De Rham affineness of the Nygaard filtered prismatization in positive characteristic

TL;DR

This paper demonstrates that the Nygaard filtered prismatization of an animated algebra over a perfect ring in characteristic p is isomorphic to a relative spectrum of a Rees algebra, revealing a de Rham affineness property.

Contribution

It introduces and formalizes the concept of de Rham affineness for the Nygaard filtered prismatization, connecting it to the structure of ring stacks and functorial properties.

Findings

Nygaard filtered prismatization is isomorphic to a relative spectrum of a Rees algebra.

De Rham affineness is characterized as a structural property of certain functors to stacks.

The concept of transmutation links de Rham affine functors to ring stacks.

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 result of independent interest: a functor to stacks which is de Rham affine often arises via ring stacks through transmutation.