### Abstract

We formalize mainstream structures in algebraic geometry culminating in Grothendieck's schemes: presheaves of rings, sheaves of rings, ringed spaces, locally ringed spaces, affine schemes and schemes. We prove that the spectrum of a ring is a locally ringed space, hence an affine scheme. Finally, we prove that any affine scheme is a scheme.

### License

BSD License

### Topics

- Mathematics/Algebra
- Mathematics/Geometry

### Session Grothendieck_Schemes

- Set_Extras
- Group_Extras
- Topological_Space
- Comm_Ring
- Scheme

### Depends on

- A Case Study in Basic Algebra

## Cite

×

Grothendieck_Schemes-AFP

    @article{Grothendieck_Schemes-AFP,
      author  = {Anthony Bordg and Lawrence C. Paulson and Wenda Li},
      title   = {Grothendieck's Schemes in Algebraic Geometry},
      journal = {Archive of Formal Proofs},
      month   = {March},
      year    = {2021},
      note    = {\url{https://isa-afp.org/entries/Grothendieck_Schemes.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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