Prismatization over $\mathbf{Z}$

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 in [Dri20] and then given by Bhatt-Lurie in [BL22b]) to all schemes over Spec(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 Q this construction recovers (a version of) the filtered de Rham stack. ContentsIntroduction 1. Preliminaries 1.1. Conventions 2. de Rham cohomology over Q 2.1. Reminders on filtrations and the Cartier stack 2.2. The de Rham stack 2.3. Cohomology of X dR 3. Witt vectors 3.1. Reminders 3.2. The map V(1) : V → W 3.3. The ring scheme W 4. The stack Σ Z 4.1. Hodge-Tate elements and ∆ Z 4.2. The stack Σ Z 5. Prismatization over Z 5.1. The ring stacks W E 5.2. Definition and algebraicity of X ∆ Z 5.3. The Hodge-Tate stack X HT 5.4. Presentations of X ∆ Z 5.5. Comparisons and the cohomology of X ∆ Z References