The prismatization of $p$-adic formal schemes

Abstract

In this note, we introduce and study the Cartier--Witt stack ${\mathrm{WCart}}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on ${\mathrm{WCart}}_X$ in favorable situations.

preliminary version, 52 pages, comments welcome