Functorial desingularization of quasi-excellent schemes in characteristic zero: the non-embedded case

TL;DR

This paper proves that all noetherian quasi-excellent schemes in characteristic zero can be strongly desingularized in a functorial manner, extending to stacks, formal schemes, and analytic spaces, with methods adaptable to non-compact cases.

Contribution

It establishes a functorial strong desingularization for quasi-excellent schemes in characteristic zero, including non-embedded cases, and extends the results to various related spaces.

Findings

Existence of functorial desingularization for quasi-excellent schemes

Extension of desingularization to stacks, formal schemes, and analytic spaces

Method for non-compact spaces using blow-up hyper-sequences

Abstract

We prove that any noetherian quasi-excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms. We show that as an easy formal consequence of this result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi-excellent stacks and formal schemes, and complex and non-archimedean analytic spaces. Moreover, these functors easily generalize to non-compact setting by use of converging blow up hyper-sequences with regular centers.