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

TL;DR

This paper generalizes Chevalley's theorem by introducing 'good maps' in topology, providing a purely topological characterization of constructibility that extends beyond Noetherian spaces, with applications in algebraic and non-Archimedean geometry.

Contribution

It defines 'good' maps as a topological generalization of constructibility preservation, applicable beyond Noetherian spaces, and proves that morphisms locally of finite type are good.

Findings

Every morphism locally of finite type is good

Generalization of Chevalley's theorem for Noetherian spaces

Elementary proof of Jacobson ascent

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.