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Functorial flatification of proper morphisms

David Rydh

TL;DR

The paper develops a functorial flatification framework for proper morphisms $f:X\to S$ of noetherian schemes by constructing a canonical, base-change-friendly sequence of blow-ups that makes the strict transform flat on the blown-up base. Central tools include the universal flattening $Flat_{\mathcal{F}/S}$ with its canonical flattening filtration, a precise dévissage-based flatness criterion, and a resolution of monomorphisms to obtain local regular embeddings; these yield a smooth-centers variant in characteristic zero and extend to stacks. A parallel functorial étalification is proved in characteristic zero using Kummer blow-ups, yielding étale morphisms after controlled root-stack enhancements. The framework yields several powerful applications: functorial cofinality of blow-ups for modifications, functorial resolution of indeterminacy loci, and a general Chow lemma, with implications for stack-desingularization and weak factorization in characteristic zero. Together, the results provide a canonical, functorial approach to flattening and étalifying morphisms in a broad algebraic-stack setting, compatible with base changes and serving as a foundation for further birational and desingularization theories.

Abstract

For proper morphisms, we give a functorial flatification algorithm by blow-ups in the spirit of Hironaka's flatification algorithm. In characteristic zero, this gives functorial flatification by blow-ups in smooth centers. We also give a functorial étalification algorithm by Kummer blow-ups in characteristic zero.

Functorial flatification of proper morphisms

TL;DR

The paper develops a functorial flatification framework for proper morphisms of noetherian schemes by constructing a canonical, base-change-friendly sequence of blow-ups that makes the strict transform flat on the blown-up base. Central tools include the universal flattening with its canonical flattening filtration, a precise dévissage-based flatness criterion, and a resolution of monomorphisms to obtain local regular embeddings; these yield a smooth-centers variant in characteristic zero and extend to stacks. A parallel functorial étalification is proved in characteristic zero using Kummer blow-ups, yielding étale morphisms after controlled root-stack enhancements. The framework yields several powerful applications: functorial cofinality of blow-ups for modifications, functorial resolution of indeterminacy loci, and a general Chow lemma, with implications for stack-desingularization and weak factorization in characteristic zero. Together, the results provide a canonical, functorial approach to flattening and étalifying morphisms in a broad algebraic-stack setting, compatible with base changes and serving as a foundation for further birational and desingularization theories.

Abstract

For proper morphisms, we give a functorial flatification algorithm by blow-ups in the spirit of Hironaka's flatification algorithm. In characteristic zero, this gives functorial flatification by blow-ups in smooth centers. We also give a functorial étalification algorithm by Kummer blow-ups in characteristic zero.
Paper Structure (13 sections, 16 theorems, 3 equations)

This paper contains 13 sections, 16 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Functorial flatification
  3. Universal flattening and flattening stratifications
  4. Flatifying modules
  5. Resolving monomorphisms
  6. Dévissage
  7. Flattening in the local case
  8. Proof of the main theorem
  9. Functorial étalification
  10. Applications of functorial flatification
  11. Cofinality of blow-ups among modifications
  12. Resolution of the indeterminacy locus
  13. A general Chow lemma

Key Result

Theorem A

Let $f\colon X\to S$ be a proper morphism of noetherian schemes. Let $U\subseteq S$ be the largest open substack such that $f|_U$ is flat. Then there exists a sequence of blow-ups $\widetilde{S}\to S$ with centers disjoint from $U$ such that the strict transform $\widetilde{f}\colon \widetilde{X}\to

Theorems & Definitions (30)

  • Theorem A
  • Theorem B
  • Theorem C: Functorial étalification of proper morphisms in characteristic zero
  • Theorem 2.0.1
  • Remark 2.0.2
  • Example 2.0.3: Non-noetherian counter-example
  • Remark 2.1.1
  • Proposition 2.1.2
  • Example 2.1.3: Hironaka hironaka_flattening
  • Example 2.1.4: Kresch kresch_email-aug11-2010
  • ...and 20 more