Alphabeta Math
RemarkRemark: 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.

Inverse image of sheaves and pullback of modules are not the same construction

The notation f−1 and f∗ describes two different operations.

The inverse image f−1G only uses the underlying continuous map f:X→Y and is defined for any sheaf on Y (Inverse image presheaf and inverse image sheaf). It changes the topological base space but does not yet change scalars.

The pullback f∗G is defined only after choosing a morphism of ringed spaces, and for modules it adds the scalar extension

OX⊗f−1OY−

to the inverse image (Pullback of a module along a morphism of ringed spaces).

Thus f−1 is the sheaf-theoretic inverse image, whereas f∗ is the module-theoretic pullback built from f−1 plus tensoring.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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