$F$-zips with additional structure

Richard Pink, Torsten Wedhorn and Paul Ziegler

Vol. 274 (2015), No. 1, 183–236

DOI: 10.2140/pjm.2015.274.183

Abstract

Let ${\mathbb{F}}_q$ be a fixed finite field of cardinality $q$. An $F$-zip over a scheme $S$ over ${\mathbb{F}}_q$ is a certain object of semilinear algebra consisting of a locally free sheaf of ${\mathcal{O}}_S$-modules with a descending filtration and an ascending filtration and a ${Frob}_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called “$F$-zips with a $G$-structure”, for an arbitrary reductive linear algebraic group $G$ over ${\mathbb{F}}_q$.

These objects come in two incarnations. One incarnation is an exact ${\mathbb{F}}_q$-linear tensor functor from the category of finite dimensional representations of $G$ over ${\mathbb{F}}_q$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G_k$ for a finite extension $k$ of ${\mathbb{F}}_q$ that determines the ranks of the graded pieces of the filtrations. The other incarnation is a certain $G$-torsor analogue of the notion of $F$-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form $\left[E_{G,\chi }\setminus G_k\right]$ that was studied in our earlier paper (Doc. Math. 16 (2011), 253–300). By the results of this work they are therefore smooth algebraic stacks of dimension $0$ over $k$. Using our previous work we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others.

For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary $F$-zips, a part of which has been described before by Moonen and Wedhorn (Int. Math. Res. Not. 2004:72, 3855–3903). The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof) and to truncated Barsotti–Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl–Oort stratification of the special fibers of arbitrary Shimura varieties.

Keywords

F-zips, Ekedahl–Oort strata, De Rham cohomology

Mathematical Subject Classification 2010

Primary: 14F40, 14L30, 18D10, 20G40

Secondary: 14L05

Milestones

Received: 5 September 2013

Revised: 3 April 2014

Accepted: 4 April 2014

Published: 2 March 2015