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.

Every open perfect complex is a retract of a restriction

PDF p. 7: Zariski locality; PDF p. 15: Localization diagram

Statement

Assume the Axiom of Choice and fix the suitable universes of the supplied derived and K-theory constructions. Let $X$ be qcqs and $U \subseteq X$ quasi-compact open with complement $T$ . Every perfect object on $U$ is a direct summand of the restriction of a perfect object on $X$ . Consequently the functor $h P e r f (X) / h {P e r f}_{T} (X) ⟶ h P e r f (U)$ is an equivalence after idempotent completion.

Facts & Assumptions

Given: A perfect object $F$ on $U$ . Assume the Axiom of Choice and use the universes fixed in the Statement.

Goal: Prove density up to retracts with no assertion that $F$ itself extends.

[A1]

Choice is inherited when invoking the full-faithfulness and idempotent-completeness interfaces in [F2]. The Axiom of Choice

[F1]

External extension and morphism input for perfect complexes gives the classical Euler extension criterion and $[F [1]] = - [F]$ .

[F2]

Full faithfulness of the perfect triangulated quotient gives full faithfulness of the actual quotient functor. Perfect complexes on schemes and their support gives idempotent completeness of the target and its homotopy category.

Proof

1.1F1given

In the source-side Euler group of [F1], $[F \oplus F [1]] = 0$ . The zero class belongs to the image of restriction from $X$ , so [F1] gives perfect $E$ on $X$ with $j^{*} E ≃ F \oplus F [1]$ in the ordinary derived category. The canonical inclusion and projection exhibit $F$ as a summand of $j^{*} E$ . No equality with the currently defined $K_{0}$ is used.

2.1A1F2step 1.1∎

A fully faithful functor extends fully faithfully to idempotent completions: on a retract of $A$ determined by an idempotent $e$ , and a retract of $B$ determined by $f$ , its Hom group is the subgroup of maps $u : A \to B$ satisfying $u = f u e$ , which full faithfulness identifies. By step 1.1 every target object is a retract of an image object, so the extension is essentially surjective. The target is already idempotent complete by [F2]. This proves the stated equivalence after idempotent completion.

Depends on

Used by · 2

Dependency tree · 3 levels

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