Stratifications of flag spaces and functoriality
Wushi Goldring, Jean-Stefan Koskivirta

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 -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 . The fiber product with a Shimura variety of Hodge-type is a generalization of flag spaces considered by Ekedahl-Van der Geer. For large , we prove that all strata are affine. We prove a theorem on discreteness of fibers for finite morphisms between stacks of -zips. This allows us to prove that the zip stratification is principally pure for all primes , for zip data of Hodge-type. This provides a second proof, entirely in characteristic , of the existence of generalized Hasse invariants for Ekedahl-Oort strata in the good reduction of Shimura varieties of Hodge-type.
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