Prismatic G-displays and descenttheory

For a smooth affine group scheme G over the ring of p-adic integers p and a cocharacter of G, we study G--displays over the prismatic site of Bhatt and Scholze.In particular, we obtain several descent results for them.If G = GL n , then our G--displays can be thought of as Breuil-Kisin modules with some additional conditions.The relation between our G--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 O E of any finite extension E of p by using O E -prisms, which are O E -analogues of prisms.1. Introduction 1685 2. Preliminaries on O E -prisms 1691 3. Displayed Breuil-Kisin modules 1708 4. Display group 1714 5. Prismatic G--displays 1722 6. Prismatic G--displays over complete regular local rings 1738 7. p-divisible groups and prismatic Dieudonn crystals 1752 8. Comparison with prismatic F-gauges 1756