Purity of G-zips

TL;DR

This paper studies the geometric structure of G-zips over schemes in characteristic p, proving that their stratification into isomorphism classes is affine and exploring various applications of this purity property.

Contribution

It establishes the affineness of G-zip strata and demonstrates several geometric applications of this purity result.

Findings

G-zip strata are affine schemes.

Purity of G-zip stratification is proven.

Applications include insights into moduli spaces and algebraic group actions.

Abstract

Let $k$ be a perfect field of characteristic $p>0$, and $S$ an scheme over $k$. An $F$-zip is basically a locally free $O_S$-module of finite rank endowed with two filtration and an Frobenius-linear isomorphism between their graded pieces. The natural generalization of this notion for a reductive algebraic group $G/k$ is an "$F$-zip with $G$-structure", a so-called $G$-zip introduced by R. Pink, T. Wedhorn, P. Ziegler. A $G$-zip $I$ over $S$ yields the stratification of the base scheme in loci, where $I$ has locally a constant isomorphism class for the fppf topology. We show that these strata are affine and give a number of geometric applications of this purity result.