LemmaStatement: AI-generatedProof: AI-generated

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.

Restriction, stalks and cohomological support

PDF p. 16: Perfect complexes with support

Statement

Assume the Axiom of Choice. Let $X$ be a scheme, $U \subseteq X$ open, and $x \in X$ . Restriction and taking the stalk at $x$ induce exact functors $D (X) \to D (U)$ and $D (X) \to D (O_{X, x})$ . For $E \in D (X)$ put $Supp (E) = ⋃_{n} {x : H^{n} (E)_{x} \neq 0}$ . This support is invariant under equivalence. For $T \subseteq X$ , it is contained in $T$ exactly when the stalk complex at every point outside $T$ is acyclic. If $T$ is closed, this is equivalent to $E ∣_{X ∖ T} ≃ 0$ . Support is unchanged by shifts, is contained in the union of the supports of the first two objects of a cofiber sequence, and cannot increase on passing to a retract.

Facts & Assumptions

Given: $X$ , $U$ , $x$ , $T$ and $E$ as stated.

Goal: Establish exactness and the support assertions.

[A1]

Choice is used to invoke the derived enhancement in [F1], in suitable fixed universes; no extra choice is needed by the stalkwise argument. The Axiom of Choice

[F1]

External input on derived enhancements supplies the derived localizations and represents cofibers by cones.

Proof

1.1F1given

Restriction is exact because sheaf exactness is local. Stalks are exact: a germ relation or a local preimage is witnessed on a sufficiently small neighbourhood. Both operations therefore commute with cohomology and preserve quasi-isomorphisms. They preserve zero and mapping cones degreewise, and hence induce exact functors by [F1].

2.1F1step 1.1

Equivalent objects have isomorphic cohomology sheaves. By step 1.1, the cohomology of a stalk complex is the stalk of the cohomology sheaf, proving the stalkwise criterion. A sheaf vanishes exactly when all its stalks vanish, so, when $X ∖ T$ is open, the same criterion is equivalent to vanishing of all cohomology sheaves of the restriction, hence to its being zero in the derived localization.

3.1F1step 1.1step 2.1∎

Shifts merely reindex cohomology. The long exact cohomology sequence of a cone shows that a cofiber stalk is acyclic when the two preceding stalks are. A retract induces retracts on cohomology stalks, so its support is contained in that of the original object. These observations prove the remaining assertions.

Depends on

Used by · 4

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.