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Toolkit for the algebraic geometer

Sourayan Banerjee, Oliver Lorscheid, Alejandro Martínez Méndez, Alejandro Vargas

Abstract

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model spaces via sheaf theory and recover a posteriori the underlying topological spaces that are often present in the construction of such geometric objects. We apply this formalism to semiring schemes and reason why the usual definition of semiring schemes has to be considered as the good approach to the geometry of semirings.

Toolkit for the algebraic geometer

Abstract

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model spaces via sheaf theory and recover a posteriori the underlying topological spaces that are often present in the construction of such geometric objects. We apply this formalism to semiring schemes and reason why the usual definition of semiring schemes has to be considered as the good approach to the geometry of semirings.

Paper Structure

This paper contains 29 sections, 14 theorems, 7 equations, 1 table.

Table of Contents

  1. Recap on sheaf theory
  2. Sites
  3. Sheaves
  4. The Yoneda embedding
  5. s-Schemes
  6. Principal opens
  7. Principal opens for the Euclidean topology
  8. Principal opens for the Zariski topology
  9. s-Presentations
  10. Morphisms of s-schemes
  11. Smooth manifolds as s-schemes
  12. Usual schemes as s-schemes
  13. Points for s-schemes
  14. Stone Duality
  15. Open subschemes
  16. ...and 14 more sections

Key Result

Lemma 1

Let ${\mathcal{P}}$ be a class of morphisms and ${\mathcal{T}}=\langle {\mathcal{P}} \rangle_\textup{can}$. Then ${\mathcal{P}}_{\mathcal{T}}\subset{\mathcal{P}}$, with equality if and only if ${\mathcal{P}}$ satisfies properties PO1--PO3. Conversely, a Grothendieck pretopology ${\mathcal{T}}$ is co

Theorems & Definitions (39)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 1
  • Remark 1
  • Example 5
  • Definition 1
  • Remark 2
  • Example 6: Atlas of a covering family
  • ...and 29 more