The compatibility with the duality for partial Hasse invariants

TL;DR

This paper proves the compatibility of the Hasse invariant with duality for p-divisible groups under Pappas-Rapoport conditions, expressing it as a product of partial invariants and analyzing their duality behavior.

Contribution

It provides a natural proof of Hasse invariant compatibility with duality and studies its behavior in p-divisible groups with ramified endomorphism actions under Pappas-Rapoport conditions.

Findings

Hasse invariant is compatible with duality for p-divisible groups.

Hasse invariant decomposes into partial invariants under Pappas-Rapoport conditions.

Dual p-divisible groups satisfy the same Pappas-Rapoport conditions.

Abstract

We give a simple and natural proof for the compatibility of the Hasse invariant with duality. We then study a $p$-divisible group with an action of the ring of integers of a finite ramified extension of $\mathbb{Q}_p$. We suppose that it satisfies the Pappas-Rapoport condition ; in that case the Hasse invariant is a product of partial Hasse invariants, each of which can be expressed in terms of primitive Hasse invariants. We then show that the dual of the $p$-divisible group naturally satisfies a Pappas-Rapoport condition, and prove the compatibility with the duality for the partial and primitive Hasse invariants.