The Stacks project

6.27 Skyscraper sheaves and stalks

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.

Lemma 6.27.2. Let $X$ be a topological space, $x \in X$ a point, and $A$ a set. For any point $x' \in X$ the stalk of the skyscraper sheaf at $x$ with value $A$ at $x'$ is

\[ (i_{x, *}A)_{x'} = \left\{ \begin{matrix} A & \text{if} & x' \in \overline{\{ x\} } \\ \{ *\} & \text{if} & x' \not\in \overline{\{ x\} } \end{matrix} \right. \]

A similar description holds for the case of abelian groups, algebraic structures and sheaves of modules.

Proof. Omitted. $\square$

Lemma 6.27.3. Let $X$ be a topological space, and let $x \in X$ a point. The functors $\mathcal{F} \mapsto \mathcal{F}_ x$ and $A \mapsto i_{x, *}A$ are adjoint. In a formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(\mathcal{F}_ x, A) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X)}(\mathcal{F}, i_{x, *}A). \]

A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\mathcal{F}_ x, A) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, i_{x, *}A). \]

Proof. Omitted. Hint: The stalk functor can be seen as the pullback functor for the morphism $i_ x : \{ x\} \to X$. Then the adjointness follows from adjointness of $i_ x^{-1}$ and $i_{x, *}$ (resp. $i_ x^*$ and $i_{x, *}$ in the case of sheaves of modules). $\square$

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