Prismatization over $\mathbf{Z}$

TL;DR

This paper extends the prismatization functor from p-adic formal schemes to all schemes over the integers, establishing key properties and connecting it to the filtered de Rham stack for smooth schemes over Q.

Contribution

It introduces a new extension of the prismatization functor to all schemes over Z, with foundational properties and links to known geometric structures.

Findings

Proves algebraicity and flatness of the extended functor

Shows the perfectness of cohomology in this context

Recovers the filtered de Rham stack for smooth schemes over Q

Abstract

The aim of this article is to given an extension of the prismatization functor for $p$-adic formal schemes (whose construction was first sketched by Drinfeld and then given by Bhatt-Lurie) to all schemes over $\mathrm{Spec}(\mathbf{Z})$. We then prove some basic properties of this extension (algebraicity, flatness for syntomic morphisms, perfectness of cohomology) and show that for smooth schemes over $\mathbf{Q}$ this construction recovers (a version of) the filtered de Rham stack.