let $s\in \mathcal{F}(U)$. The support of $s$ is $\mathrm{Supp}(s):=\{x\in U:s_x\mathrm{\ne }0\}\subseteq U,$ where $s_x\in {\mathcal{F}}_x$ denotes the germ of $s$ at $x$ in the stalk ${\mathcal{F}}_x$ (The stalk of a presheaf at a point, Germs of sections). A section $s\in \mathcal{F}(X)$ is compactly supported when its support is a compact subspace of $X$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right). For a subspace $Z\subseteq X$ and a presheaf $\mathcal{G}$ on $Z$, the support of $t\in \mathcal{G}(Z)$ is defined by the same condition using the stalks of $\mathcal{G}$. The support is not assumed closed in $X$ nor compact.

When $\mathcal{F}$ is a sheaf of abelian groups (in particular of $k$-vector spaces) the following three consistency facts hold; they are immediate consequences of the definitions above and are recorded here because every later compact-support statement uses them.

1. $\mathrm{Supp}(s)$ is closed in $U$. Indeed, if $x\mathrm{\notin }\mathrm{Supp}(s)$ then $s_x=0$ in the stalk; equality of germs means that $s$ and the zero section agree on some open neighbourhood $W\subseteq U$ of $x$ (Germs of sections), so $s$ vanishes on the open neighbourhood $W$ of $x$, and $U\setminus \mathrm{Supp}(s)$ is therefore open in $U$.
2. $\mathrm{Supp}(s)=\mathrm{\varnothing }$ if and only if $s=0$. If every germ of $s$ is zero then $s=0$ by the locality axiom of a sheaf applied to the cover of $U$ by its open subsets (A section of a sheaf of groups is zero exactly when all of its germs are zero); the converse is immediate, since the zero section has zero germ at every point.
3. $\mathrm{Supp}(s{\mathrm{\mid }}_V)=\mathrm{Supp}(s)\cap V$ for every open $V\subseteq U$. Restriction is compatible with the germ maps, so $(s{\mathrm{\mid }}_V{)}_x=s_x$ for every $x\in V$ (Germs of sections).

For a sheaf, a section whose support is contained in a compact subset $C\subseteq U$ is compactly supported: its support is closed in $C$ by fact 1, hence compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact). For a global section ($U=X$), this applies to any compact subset of $X$. For a local section, containment in a compact subset of $X$ alone does not suffice: the constant nonzero section of the constant sheaf (The constant sheaf is the sheaf of locally constant functions) on $U=(0,1)\subseteq X=[0,1]$ has support $U$, which is not compact although it is contained in the compact $X$ (A subset of ${\mathbb{R}}^n$ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology, $n=1$). The open cover $\{(1\mathrm{/}n,1):n\ge 2\}$ of $U$ has no finite subcover. No choice principle is used: germs, stalks, supports and restrictions are elementary.

• For a sheaf $\mathcal{F}$ of abelian groups and sections $s,t\in \mathcal{F}(U)$ one has $\mathrm{Supp}(s+t)\subseteq \mathrm{Supp}(s)\cup \mathrm{Supp}(t)$ and $\mathrm{Supp}(-s)=\mathrm{Supp}(s)$: if both germs vanish at $x$ then $s$ and $t$ vanish on a common open neighbourhood of $x$, hence so does $s+t$, and if $s$ vanishes on a neighbourhood of $x$ then so does $-s$. These inclusion facts are used silently in the patching arguments of the compact-support calculus.