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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Extension by zero for abelian sheaves on an open subspace

Definition

Let $j : U ↪ X$ be the inclusion of an open subspace, and let $F$ be a sheaf of abelian groups on $U$ .

For an open set $V \subseteq X$ , define $(j_{!} F) (V) : = {s \in F (V \cap U) : Supp (s) is closed in V} .$ Equivalently, $s$ lies in $(j_{!} F) (V)$ exactly when for every $x \in V ∖ U$ there exists an open neighbourhood $W_{x} \subseteq V$ with

s ∣_{W_{x} \cap U} = 0.

Restriction maps are inherited from $F$ , so $j_{!} F$ is a subsheaf of $j_{*} F$ . When $V \subseteq U$ , the support condition is vacuous and

(j_{!} F) (V) = F (V) .

This is the extension by zero of $F$ along $j$ . It is distinct in general from the ordinary direct image $j_{*} F$ .

Extension by zero for abelian sheaves on an open subspace

Definition

Depends on

Dependency tree · 2 levels

Sources