Let $X$ be a non-empty topological space. What are some interesting examples of a sheaf of sets $F$ over $X$ such that for some non-empty open $U \subset X$, $F(U) = \emptyset$?

Here is an example that I think is interesting:

Let $X$ be a topological space and $U \subset X$ be an open subset. Then one can define a sheaf of sets $F$ on $X$ as follows:

  • If $V \subset X$ is open and $V \subset U$ then $F(V) = \{e\}$, where $\{e\}$ is any one element set.
  • If $V \subset X$ is open and $V \not\subset U$ then $F(V) = \emptyset$.

The restriction maps are obviously the identity map on $\{e\}$ and the empty maps. Ravi Vakil in his online notes on Algebraic Geometry calls this sheaf "the indicator sheaf".

This sheaf is not trivial in the sense that one can use it to show that if a morphism of sheaves of sets on $X$ is a monomorphism in the category of sheaves of sets on $X$, then all its component maps are injective.

If I am not mistaken, one can also define a skyscraper sheaf at a point $p \in X$ such that if $U$ is an open subset of $X$ containing $p$, then the set of sections over $U$ is empty.

What are some other interesting examples of sheaves of sets with no sections over a non-empty open subset?

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    $\begingroup$ The skyscraper sheaf at a point $p$ is a sheaf such that the set of sections over an open set not containing $p$ is empty. $\endgroup$ – Zhen Lin Nov 23 '11 at 8:41
  • $\begingroup$ Well then the set of sections over the empty set would be empty, which is not what is usually the condition assumed. $\endgroup$ – Rankeya Nov 23 '11 at 8:47
  • $\begingroup$ So this the definition of a skyscraper sheaf that I am using (this is the definition that Ravi Vakil uses in his online notes): Suppose $X$ is a topological space with a point $p \in X$, and $S$ is a set. Then $i_{p,{*}}S$ defined by $i_{p,{*}}S(U) = S$ if $p \in U$ and $i_{p,{*}}S(U) = \{t\}$ for $p \notin U$ and some singleton $\{t\}$, is a sheaf called the skyscraper sheaf. All I am observing is that we can set $S$ to be the empty set. $\endgroup$ – Rankeya Nov 23 '11 at 8:50
  • $\begingroup$ Apologies, I meant trivial (in the sense of being a singleton) rather than empty. Sorry for the confusion. $\endgroup$ – Zhen Lin Nov 23 '11 at 15:44
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    $\begingroup$ @QiL: Of course, $F(X)$ could be empty! Regards, $\endgroup$ – Matt E Nov 23 '11 at 18:36

A lot of sheaves can be thought of as functions satisfying some differential equation(s) (e.g. the sheaf of holomorphic functions can be described as those satisfying the Cauchy-Riemann equation). Often solutions to these differential equations will exist locally by general facts, but these general facts won't necessarily give you global solutions, especially if you're on a compact manifold. For instance, the sheaf of holomorphic functions on a compact complex manifold restricted to be zero at one point and one at another is quite big, but it has no global sections.


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