CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Extension by zero can be strictly smaller than direct image on a punctured interval

Statement refuted

For every open immersion $j : U ↪ X$ and every sheaf of abelian groups $F$ on $U$ , one has $j_{!} F = j_{*} F$ .

Facts & Assumptions

Given: The open immersion $j : U = (- 1, 0) \cup (0, 1) ↪ X = (- 1, 1)$ and the constant sheaf $\underline{Z}$ on $U$ .

[F1]

For an open immersion, direct image is computed by intersection: $(j_{*} \underline{Z}) (V) = \underline{Z} (V \cap U)$ (Direct image along an open immersion is restriction-compatible intersection).

[F2]

Extension by zero consists of sections whose support is closed in the test open (Extension by zero for abelian sheaves on an open subspace).

Counterexample

technique · direct

1.1F1givenchoose

By [F1], the global section set of $j_{*} \underline{Z}$ is $(j_{*} \underline{Z}) (X) = \underline{Z} (U) ≅ Z \times Z,$ because $U$ has two connected components. Let $s$ be the section that is $1$ on both components.

2.1F2step 1.1

The germ of $s$ is nonzero at every point of $U$ , so its support is all of $U$ . But $U = (- 1, 0) \cup (0, 1)$ is not closed in $X = (- 1, 1)$ , since its closure contains $0$ . Therefore [F2] shows that $s \notin (j_{!} \underline{Z}) (X)$ .

3.1step 1.1step 2.1∎

Thus $s$ lies in $j_{*} \underline{Z}$ but not in $j_{!} \underline{Z}$ , so the two sheaves are not equal. This refutes the statement.

Depends on

Used by

The objectwise cokernel presheaf can fail to be a sheaf Counterexample

Dependency tree · two levels

5 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

Ravi Vakil, The Rising Sea, nearby discussion around Section 2.7 (standard reference, not scraped)
The Stacks Project, Section 6.31 (standard reference, not scraped)