Definition 17.5.1. Let (X, \mathcal{O}_ X) be a ringed space. Let \mathcal{F} be a sheaf of \mathcal{O}_ X-modules.
The support of \mathcal{F} is the set of points x \in X such that \mathcal{F}_ x \not= 0.
We denote \text{Supp}(\mathcal{F}) the support of \mathcal{F}.
Let s \in \Gamma (X, \mathcal{F}) be a global section. The support of s is the set of points x \in X such that the image s_ x \in \mathcal{F}_ x of s is not zero.
Of course the support of a local section is then defined also since a local section is a global section of the restriction of \mathcal{F}.
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.
The support of s \in \mathcal{F}(U) is closed in U.
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).
The support of s + s' is contained in the union of the supports of s, s' \in \mathcal{F}(U).
The support of \mathcal{F} is the union of the supports of all local sections of \mathcal{F}.
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
In general the support of a sheaf of modules is not closed. Namely, the sheaf could be an abelian sheaf on \mathbf{R} (with the usual archimedean topology) which is the direct sum of infinitely many nonzero skyscraper sheaves each supported at a single point p_ i of \mathbf{R}. Then the support would be the set of points p_ i which may not be closed.
Another example is to consider the open immersion j : U = (0 , \infty ) \to \mathbf{R} = X, and the abelian sheaf j_!\underline{\mathbf{Z}}_ U. By Sheaves, Section 6.31 the support of this sheaf is exactly U.
Lemma 17.5.3. Let X be a topological space. The support of a sheaf of rings is closed.
Proof. This is true because (according to our conventions) a ring is 0 if and only if 1 = 0, and hence the support of a sheaf of rings is the support of the unit section. \square
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