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On Gluing Data, Finite Ringed Spaces and schemes

Rita Fioresi, Angelica Simonetti, Ferdinando Zanchetta

Abstract

From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link important geometric objects, such as differentiable manifolds or schemes, to certain finite ringed spaces arising from sheaves on 2 dimensional semisimplicial sets, thus opening the door to their applications in fields such as discrete differential geometry.

On Gluing Data, Finite Ringed Spaces and schemes

Abstract

From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link important geometric objects, such as differentiable manifolds or schemes, to certain finite ringed spaces arising from sheaves on 2 dimensional semisimplicial sets, thus opening the door to their applications in fields such as discrete differential geometry.
Paper Structure (16 sections, 17 theorems, 33 equations, 2 figures)

This paper contains 16 sections, 17 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Semisimplicial sets, graphs and sheaves
  3. The categories of (semi)simplicial sets and digraphs
  4. Sheaves on preordered sets
  5. Finite ringed spaces, gluing data and semisimplicial sets
  6. Gluing data
  7. Gluing data for 2-dimensional semisimplicial sets
  8. Gluing Ringed Spaces
  9. Finite ringed spaces, schemes and semisimplicial sets
  10. Finite ringed spaces and schemes
  11. The category $\mathcal{C}_{\mathrm{Sch}}^2$
  12. The category $\mathcal{C}_{\mathrm{Sch}}^2$ and weak equivalences
  13. Schematic Localization of $\mathcal{C}_{\mathrm{Sch}}^2$
  14. The equivalence of categories between $\mathcal{C}_{\mathrm{Sch}}^2[\mathcal{W}^{-1}]$ and schemes
  15. Finite spaces and manifolds
  16. ...and 1 more sections

Key Result

Theorem 1.1

The category $\mathcal{C}_{\mathrm{Sch}}^2[\mathcal{W}^{-1}]$ is equivalent to the category of quasi-compact and semi-separated schemes.

Figures (2)

  • Figure 1: The category $\Xi$ and a gluing cube
  • Figure 2: All the simplices of the degenerate expansion $\boldsymbol{G}_\bullet$ of $G_\bullet: u\rightarrow v$

Theorems & Definitions (68)

  • Theorem 1.1: \ref{['main-res']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.8
  • Theorem 2.11
  • Remark 2.12
  • ...and 58 more