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De Rham affineness of the Nygaard filtered prismatization in positive characteristic

Shubhankar Sahai

TL;DR

The article proves a de Rham affineness property for the Nygaard filtered prismatization in positive characteristic, showing that the relative Nygaard prismatization $R^{\mathrm{Nyg}}$ is naturally isomorphic to the relative spectrum of the Rees algebra of Nygaard-filtered prismatic cohomology over $k^{\mathrm{Nyg}}$. It formalizes de Rham affineness as a structural principle, demonstrates the main equivalence via a reduction to a polynomial ring and a quasisyntomic cover, and develops the Rees stack framework to encode Nygaard and conjugate filtrations across essential loci of $k^{\mathrm{Nyg}}$. The work not only provides a concrete identification but also lays groundwork for subsequent studies on syntomification and mixed-characteristic generalizations, with plans to apply these ideas in forthcoming papers Sah25Nyg and Sah25Syn. Overall, this establishes a robust, functorial, and descent-friendly description of Nygaard-prismatic data as an affine object in the derived setting.

Abstract

Let $k$ be a perfect ring of characteristic $p>0$, and let $R$ be an animated $k$-algebra. This note aims to show that the Nygaard filtered prismatization $R^{\mathrm{Nyg}}$ of $R$ is naturally isomorphic, as a stack over $k^{\mathrm{Nyg}}$, to the relative spectrum over $k^{\mathrm{Nyg}}$ of the Rees algebra of the Nygaard filtered prismatic cohomology of $R$ relative to $k$. In doing so, we axiomatise the functorial affineness property displayed by the relative Nygaard filtered prismatization, and dub it de Rham affineness after the fundamental example of the functor sending an animated ring to its relative de Rham stack. While we treat this concept as an organising tool for the author's forthcoming work on the syntomification of Frobenius liftable schemes, we are able to frame some questions based on a structural theorem of independent interest: a functor to stacks which is de Rham affine often arises via ring stacks through transmutation.

De Rham affineness of the Nygaard filtered prismatization in positive characteristic

TL;DR

The article proves a de Rham affineness property for the Nygaard filtered prismatization in positive characteristic, showing that the relative Nygaard prismatization is naturally isomorphic to the relative spectrum of the Rees algebra of Nygaard-filtered prismatic cohomology over . It formalizes de Rham affineness as a structural principle, demonstrates the main equivalence via a reduction to a polynomial ring and a quasisyntomic cover, and develops the Rees stack framework to encode Nygaard and conjugate filtrations across essential loci of . The work not only provides a concrete identification but also lays groundwork for subsequent studies on syntomification and mixed-characteristic generalizations, with plans to apply these ideas in forthcoming papers Sah25Nyg and Sah25Syn. Overall, this establishes a robust, functorial, and descent-friendly description of Nygaard-prismatic data as an affine object in the derived setting.

Abstract

Let be a perfect ring of characteristic , and let be an animated -algebra. This note aims to show that the Nygaard filtered prismatization of is naturally isomorphic, as a stack over , to the relative spectrum over of the Rees algebra of the Nygaard filtered prismatic cohomology of relative to . In doing so, we axiomatise the functorial affineness property displayed by the relative Nygaard filtered prismatization, and dub it de Rham affineness after the fundamental example of the functor sending an animated ring to its relative de Rham stack. While we treat this concept as an organising tool for the author's forthcoming work on the syntomification of Frobenius liftable schemes, we are able to frame some questions based on a structural theorem of independent interest: a functor to stacks which is de Rham affine often arises via ring stacks through transmutation.
Paper Structure (14 sections, 34 theorems, 75 equations, 1 figure)

This paper contains 14 sections, 34 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background.
  3. Main result
  4. Overview of the Note.
  5. Forthcoming work.
  6. Conventions
  7. Acknowledgements.
  8. Generalities on de Rham affineness
  9. Some loci of $k^\mathrm{Nyg}$
  10. The Rees stack R_nyg(F^*Delta_R/k)
  11. Proof of de Rham affineness of the Nygaard filtered prismatization in positive characteristic
  12. Testing faithful flatness on a constructible stratification
  13. Testing connectivity modulo $J$ for $J$-complete objects
  14. Colimit preservation and Künneth for derived Nygaard filtered prismatic cohomology

Key Result

Theorem 1.2

When $X/\overline{A}$ is affine, then there is an equivalence of $(p,I)$-adic formal stacks over $A$

Figures (1)

  • Figure 1: The diagram of $W$-module schemes over $k^\mathrm{Nyg}$Bhatt22.

Theorems & Definitions (104)

  • Example 1.1: The de Rham gerbe
  • Theorem 1.2: Bhatt-Lurie
  • Example 1.3
  • Remark 1.4
  • Remark 1.5: The property of de Rham affineness
  • Definition 1.6: De Rham affineness
  • Remark 1.7: Colimit preserving de Rham affine functors arise via transmutation
  • Remark 1.8: Toën's affineness
  • Example 1.9
  • Example 1.10
  • ...and 94 more