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Extending Chevalley's Theorem: A Topological Characterization of Constructibility and its Generalization Beyond Noetherian Spaces

Jiawei Sheng

TL;DR

The paper introduces the topological notion of good maps to capture a purely topological manifestation of constructibility preservation, providing a Chevalley-type criterion in Noetherian spaces and a broader generalization beyond Noetherian contexts. It proves that every morphism locally of finite type is good, yielding a generalized Chevalley theorem for schemes with Noetherian underlying spaces and establishing Jacobson ascent. The framework applies to sober categories such as adic and perfectoid spaces and remains stable under formal completion, suggesting extensions to formal and non-Archimedean geometry. An illustrative example with schemes over Spec(O_{C_p}) demonstrates the practical relevance in arithmetic geometry.

Abstract

We introduce the notion of a good map between topological spaces: a continuous map $f:X \to Y$ is *good* if for every non-empty irreducible locally closed subset $U \subseteq X$, there exists a non-empty open subset $W \subseteq Y$ such that $W \cap f(U) = W \cap \overline{f(U)} \neq \varnothing$. In Noetherian spaces, this condition is equivalent to preserving constructible subsets (Theorem 2.5), giving a purely topological characterization of Chevalley's theorem. Without the Noetherian assumption, the good property continues to make sense and serves as a reasonable generalization. We establish basic properties of good maps and introduce a weaker variant, *weak good maps*. In algebraic geometry, we prove that **every morphism locally of finite type is good** (Theorem 4.1). From this we obtain a generalization of Chevalley's theorem for morphisms locally of finite type whose underlying topological spaces are Noetherian (Theorem 4.3), and an elementary proof of Jacobson ascent (Corollary 4.4). The theory is developed for sober spaces and therefore applies not only to schemes but also to other geometric categories such as **adic spaces** and **perfectoid spaces**. The good property is stable under formal completion, suggesting extensions to formal and non-Archimedean geometry.

Extending Chevalley's Theorem: A Topological Characterization of Constructibility and its Generalization Beyond Noetherian Spaces

TL;DR

The paper introduces the topological notion of good maps to capture a purely topological manifestation of constructibility preservation, providing a Chevalley-type criterion in Noetherian spaces and a broader generalization beyond Noetherian contexts. It proves that every morphism locally of finite type is good, yielding a generalized Chevalley theorem for schemes with Noetherian underlying spaces and establishing Jacobson ascent. The framework applies to sober categories such as adic and perfectoid spaces and remains stable under formal completion, suggesting extensions to formal and non-Archimedean geometry. An illustrative example with schemes over Spec(O_{C_p}) demonstrates the practical relevance in arithmetic geometry.

Abstract

We introduce the notion of a good map between topological spaces: a continuous map is *good* if for every non-empty irreducible locally closed subset , there exists a non-empty open subset such that . In Noetherian spaces, this condition is equivalent to preserving constructible subsets (Theorem 2.5), giving a purely topological characterization of Chevalley's theorem. Without the Noetherian assumption, the good property continues to make sense and serves as a reasonable generalization. We establish basic properties of good maps and introduce a weaker variant, *weak good maps*. In algebraic geometry, we prove that **every morphism locally of finite type is good** (Theorem 4.1). From this we obtain a generalization of Chevalley's theorem for morphisms locally of finite type whose underlying topological spaces are Noetherian (Theorem 4.3), and an elementary proof of Jacobson ascent (Corollary 4.4). The theory is developed for sober spaces and therefore applies not only to schemes but also to other geometric categories such as **adic spaces** and **perfectoid spaces**. The good property is stable under formal completion, suggesting extensions to formal and non-Archimedean geometry.

Paper Structure

This paper contains 19 sections, 12 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Good Maps
  3. Definition
  4. An equivalent characterization
  5. Characterization in Noetherian Spaces
  6. Basic Properties
  7. Local Nature of Good Maps
  8. Weak Good Maps
  9. Definition
  10. Properties
  11. Applications in Algebraic Geometry
  12. Every Morphism Locally of Finite Type is Good
  13. Generalized Chevalley Theorem for Noetherian Underlying Spaces
  14. A Key Example: Schemes over $\operatorname{Spec}(\mathcal{O}_{\mathbb{C}_{p}})$
  15. Jacobson Ascent
  16. ...and 4 more sections

Key Result

Proposition 2.3

A continuous map $f:X\to Y$ is good if and only if for every non-empty irreducible closed subset $Z\subseteq X$ and every non-empty open subset $U\subseteq Z$ (in the subspace topology), the image $f(U)$ contains a non-empty open subset of $\overline{f(Z)}$ (in the subspace topology of $\overline{f(

Theorems & Definitions (29)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4: Constructibility criterion
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7: Goodness is finite local on the source
  • ...and 19 more