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Reduction modulo $p$ of the Noether problem

Emiliano Ambrosi, Domenico Valloni

TL;DR

The paper shows that, for a smooth proper family $X\to\mathrm{Spec}(R)$ with mixed characteristic, stable rationality of the special fiber $X_k$ forces the $p$-torsion in $H^3(X_K,\mathbb{Z}_p)$ to vanish, tying birational invariants across characteristics via integral $p$-adic Hodge theory. The authors prove a crucial vanishing criterion on crystalline cohomology under concrete hypotheses on $X$ and use a detailed factorisation of the de Rham cycle class map to deduce this. They apply the result to the Noether problem, obtaining obstructions for constructing smooth reductions compatible with stable rationality on the generic fiber, and discuss consequences for Artin–Mumford-type invariants of $p$-groups. Finally, they construct mixed-characteristic examples where the special fiber is rational but the generic fiber is stably irrational, highlighting the subtle interplay between reduction behavior and birational geometry in mixed characteristic.

Abstract

Let $R$ be a complete valuation ring of mixed characteristic $(0,p)$ with algebraically closed fraction field $K$ and residue field $k$. Let $X/R$ be a smooth projective morphism. We show that if $X_k$ is stably rational, then $H^3(X_K, \mathbb Z_p)$ is torsion-free. The proof uses integral $p$-adic Hodge theory of Bhatt-Morrow-Scholze and the study of differential forms in positive characteristic. We then apply this result to study the Noether problem for finite $p$-groups.

Reduction modulo $p$ of the Noether problem

TL;DR

The paper shows that, for a smooth proper family with mixed characteristic, stable rationality of the special fiber forces the -torsion in to vanish, tying birational invariants across characteristics via integral -adic Hodge theory. The authors prove a crucial vanishing criterion on crystalline cohomology under concrete hypotheses on and use a detailed factorisation of the de Rham cycle class map to deduce this. They apply the result to the Noether problem, obtaining obstructions for constructing smooth reductions compatible with stable rationality on the generic fiber, and discuss consequences for Artin–Mumford-type invariants of -groups. Finally, they construct mixed-characteristic examples where the special fiber is rational but the generic fiber is stably irrational, highlighting the subtle interplay between reduction behavior and birational geometry in mixed characteristic.

Abstract

Let be a complete valuation ring of mixed characteristic with algebraically closed fraction field and residue field . Let be a smooth projective morphism. We show that if is stably rational, then is torsion-free. The proof uses integral -adic Hodge theory of Bhatt-Morrow-Scholze and the study of differential forms in positive characteristic. We then apply this result to study the Noether problem for finite -groups.
Paper Structure (15 sections, 5 theorems, 30 equations)

This paper contains 15 sections, 5 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. Applications to the Noether problem
  4. Acknowledgement
  5. Proof of Theorem -
  6. Notation
  7. Preliminary reductions
  8. Factorisation of the cycle class map
  9. Studying the factorisation
  10. Proof of (1)
  11. Proof of (2)
  12. Proof of (3)
  13. A stably irrational variety reducing to a rational variety
  14. A general lemma
  15. An example

Key Result

Theorem 1.2

Let $X/R$ be a smooth proper morphism, and assume that the special fiber $X_k$ is stably rational. Then the $p$-torsion of $H^3_{\mathrm{\acute et}}(X_K, \mathbb{Z}_p)$ vanishes.

Theorems & Definitions (6)

  • Theorem 1.2
  • Theorem 1.1.5
  • Corollary 1.2.1
  • Remark 1.2.2
  • Lemma 2.4.2.1
  • Lemma 3.1.1