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LemmaStatement: Literature-sourcedProof: Literature-sourced
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.

External inputs for flat descent of line bundles

PDF p. 4: Weight one units and Picard identification

Statement

For a scheme, an fpqc covering admits effective and fully faithful descent of quasi-coherent modules. If the descended module pulls back to a finite locally free module on every member, it is finite locally free. A jointly surjective étale family is fpqc. On a site satisfying the four hypotheses of Stacks Theorem 59.17.4 quoted in the source record, the structure presheaf is a sheaf.

External input

These are imported established descent and sheaf results with the hypotheses and reading limits in the source records. Their proofs are not locally reconstructed.

Depends on

Used by · 1

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

The Stacks Project, Fpqc descent of quasi-coherent sheaves · Proposition 35.5.2

For an fpqc covering of a scheme, descent data for quasi-coherent modules are effective and the pullback functor to descent data is fully faithful.

Hypotheses: The family is an fpqc covering and the modules in the descent datum are quasi-coherent.

Reading limits: The proposition statement and the beginning of its proof were read; the underlying affine faithfully flat descent proof was not reconstructed.

The Stacks Project, Descent of finiteness properties of modules · Lemma 35.7.6

A quasi-coherent module whose pullbacks to an fpqc covering are finite locally free is finite locally free.

Hypotheses: The module is quasi-coherent and the family is an fpqc covering.

Reading limits: The lemma statement and its short proof were consulted; the cited algebraic descent results were not reconstructed.

The Stacks Project, Meta theorem on quasi-coherent sheaves · Theorem 59.17.4, first conclusion

The structure presheaf is a sheaf on a site satisfying the listed full-subcategory, Zariski-refinement, terminal-object and fpqc-cover hypotheses.

Hypotheses: The site is a full subcategory of schemes over the base, contains the base, has refinements of Zariski covers, and has fpqc covering families.

Reading limits: The theorem statement and proof note were read; its omitted formal arguments were not reconstructed.

The Stacks Project, The fppf topology · Lemma 34.7.2

Every étale covering is an fppf covering.

Hypotheses: A jointly surjective family of étale scheme morphisms.

Reading limits: The statement and short proof were read; the cited morphism lemmas were not reconstructed.

The Stacks Project, The V topology · Lemma 34.10.10, proof

Every fppf covering is an fpqc covering.

Hypotheses: An fppf covering of schemes.

Reading limits: The lemma statement and proof were read; the cited fpqc-topology lemma was not reconstructed.

Sources