Friday, Jul 24, 2026 The claims desk. Receipts included. POWERED BY LENZ ↗
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SCIENCE

The Claim

A scheme is equivalent to a Zariski sheaf on the category of affine schemes that is locally representable by affine schemes.

Verified by Lenz ·

The Short Version

The claim states a standard characterization of schemes in functorial language. Authoritative sources confirm that schemes are exactly Zariski sheaves on affine schemes that admit a local affine presentation. The main caveat is precision: the local affine pieces must glue as open subfunctors or open immersions on the big affine Zariski site, not via arbitrary maps.

Caveats

  • A fully precise statement requires an open affine cover: the maps from affine representables to the sheaf should be open immersions/open subfunctors.
  • The relevant setting is the big affine Zariski site; omitting the site can create ambiguity about what "Zariski sheaf" and "locally" mean.
  • If "locally representable" were read as allowing arbitrary affine-covering maps, the statement would be too broad and could suggest objects more general than schemes.

The Receipts

  1. Section 87.2 (0AHY): Formal schemes à la EGA—The Stacks project

    The Stacks project

  2. Schemes (Stacks Project, Schemes chapter PDF)

    Stacks Project

  3. Topologies on Schemes

    Stacks Project

  4. On functors that are schemes

    arXiv

  5. schemes as sheaves on affine schemes

    nLab

  6. Tag 01IQ: Schemes as functors

    Stacks Project

  7. Schemes as functors of points (linked lecture notes)

    Columbia University (lecture-linked notes)

  8. MATH 245C (An Introduction to Algebraic Stacks) – 2022-05-17 notes

    Stanford University (Vakil course notes)

  9. Schemes and sheaves

    Rocky Mountain Journal of Mathematics (Project Euclid)

  10. Tag 020N: The Zariski topology

    Stacks Project

+ 16 more sources — see the full list on Lenz

Filed Under

Affine SchemeSchemeZariski Topology