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On maximal proper subgroups of field automorphism groups
Selecta Mathematica ( IF 1.2 ) Pub Date : 2006 , DOI: 10.1007/s00029-009-0520-2
M. Rovinsky

Let G be the automorphism group of an extension \(F\mid k\) of algebraically closed fields of characteristic zero of transcendence degree n, 1 ≤ n ≤ ∞. In this paper we
  • construct some maximal closed non-open subgroups G v , and some (all, in the case of countable transcendence degree) maximal open proper subgroups of G;
  • describe, in the case of countable transcendence degree, the automorphism subgroups over the intermediate subfields (a question of Krull, [K2, §4, question 3b)]);
  • construct, in the case n = ∞, a fully faithful subfunctor ( − ) v of the forgetful functor from the category \({\mathcal{S}}m_G\) of smooth representations of G to the category of smooth representations of G v ;
  • construct, using the functors ( − ) v , a subfunctor Γ of the identity functor on \({\mathcal{S}}m_{G}\), coincident (via the forgetful functor) with the functor Γ on the category of admissible semilinear representations of G constructed in [R3] in the case n = ∞ and \(k = \overline{\mathbb{Q}}\).
The study of open subgroups is motivated by the study of (the stabilizers of) smooth representations undertaken in [R1, R3]. The functor Γ is an analogue of the global sections functor on the category of sheaves on a smooth proper algebraic variety. Another result is that ‘interesting’ semilinear representations are ‘globally generated’.


中文翻译:

关于场自同构群的最大适当子群

G ^是一个扩展的自同构组\(F \中间ķ\)超越次数的特征零的代数闭域的Ñ,1≤  Ñ ≤∞。在本文中,我们
  • 构造一些最大的封闭的非开放子群G v,以及一些(在超越度可数的情况下)全部的G的最大开放的适当子群;
  • 在可数超越程度的情况下,描述中间子域上的自同构子群(Krull问题,[K2,§4,问题3b)]);
  • 构建体中,在的情况下Ñ  =∞,完全忠实subfunctor( - )v从类别健忘函子\({\ mathcal {S}} m_G \)的平滑表示的ģ到的平滑表示的类别ģ v ;
  • 使用函子(−)v构造\({\ mathcal {S}} m_ {G} \)上身份函子的子函子Γ ,(通过健忘函子)与函子Γ在可允许的类别上重合在n  =∞和\(k = \ overline {\ mathbb {Q}} \)的情况下,用[R3]构造的G的半线性表示。
[R1,R3]中进行的光滑表示(的稳定器)的研究激发了开放子组的研究。函子Γ是光滑的适当代数形式上的滑轮类上的整体截面函子的类似物。另一个结果是“有趣的”半线性表示是“全局生成的”。
更新日期:2020-09-24
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