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All subterminal schemes

Jaime Benabent Guerrero

TL;DR

The paper solves the classification problem for subterminal schemes by leveraging the established classification of solid rings (types (1)–(4)) and showing that any affine subterminal scheme is the spectrum of a solid ring. It then analyzes the permissible point structures, stalks, and local-topology constraints, proving that all subterminal schemes can be assembled as colimits of spectra of solid rings, parameterized by data $(e,q,C)$ encoding prime-wise exponents, the presence of a generic point, and a finite-symmetric-difference equivalence class of prime subsets. The resulting full classification specifies the exact point/ stalk configurations and the global topologies, giving an explicit and constructive description of all subterminal schemes. This advances understanding of morphism-uniqueness constraints in Schemes and provides a concrete framework for constructing and recognizing subterminal objects in the broader category of schemes.

Abstract

We classify all subterminal schemes by characterizing their point structure, stalks, and topologies. This extends our previous classification of subterminal affine schemes, which correspond to spectra of solid rings.

All subterminal schemes

TL;DR

The paper solves the classification problem for subterminal schemes by leveraging the established classification of solid rings (types (1)–(4)) and showing that any affine subterminal scheme is the spectrum of a solid ring. It then analyzes the permissible point structures, stalks, and local-topology constraints, proving that all subterminal schemes can be assembled as colimits of spectra of solid rings, parameterized by data encoding prime-wise exponents, the presence of a generic point, and a finite-symmetric-difference equivalence class of prime subsets. The resulting full classification specifies the exact point/ stalk configurations and the global topologies, giving an explicit and constructive description of all subterminal schemes. This advances understanding of morphism-uniqueness constraints in Schemes and provides a concrete framework for constructing and recognizing subterminal objects in the broader category of schemes.

Abstract

We classify all subterminal schemes by characterizing their point structure, stalks, and topologies. This extends our previous classification of subterminal affine schemes, which correspond to spectra of solid rings.

Paper Structure

This paper contains 12 sections, 6 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Spectra of solid rings
  3. Solid rings of type (1)
  4. Solid rings of type (2)
  5. Solid rings of type (3)
  6. Solid rings of type (4)
  7. All subterminal schemes
  8. Point structure and stalks
  9. Possible topologies
  10. Open sets excluding ${\mathbb{Q}}$
  11. Open sets containing ${\mathbb{Q}}$
  12. The Full Classification of Subterminal Schemes

Key Result

Lemma 2.1

Let $S$ be a scheme such that for every affine scheme ${\textup{Spec}}(A)$, there exists at most one morphism ${\textup{Spec}}(A) \to S$. Then $S$ is a subterminal object in the category of schemes.

Theorems & Definitions (12)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 2 more