Semilinear representations of symmetric groups and of automorphism groups of universal domains (Q1652898)

Let \(K\) be a field and let \(G\) be a group of automorphisms of \(K\), endowed with the compact-open topology. The goal of this paper is to study the smooth semilinear representations of the group \(\mathcal{S}_\Psi\) of all permutations of an infinite set \(\Psi\). Suppose \(k\) is a field and \(k(\Psi)\) is the field freely generated by \(\Psi\) over \(k\). The author describes the Gabriel spectrum of the category of smooth \(k(\Psi)\)-semilinear representations of \(\mathcal{S}_\Psi\). In particular, it is shown that for any smooth \(\mathcal{S}_\Psi\)-field \(K\) any smooth finitely generated \(K\)-semilinear representation of \(\mathcal{S}_\Psi\) is Noetherian. Moreover, for any \(\mathcal{S}_\Psi\)-invariant subfield \(K\) of \(k(\Psi)\), the object \(k(\Psi)\) is an injective cogenerator of the category of smooth \(K\)-semilinear representations of \(\mathcal{S}_\Psi\). Additional properties are proven concerning the subfield of \(k(\Psi)\) of rational homogeneous functions of degree 0, respectively the subfield generated over \(k\) by \(x-y\) for all \(x,y\in \Psi\).