Skyscraper sheaf?

Question

I was wondering whether someone could provide me with an easy definition of a skyscraper sheaf and, more importantly, with a visualization of it.

Thanks in advance.

Answer 1

This is my take: a skyscraper sheaf $\mathcal F$ on a topological space $X$ is given by a specific point $x\in X$ and a ring/group/whatever $A$.

We then have, for any open $U\subseteq X$, that $$\mathcal F(U)=\cases{0& if $x\notin U$\\A& if $x\in U$}$$ So it's basically saying "Everything that matters about this sheaf is in the point $x$", hence the idea that you've placed a lone skyscraper at the point $x$, which is where the name comes from.