De Rham affineness of the Nygaard filtered prismatization in positive characteristic

1. Problem Statement and Context

The Core Problem:
The paper addresses the structural relationship between prismatic cohomology (specifically its Nygaard filtration) and the geometry of stacks in positive characteristic. While the theory of prismatic cohomology (developed by Bhatt, Morrow, and Scholze) provides a powerful cohomological framework for $p$-adic geometry, a fundamental question remains: Can the geometric objects associated with these cohomology theories (prismatizations) be reconstructed directly from their cohomological data?

In the context of de Rham cohomology, Mondal [Mon22] showed that for a smooth affine scheme $X$, the de Rham stack $(X/k)_{dR}$ is isomorphic to the spectrum of the de Rham cohomology ring $dR_{X/k}$. This property, termed de Rham affineness, allows one to pass between geometric stacks and algebraic rings.

The author seeks to establish an analogous result for the Nygaard filtered prismatization in positive characteristic. Specifically, given an animated $k$-algebra $R$ (where $k$ is a perfect ring of characteristic $p$), does the stack $R_{Nyg}$ (the Nygaard filtered prismatization) arise naturally as the relative spectrum of the Rees algebra associated with the Nygaard filtered prismatic cohomology of $R$?

Key Challenges:

• Derived Geometry: The objects involved are "animated" (derived) rings and stacks, requiring the language of $\infty$-categories.
• Filtrations: The Nygaard filtration is a complex structure on prismatic cohomology that must be encoded geometrically via the stack $k_{Nyg}$ (the Nygaard stack).
• Reconstruction: Proving that the functor $R \mapsto R_{Nyg}$ is "affine" relative to the base stack $k_{Nyg}$, meaning it is representable by a derived algebra over $k_{Nyg}$.

2. Methodology

The paper employs a blend of Derived Algebraic Geometry (DAG), Prismatic Cohomology, and Stack Theory.

• Framework: The author works within the $\in$-category of animated rings ($\mathcal{A}\text{Ring}$) and stacks. All functors and limits/colimits are derived.
• The Base Stack ($k_{Nyg}$): The paper utilizes the stack $k_{Nyg}$, which serves as a moduli space for the Nygaard filtration. It is defined as the quotient stack $\text{Spf}(W(k)\langle u, t \rangle / (ut-p)) / \mathbb{G}_m$. This stack interpolates between various cohomological theories (crystalline, de Rham, Hodge-Tate) via its loci (e.g., $t=0$, $u=0$, $t \neq 0$).
• The Rees Construction: The central algebraic tool is the Rees algebra. The author takes the Nygaard filtered prismatic cohomology $F^* \Delta_{R/k}$ (a filtered derived algebra) and constructs its Rees algebra, $\text{Rees}(F^* \Delta_{R/k})$, which is a graded derived algebra over the base ring of $k_{Nyg}$.
• The "De Rham Affine" Property: The author axiomatizes the property of a functor $X: \mathcal{A}\text{Ring}_k \to \text{Stacks}_{k_{Nyg}}$ being de Rham affine. This means $X(R) \simeq \text{Spec}_{k_{Nyg}}(T(R))$ for some functor $T$ taking values in derived algebras over $k_{Nyg}$.
• Proof Strategy:
  1. Construction: Define a natural map $\eta_R: R_{Nyg} \to \text{Spec}_{k_{Nyg}}(\text{Rees}(F^* \Delta_{R/k}))$.
  2. Reduction: Use the fact that both functors preserve colimits (specifically sifted colimits) to reduce the problem to checking the isomorphism on a specific generator, $R = k[x]$.
  3. Descent: Employ a Cech nerve argument using a quasisyntomic cover $S = k[x^{1/p^\infty}]$. The proof relies on showing that the map becomes an fpqc epimorphism after base change to $S$.
  4. Flatness Analysis: The core technical difficulty involves proving that the morphism of derived algebras induced by the cover is p-completely faithfully flat. This is achieved by analyzing the connectivity of the fiber of the map using Bar resolutions and testing flatness on a constructible stratification of the base stack (loci where $u=0$, $t=0$, etc.).

