Classification of BTn-groups over perfectoid rings

• Main Author: Henkel, Timo
• Format: Others
• Language: en
• Published: 2020
• Online Access: https://tuprints.ulb.tu-darmstadt.de/14223/1/Genehmigte%20Dissertation.pdf; Henkel, Timo <http://tuprints.ulb.tu-darmstadt.de/view/person/Henkel=3ATimo=3A=3A.html> (2020): Classification of BTn-groups over perfectoid rings. (Publisher's Version)Darmstadt, Technische Universität, DOI: 10.25534/tuprints-00014223 <https://doi.org/10.25534/tuprints-00014223>, [Ph.D. Thesis]

In this work, we investigate p-divisible groups over integral perfectoid rings by focussing on the relevant p-power torsion subgroups, which are instances of BTn-groups. We use results of Lau and Anschütz-Le Bras to show that such groups can be described by semi linear algebra objects which live over the tilt of the ground ring. These objects are called BKn-modules. From this classification, we deduce that in our setting every BTn-group can be lifted to a p-divisible group. In the case of local perfectoid rings, we find an explicit description of this data in terms of orbits with respect to a certain group operation. By the connection between BT1-groups and F-Zips, this contains the classification of F-Zips ober a perfect field of characteristic p as a special case. We also deal with globalization aspects of this results. We show that BKn modules can be glued with respect to a certain topology which is fine enough to depict the classifying stack of BKn-modules as a quotient stack. Moreover, we consider our constructions with respect to the finer p-complete arc topology. This topology has a basis consisting of products of perfectoid valuation rings of rank at most 1. Finally, we show globalization results for this topology. In particular, BKn-modules over a perfect ring can be glued together and the resulting stack has a description as a quotient stack. Assuming a conjecture, analogous results are proved for general perfectoid rings.