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：

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关于场自同构群的最大适当子群

让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”。