ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge 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.

Pullback carries a free module to the corresponding free module

Example

Let

(f, f^{♯}) : (X, O_{X}) ⟶ (Y, O_{Y})

be a morphism of ringed spaces. For every integer $n \geq 0$ , one has a canonical isomorphism

f^{*} (O_{Y}^{n}) ≅ O_{X}^{n} .

Facts & Assumptions

Given: A morphism of ringed spaces $(f, f^{♯}) : (X, O_{X}) \to (Y, O_{Y})$ and an integer $n \geq 0$ .

[F1]

Pullback is defined by $f^{*} G = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} G$ (Pullback of a module along a morphism of ringed spaces).

[L1]

Pullback is a left adjoint and therefore preserves finite coproducts (Pullback of modules is left adjoint to pushforward).

Verification

technique · direct

1.1L1given

The sheaf $O_{Y}^{n}$ is the finite direct sum of $n$ copies of $O_{Y}$ , with the case $n = 0$ giving the zero sheaf. Since [L1] makes $f^{*}$ a left adjoint, it preserves these finite direct sums. Thus $f^{*} (O_{Y}^{n}) ≅ (f^{*} O_{Y})^{n} .$

2.1F1step 1.1∎

Applying [F1] to $G = O_{Y}$ gives $f^{*} O_{Y} = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} O_{Y} ≅ O_{X},$ because tensoring a module over a ring with the ring itself leaves the module unchanged. Substituting this into step 1.1 yields $f^{*} (O_{Y}^{n}) ≅ O_{X}^{n} .$

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

The Stacks Project, Section 6.26 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Chapter 6 (standard reference, not scraped)