Prismatic G-displays and descent theory

Abstract

For a smooth affine group scheme $G$ over the ring of $p$-adic integers ${\mathbb{Z}}_p$ and a cocharacter $\mu$ of $G$, we study $G$-$\mu$-displays over the prismatic site of Bhatt and Scholze. In particular, we obtain several descent results for them. If $G={\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$, then our $G$-$\mu$-displays can be thought of as Breuil–Kisin modules with some additional conditions. The relation between our $G$-$\mu$-displays and prismatic $F$-gauges introduced by Drinfeld and Bhatt–Lurie is also discussed.

In fact, our main results are formulated and proved for smooth affine group schemes over the ring of integers ${\mathcal{𝒪}}_E$ of any finite extension $E$ of ${\mathbb{Q}}_p$ by using ${\mathcal{𝒪}}_E$-prisms, which are ${\mathcal{𝒪}}_E$-analogues of prisms.