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

TL;DR

This paper investigates the structure of smooth semilinear representations of the infinite permutation group, revealing their local noetherianity, injective cogenerators, and connections to algebraic geometry and classical groups.

Contribution

It characterizes the categories of semilinear representations for non-precompact permutation groups, describes their Gabriel spectra, and links invariant subfields to algebraic torsors and classical group representations.

Findings

Categories are locally noetherian and morphisms are locally split.

The object $F_{k,S}$ is an injective cogenerator in the category.

Finite-dimensional irreducible representations are trivial, with notable exceptions related to $PGL_{2,k}$.

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 of `generically $F$-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 $Sm_L(G)$.