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Prismatic Kunz's theorem

Ryo Ishizuka, Kei Nakazato

TL;DR

This work extends Kunz's regularity criterion to mixed characteristic via prismatic cohomology by introducing a Frobenius lift on prismatic complexes and proving its faithful flatness characterizes regularity for complete Noetherian local rings of residue characteristic $p$. The authors develop the theory of animated prisms and derived quotients to establish faithful flatness of prismatic complexes, prove a deformation theory for regular prisms, and derive a prismatic Kunz's theorem that yields both new characterizations of regularity and practical stability results under localization. Key components include the faithful-flatness results for animated LCIs, the deformation criteria for regular prisms, and the reduction to classical Kunz in suitable cases. The results connect $p$-adic Kunz, derived algebraic geometry, and prismatic cohomology to provide a robust framework for understanding regularity in mixed characteristic with potential applications to prism theory and arithmetic geometry.

Abstract

In this paper, we prove "prismatic Kunz's theorem" which states that a complete Noetherian local ring $R$ of residue characteristic $p$ is a regular local ring if and only if the Frobenius lift on a prismatic complex of (a derived enhancement of) $R$ over a specific prism $(A, I)$ is faithfully flat. This generalizes classical Kunz's theorem from the perspective of extending the "Frobenius map" to mixed characteristic rings. Our approach involves studying the deformation problem of the "regularity" of prisms and demonstrating the faithful flatness of the structure map of the prismatic complex.

Prismatic Kunz's theorem

TL;DR

This work extends Kunz's regularity criterion to mixed characteristic via prismatic cohomology by introducing a Frobenius lift on prismatic complexes and proving its faithful flatness characterizes regularity for complete Noetherian local rings of residue characteristic . The authors develop the theory of animated prisms and derived quotients to establish faithful flatness of prismatic complexes, prove a deformation theory for regular prisms, and derive a prismatic Kunz's theorem that yields both new characterizations of regularity and practical stability results under localization. Key components include the faithful-flatness results for animated LCIs, the deformation criteria for regular prisms, and the reduction to classical Kunz in suitable cases. The results connect -adic Kunz, derived algebraic geometry, and prismatic cohomology to provide a robust framework for understanding regularity in mixed characteristic with potential applications to prism theory and arithmetic geometry.

Abstract

In this paper, we prove "prismatic Kunz's theorem" which states that a complete Noetherian local ring of residue characteristic is a regular local ring if and only if the Frobenius lift on a prismatic complex of (a derived enhancement of) over a specific prism is faithfully flat. This generalizes classical Kunz's theorem from the perspective of extending the "Frobenius map" to mixed characteristic rings. Our approach involves studying the deformation problem of the "regularity" of prisms and demonstrating the faithful flatness of the structure map of the prismatic complex.
Paper Structure (10 sections, 34 theorems, 45 equations)

This paper contains 10 sections, 34 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Faithful flatness of prismatic complexes
  3. Prismatic Kunz's theorem
  4. Transversal Prisms
  5. Regular Prisms
  6. Faithful Flatness of Prismatic Complexes
  7. Prismatic Kunz's Theorem
  8. The proof of prismatic Kunz's theorem
  9. Applications
  10. Preliminaries about Animated Rings

Key Result

Theorem 1.1

Let $R$ be a Noetherian ring of characteristic $p$. Then $R$ is regular if and only if the Frobenius map $F\colon R\to F_*R$ is faithfully flat if and only if the canonical map $R \to R_{\mathop{\mathrm{perf}}\nolimits}$ is faithfully flat, where $R_{\mathop{\mathrm{perf}}\nolimits}$ is the perfect

Theorems & Definitions (88)

  • Theorem 1.1: Kunz's theorem; kunz1969Characterizations
  • Theorem 1.3: \ref{['EquivRegularPrism']}
  • Theorem 1.5: \ref{['AnimatedLCIPrism']} and \ref{['NoetherianCase']}
  • Theorem 1.6: Prismatic Kunz's theorem (\ref{['RegularGivesFFlat']}, \ref{['FFlatGivesRegular']}, and \ref{['AnimatedCase']})
  • Definition 2.1: bhatt2022Prismsa
  • Definition 2.2: cf. stacks-project and bhatt2022Prismsa
  • Lemma 2.3: kedlayaNotes
  • Definition 2.4: bhatt2022Prismsa
  • Remark 2.5
  • Lemma 2.6
  • ...and 78 more