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Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria

Ofer Gabber, Rémi Lodh

TL;DR

This work extends Raynaud’s cohomological flatness criteria to higher dimensions and provides a robust logarithmic criterion: if $f_{ ext{log}}:(X,M_X) o (T,M_T)$ is log smooth over a log-regular base, then $f$ is cohomologically flat in dimension 0. It develops numerical criteria via multiplicities and divisorial cycles, establishing a Gabber–Raynaud index theorem and a gcd-variant that ensure flatness when the index (or gcd) is invertible with respect to the base. The paper then applies these ideas to torsors under abelian varieties with good reduction and to curves, deriving necessary and sufficient conditions for the existence of proper log-smooth models and for log smoothness of regular models. The results unify geometric, arithmetic, and log-geometric approaches to representability questions and regular models over discrete valuation rings, with implications for Picard functors and deformation theory. Overall, the work provides broad, local criteria that connect fibre structure, base-change behaviour, and logarithmic geometry to cohomological flatness in dimension 0.

Abstract

We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalizing classical results of Raynaud in two different ways. The first is a higher dimensional generalization of Raynaud's numerical criteria, in both the variant for the multiplicity of the special fibre and that for the index of the generic fibre. The second is a logarithmic criterion: we show that, over a log regular base, a proper flat fs log smooth morphism is cohomologically flat in dimension 0. We apply this latter result to curves and torsors under abelian varieties with good reduction, providing necessary and sufficient conditions for the log smoothness of their regular models over arbitrary discrete valuation rings.

Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria

TL;DR

This work extends Raynaud’s cohomological flatness criteria to higher dimensions and provides a robust logarithmic criterion: if is log smooth over a log-regular base, then is cohomologically flat in dimension 0. It develops numerical criteria via multiplicities and divisorial cycles, establishing a Gabber–Raynaud index theorem and a gcd-variant that ensure flatness when the index (or gcd) is invertible with respect to the base. The paper then applies these ideas to torsors under abelian varieties with good reduction and to curves, deriving necessary and sufficient conditions for the existence of proper log-smooth models and for log smoothness of regular models. The results unify geometric, arithmetic, and log-geometric approaches to representability questions and regular models over discrete valuation rings, with implications for Picard functors and deformation theory. Overall, the work provides broad, local criteria that connect fibre structure, base-change behaviour, and logarithmic geometry to cohomological flatness in dimension 0.

Abstract

We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalizing classical results of Raynaud in two different ways. The first is a higher dimensional generalization of Raynaud's numerical criteria, in both the variant for the multiplicity of the special fibre and that for the index of the generic fibre. The second is a logarithmic criterion: we show that, over a log regular base, a proper flat fs log smooth morphism is cohomologically flat in dimension 0. We apply this latter result to curves and torsors under abelian varieties with good reduction, providing necessary and sufficient conditions for the log smoothness of their regular models over arbitrary discrete valuation rings.
Paper Structure (15 sections, 38 theorems, 47 equations)

This paper contains 15 sections, 38 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Cohomological flatness
  4. Definition and basic properties
  5. Converse to Artin's theorem
  6. Numerical criteria
  7. Multiplicities
  8. Raynaud's condition (N)
  9. Divisorial cycles
  10. Proof of Theorem \ref{['gabberraynaudindex']}
  11. Logarithmic criterion
  12. Proof of Theorem \ref{['main']}
  13. An example
  14. Application: torsors under abelian varieties with good reduction
  15. Application: curves

Key Result

Theorem 1

Assume $f:X\to T$ is a finitely presented morphism of schemes that is proper and flat. If $f$ is cohomologically flat in dimension 0, then the Picard functor $\mathop{\mathrm{\mathbf{Pic}}}\nolimits_{X/T}$ is representable by an algebraic space.

Theorems & Definitions (78)

  • Theorem 1: Artin artinformalIartinversal
  • Theorem 2: Raynaud raynaud
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • ...and 68 more