Frobenius semisimplicity for convolution morphisms
Frobenius semisimplicity for convolution morphisms
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卷积态射的 Frobenius 半单纯性
DOI：
10.1007/s00209-017-1946-4
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发表时间：
2016
影响因子：
0.8
通讯作者：
Li Li
中科院分区：
文献类型：
--
作者：
M. A. Cataldo;T. Haines;Li Li
本文关注有限域上的混合$$\ell $-adic复形的性质，与Frobenius自同构的作用有关。我们建立了有限域上直像复形在簇的真态射下的半单性和Frobenius半单性的纤维准则。我们猜想域上的交复形的直像总是半单的和Frobenius半单的;这个猜想意味着Beilinson-Bernstein-Deligne-Gabber分解定理的一个强形式在有限域上是有效的。我们证明了我们的猜想（广义）卷积态射与部分仿射旗品种分裂连通约化群在有限域上。作为一个重要的工具，我们发展了一个新的图式理论的大细胞环群。通过适当的变换，主要结果在任何代数闭基域上都是有效的。
This article concerns properties of mixed $$\ell $$ℓ-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson–Bernstein–Deligne–Gabber is valid over finite fields. We prove our conjecture for (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields. As a crucial tool, we develop a new schematic theory of big cells for loop groups. With suitable reformulations, the main results are valid over any algebraically closed ground field.