Noetherian approximation of algebraic spaces and stacks

TL;DR

This paper demonstrates that non-noetherian algebraic spaces and stacks with certain finiteness conditions can be approximated by noetherian objects, extending classical theorems to broader contexts.

Contribution

It introduces a method for approximating algebraic spaces and stacks with noetherian models, generalizing key theorems to non-noetherian settings.

Findings

Every quasi-compact algebraic space/stack with quasi-finite diagonal can be approximated by a noetherian model.

Stacks etale-locally is a global quotient can be approximated.

Existence of finite generically flat covers for quasi-compact algebraic stacks.

Abstract

We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.