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-Scholze. In particular, we obtain several descent results for them. If $G={\mathrm{GL}}_n$, 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 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.

78 pages. Final version. To appear in Algebra and Number Theory

References in corpus (1)

• Remarks on derived complete modules and complexes

Cited by in corpus (1)

• A Tannakian framework for prismatic $F$-crystals