ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A short exact sequence of abelian groups gives a short exact sequence of skyscraper sheaves

Example

Let

0 \to A^{'} \to A \to A^{''} \to 0

be a short exact sequence of abelian groups, and let $x \in X$ . Then applying the skyscraper construction at $x$ gives a short exact sequence of sheaves

0 \to i_{x, *} A^{'} \to i_{x, *} A \to i_{x, *} A^{''} \to 0.

Facts & Assumptions

Given: A short exact sequence of abelian groups $0 \to A^{'} \to A \to A^{''} \to 0$ and a point $x \in X$ .

[F1]

A stalk is the colimit of sections over neighbourhoods of the point, and for the skyscraper sheaf those section groups are $A$ on neighbourhoods containing $x$ and $0$ on neighbourhoods omitting $x$ (The stalk of a presheaf at a point, A skyscraper sheaf of abelian groups at a point).

[L1]

Exactness of a sequence of abelian sheaves is equivalent to exactness on every stalk (A sequence of abelian sheaves is exact exactly when it is exact on every stalk).

[L2]

Short exactness of the displayed sheaf sequence is exactness in the sense of sheaves (Exact sequences of sheaves).

Verification

technique · direct

1.1F1given

Fix a point $y \in X$ . If every neighbourhood of $y$ contains $x$ , then [F1] shows that each stalk in the displayed skyscraper sequence is the original group at $y$ , so the stalk sequence is exactly $0 \to A^{'} \to A \to A^{''} \to 0.$ If some neighbourhood of $y$ omits $x$ , then every smaller neighbourhood also omits $x$ , so [F1] makes all three stalks equal to $0$ and the stalk sequence is $0 \to 0 \to 0 \to 0.$ In either case the stalk sequence is exact.

2.1L1L2step 1.1∎

The stalk sequence is therefore exact at every point, so [L1] implies that the skyscraper sequence is exact as a sequence of sheaves. By [L2], this is exactly the claimed short exact sequence of skyscraper sheaves.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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 and Section 17.3 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Example 2.2.12 and Exercise 2.6.D (standard reference, not scraped)