137 research outputs found
On maximal proper subgroups of field automorphism groups
Let be the automorphism group of an extension of algebraically
closed fields of characteristic zero and of transcendence degree , .
In this paper we (i) construct some maximal closed non-open subgroups ,
and some (all, in the case of countable transcendence degree) maximal open
proper subgroups of ; (ii) describe, in the case of countable transcendence
degree, the automorphism subgroups over the intermediate subfields (a question
of Krull, \cite[\S4, question 3b)]{krull}); (iii) construct, in the case
, a fully faithful subfunctor of the forgetful functor from
the category of smooth representations of to the category of smooth
representations of ; (iv) construct, using the functors , a
subfunctor of the identity functor on the category of smooth
representations of , coincident (via the forgetful functor) with the functor
on the category of smooth admissible semilinear representations of
constructed in \cite{adm} in the case and .
The study of open subgroups is motivated by the study of (the stabilizers of
the) smooth representations undertaken in \cite{repr,adm}. The functor
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'.Comment: final versio
On semilinear representations of the infinite symmetric group
In this note the smooth (i.e. with open stabilizers) linear and {\sl
semilinear} representations of certain permutation groups (such as infinite
symmetric group or automorphism group of an infinite-dimensional vector space
over a finite field) are studied. Many results here are well-known to the
experts, at least in the case of {\sl linear representations} of symmetric
group. The presented results suggest, in particular, that an analogue of
Hilbert's Theorem 90 should hold: in the case of faithful action of the group
on the base field the irreducible smooth semilinear representations are
one-dimensional (and trivial in appropriate sense).Comment: 19 pages, significant changes; an analogue of Hilbert's Theorem 90
for infinite symmetric groups moved to arXiv:1508.0226
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