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Localization and coalescence of condensed ringed spaces

Naoto Fukutomi

Abstract

Gillam proved that the category of locally ringed spaces admits a fully faithful embedding into a certain category, which has a right adjoint that maps some simple objects to the spectra of rings. In this paper, we use condensed mathematics to define an analogous category and its coreflective full subcategories, and prove that one of their coreflections maps certain simple objects to the adic spectra of certain Huber pairs.

Localization and coalescence of condensed ringed spaces

Abstract

Gillam proved that the category of locally ringed spaces admits a fully faithful embedding into a certain category, which has a right adjoint that maps some simple objects to the spectra of rings. In this paper, we use condensed mathematics to define an analogous category and its coreflective full subcategories, and prove that one of their coreflections maps certain simple objects to the adic spectra of certain Huber pairs.
Paper Structure (91 sections, 150 theorems, 278 equations)

This paper contains 91 sections, 150 theorems, 278 equations.

Table of Contents

  1. Review of condensed mathematics
  2. Light profinite sets
  3. Condensed sets
  4. Condensed abelian groups
  5. Condensed rings
  6. Condensed modules
  7. Condensed algebras
  8. Condensed sets represented by topological spaces
  9. Coalescence
  10. The sequence space
  11. Coalescent modules
  12. The coalescence functor
  13. Internal projectivity of the sequence space
  14. Coalescence and other constructions
  15. Valuations on condensed rings
  16. ...and 76 more sections

Key Result

Theorem 1

(Gillam:paper, Theorem 2) The functor $\mathcal{M} : \mathbf{LRS} \to \mathbf{PRS}$ admits a right adjoint. For $X \in |\mathbf{PRS}|$, the image of $X$ under this right adjoint is called the localization of $X$, and is denoted by $X^{\mathrm{loc}}$.

Theorems & Definitions (341)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Definition 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.4
  • proof
  • Corollary 1.5
  • ...and 331 more