$F$-zips with additional structure

Abstract

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module 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 $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G$. 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 $[E_{G,\chi }\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results 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. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) 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.

Remark 9.20 explained, otherwise minor changes and corrections; final version, to appear in Pacific Journal of Math.; 50 pages

References in corpus (6)

• The μ-ordinary locus for Shimura varieties of Hodge type
• Discrete invariants of varieties in positive characteristic
• The $G$-stable pieces of the wonderful compactification
• Graded and Filtered Fiber Functors on Tannakian Categories
• Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
• Purity of Zip Strata with Applications to Ekedahl-Oort Strata of Shimura Varieties of Hodge Type

Cited by in corpus (24)

• The μ-ordinary locus for Shimura varieties of Hodge type
• Tautological rings of Shimura varieties and cycle classes of Ekedahl-Oort strata
• Stratifications and foliations for good reductions of Shimura varieties of Hodge type
• Generalized mu-ordinary Hasse invariants
• On some generalized Rapoport-Zink spaces
• EKOR strata for Shimura varieties with parahoric level structure
• Bruhat strata and F-zips with additional structures
• Purity of Zip Strata with Applications to Ekedahl-Oort Strata of Shimura Varieties of Hodge Type
• Ekedahl-Oort strata for good reductions of Shimura varieties of Hodge type
• Normalization of closed Ekedahl-Oort strata
• Remarks on Ekedahl-Oort stratifications
• Geometric Satake, categorical traces, and arithmetic of Shimura varieties
• Automorphic vector bundles with global sections on $G$-${\mathtt{Zip}}^{\mathcal{Z}}$-schemes
• Ekedahl-Oort stratifications of Shimura varieties via Breuil-Kisin windows
• Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic
• A Tannakian framework for $G$-displays and Rapoport-Zink spaces
• Automorphic vector bundles on the stack of $G$-zips
• Partial Hasse invariants for Shimura varieties of Hodge-type
• An alternative construction of zip period maps for Shimura varieties
• The zeta function of stacks of $G$-zips and truncated Barsotti-Tate groups
• EKOR strata on Shimura varieties with parahoric reduction
• Exterior powers of F-zips
• Lifting Chern classes by means of Ekedahl-Oort strata
• Tautological rings of Hilbert modular varieties