$F$-zips with additional structure
By R. Pink, T. Wedhorn, Paul Ziegler. 2012
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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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/80287edd5276cf8eea3d62765f66554be0dca2ac
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Citations in the corpus
- Strata Hasse invariants, Hecke algebras and Galois representations (2015, background): “rtain subset IWof the Weyl group Wof the reductive group attached to the PEL-structure. In the third stage, in a series of papers, Moonen, Wedhorn, Pink and Ziegler defined the algebraic stack G-Zipµ, [82, 107, 87, 88], whose k-points parametrize isomorphism classes of BT1’s with G-structure over k. ”
- Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type (2013, methodology): “-varieties We will describe the dimension and Zariski closure of an Ekedahl-Oort stratum. The key points are our Proposition 3.1.6 and that the partial order on JW is finer than the Bruhat order (see [41] 1.5). Keep notations as what we did before Proposition 4.4.1. We will compute the case when nis ”