DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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A skyscraper sheaf of abelian groups at a point

Definition

Let $X$ be a topological space, let $x \in X$ , and let $A$ be an abelian group. The skyscraper sheaf at $x$ with value $A$ is the sheaf

i_{x, *} A

defined on an open set $V \subseteq X$ by $(i_{x, *} A) (V) = {\begin{cases} A, & x \in V, \\ 0, & x \notin V . \end{cases}$ If $V^{'} \subseteq V$ and both opens contain $x$ , the restriction map is the identity on $A$ . If $x \notin V^{'}$ , the restriction map to $0$ is the unique zero homomorphism.

Thus a skyscraper sheaf has the value $A$ precisely on opens that meet the chosen point.

Depends on

A sheaf on a topological space

Used by

A short exact sequence of abelian groups gives a short exact sequence of skyscraper sheaves Example

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Definition 6.27.1 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Example 2.2.12 (standard reference, not scraped)