Invariant fields of rational functions and semilinear representations of symmetric groups over them

Abstract: Let $K$ be a field and $G$ be a group of its automorphisms endowed with the compact-open topology. There are many situations, where it is natural to study the category $Sm_K(G)$ of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. The category $Sm_K(G)$ is semisimple (in which case $K$ is a generator of $Sm_K(G)$) if and only if $G$ is precompact. In this note we study the case of the non-precompact group $G$ of all permutations of an infinite set $S$. It is shown that the categories $Sm_K(G)$ are locally noetherian; the morphisms are locally split'. Given a field $F$ and a subfield $k\neq F$ algebraically closed in $F$, one of principal results describes the Gabriel spectra (and related objects) of the categories $Sm_K(G)$ for some of $G$-invariant subfields $K$ of the fraction field $F_{k,S}$ of the tensor product over $k$ of the labeled by $S$ copies of $F$. In particular, the object $F_{k,S}$ turns out to be an injective cogenerator of the category $Sm_K(G)$ for any $G$-invariant subfield $K$ of $F_{k,S}$. As an application, when transcendence degree of $F|k$ is 1, a correspondence between the $G$-invariant subfields of $F_{k,S}$ algebraically closed in $F_{k,S}$ and certain systems of isogenies ofgenerically $F$-pointed' torsors over absolutely irreducible one-dimensional algebraic $k$-groups is constructed, so far only in characteristic 0. The only irreducible finite-dimensional smooth representation of $G$ is trivial. However, there is an invariant `cross-ratio' subfield $K$ of $k(S)$ such that the irreducible finite-dimensional smooth $K$-semilinear representations of $G$ correspond to the irreducible algebraic representations of $PGL_{2,k}$. In general, any smooth $G$-field $K$ admits a smooth $G$-field extension $L|K$ such that $L$ is a cogenerator of $Sm_L(G)$.