$F$ -具有附加结构的拉链
$F$-zips with additional structure
摘要：
有限域上概型$S$上的$F$ -zip是半线性代数中的一个特定对象，它由一个具有降滤子和升滤子的局部自由模以及相应分次层之间的$\Frob_q$ -twisted同构组成.本文对任意约化线性代数群$G$定义并系统地研究了“具有$G$ -结构的$F$ -拉链”.这些物体有两个化身。一个化身是从$G$的有限维表示范畴到$F$ -zips over $S$范畴的精确线性张量函子。局部地，任何这样的函子都有一个类型$χ$，它是$G$的辅字符。另一个化身是$F$ -拉链概念的某种$G$ -torsor类似物。我们证明，这两个化身定义栈是自然等价的商栈的形式$[E_{G,χ}\backslash G_k]$，在较早的论文中进行了研究。因此，根据那里得到的结果，它们是$k$上的0维光滑代数叠加。利用我们以前的结果，我们也可以分类的同构类，这样的对象在一个代数闭域，描述他们的自同构群，并确定哪些同构类可以退化成其他。对于典型群，我们可以推出相应的结果扭曲或非扭曲辛，正交，或酉$F$ -拉链。这些结果可以应用于光滑射影簇的代数de Rham上同调（或其推广，如光滑适当的Deligne-Mumford栈）和截断的Barsotti-Tate群的水平1。此外，我们希望我们系统的群论方法将有助于理解任意志村品种的特殊纤维的Ekedahl-Oort分层的类似物。
英文摘要：
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 $χ$, 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 $[E_{G,χ}\backslash G_k]$ 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.
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