3. Key Contributions

1. Definition of De Rham Affineness: The paper formalizes the concept of "de Rham affineness" for functors valued in stacks. This provides a unifying language to describe how cohomology theories (like prismatization) generate geometric stacks via their cohomological rings.
2. Main Theorem (Theorem 1.13): The author proves that for any animated $k$-algebra $R$, the Nygaard filtered prismatization $R_{Nyg}$ is naturally isomorphic to the relative spectrum of the Rees algebra of the Nygaard filtered prismatic cohomology:
   $$R_{Nyg} \simeq \text{Spec}_{k_{Nyg}}(\text{Rees}(F^* \Delta_{R/k}))$$
   This establishes that the geometric object $R_{Nyg}$ is entirely determined by the algebraic data of the filtered cohomology.
3. Transmutation and Ring Stacks: The paper clarifies the relationship between "transmutation" (a method of constructing stacks from ring stacks) and de Rham affineness. It shows that colimit-preserving de Rham affine functors arise via transmutation, offering a structural explanation for why prismatization functors behave like ring stacks.
4. Technical Appendices: The paper provides novel technical results in the appendices, including:
   • Criteria for testing faithful flatness on constructible stratifications (Appendix A).
   • Connectivity testing for $J$-complete objects (Appendix B).
   • A Künneth formula for derived Nygaard filtered prismatic cohomology (Appendix C).

4. Results and Corollaries

The main theorem yields several significant corollaries by restricting the isomorphism to specific loci of the base stack $k_{Nyg}$:

• Crystalline Case: Restricting to the locus $t \neq 0$ (where $k_{Nyg} \simeq \text{Spf}(W(k))$) recovers the classical result that the crystallization of $R$ is the spectrum of crystalline cohomology.
• Hodge-Filtered de Rham: Restricting to the locus $u=0$ (isomorphic to $\mathbb{A}^1/\mathbb{G}_m$) yields an isomorphism between the Hodge-filtered derived de Rham stack $(R/k)_{dR,+}$ and the spectrum of the Rees algebra of the Hodge-filtered de Rham cohomology. This is a filtered variant of Mondal's result.
• Conjugate Hodge-Tate: Restricting to the locus $t=0$ yields an isomorphism between the conjugate filtered Hodge-Tate stack and the spectrum of the Rees algebra of the conjugate filtered Hodge-Tate cohomology.
• Hodge Stack: Restricting to $u=t=0$ recovers the graded stack corresponding to the Hodge filtration on de Rham cohomology.

5. Significance

• Unification of Cohomology Theories: The paper demonstrates that various cohomological theories (crystalline, de Rham, Hodge-Tate) and their filtrations are not just algebraic invariants but can be realized as geometric stacks via a single, unified mechanism (the Nygaard filtered prismatization).
• Bridge to Syntomic Cohomology: The author notes that this result is a foundational step for their forthcoming work on the syntomification of Frobenius-liftable schemes. By establishing the de Rham affineness of the Nygaard stack, the author can relate the category of quasi-coherent sheaves on the syntomic stack to Fontaine-Laffaille modules, a crucial connection in $p$-adic Hodge theory.
• Structural Insight: The result suggests that "de Rham affineness" is a robust property of cohomology theories arising from prismatic geometry. It implies that the geometric complexity of these stacks is entirely encoded in the algebraic structure of their cohomology rings (specifically their Rees algebras).
• Methodological Advancement: The techniques developed for proving faithful flatness in the derived setting (using Bar resolutions and stratification) provide new tools for working with derived stacks and filtered algebras in positive characteristic.

In summary, Sahai's paper provides a rigorous proof that the Nygaard filtered prismatization is "de Rham affine," effectively solving the reconstruction problem for these stacks in positive characteristic and laying the groundwork for future applications in the study of syntomic cohomology and $p$-adic Hodge theory.