Stratifications of flag spaces and functoriality

TL;DR

This paper introduces quotient stacks of zip flags, establishes their stratification properties, and proves the existence of generalized Hasse invariants for Shimura varieties of Hodge-type in characteristic p.

Contribution

It defines new quotient stacks of zip flags, proves their stratification is principally pure, and extends results on Hasse invariants to all primes p for Hodge-type Shimura varieties.

Findings

Stratification on zip flags is principally pure under certain conditions.

All strata are affine for large p.

Established discreteness of fibers for morphisms between G-zip stacks.

Abstract

We define quotient stacks of "zip flags". They form towers above the stack of $G$-zips introduced by Moonen, Pink, Wedhorn and Ziegler. We define a stratification on the stack of zip flags, and prove that it is principally pure, under a certain assumption on $p$. The fiber product with a Shimura variety of Hodge-type is a generalization of flag spaces considered by Ekedahl-Van der Geer. For large $p$, we prove that all strata are affine. We prove a theorem on discreteness of fibers for finite morphisms between stacks of $G$-zips. This allows us to prove that the zip stratification is principally pure for all primes $p$, for zip data of Hodge-type. This provides a second proof, entirely in characteristic $p$, of the existence of generalized Hasse invariants for Ekedahl-Oort strata in the good reduction of Shimura varieties of Hodge-type.