The Stacks project

Definition 6.27.1. Let $X$ be a topological space.

1. Let $x \in X$ be a point. Denote $i_ x : \{ x\} \to X$ the inclusion map. Let $A$ be a set and think of $A$ as a sheaf on the one point space $\{ x\}$. We call $i_{x, *}A$ the skyscraper sheaf at $x$ with value $A$.

2. If in (1) above $A$ is an abelian group then we think of $i_{x, *}A$ as a sheaf of abelian groups on $X$.

3. If in (1) above $A$ is an algebraic structure then we think of $i_{x, *}A$ as a sheaf of algebraic structures.

4. If $(X, \mathcal{O}_ X)$ is a ringed space, then we think of $i_ x : \{ x\} \to X$ as a morphism of ringed spaces $(\{ x\} , \mathcal{O}_{X, x}) \to (X, \mathcal{O}_ X)$ and if $A$ is a $\mathcal{O}_{X, x}$-module, then we think of $i_{x, *}A$ as a sheaf of $\mathcal{O}_ X$-modules.

5. We say a sheaf of sets $\mathcal{F}$ is a skyscraper sheaf if there exists a point $x$ of $X$ and a set $A$ such that $\mathcal{F} \cong i_{x, *}A$.

6. We say a sheaf of abelian groups $\mathcal{F}$ is a skyscraper sheaf if there exists a point $x$ of $X$ and an abelian group $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of abelian groups.

7. We say a sheaf of algebraic structures $\mathcal{F}$ is a skyscraper sheaf if there exists a point $x$ of $X$ and an algebraic structure $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of algebraic structures.

8. If $(X, \mathcal{O}_ X)$ is a ringed space and $\mathcal{F}$ is a sheaf of $\mathcal{O}_ X$-modules, then we say $\mathcal{F}$ is a skyscraper sheaf if there exists a point $x \in X$ and a $\mathcal{O}_{X, x}$-module $A$ such that $\mathcal{F} \cong i_{x, *}A$ as sheaves of $\mathcal{O}_ X$-modules.