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

M.Rovinsky  National Research University Higher School of Economics, AG
Laboratory, HSE, 6 Usacheva str., Moscow, Russia, 119048 & Institute for
Information Transmission Problems of Russian Academy of Sciences
<EMAIL_ADDRESS>
###### Abstract.

Let $K$ be a field and $G$ be a group of its automorphisms endowed with the
compact-open topology, cf. §1.1. There are many situations, where it is
natural to study the category $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ of smooth
(i.e. with open stabilizers) $K$-semilinear representations of $G$.

According to Hilbert’s Theorem 90, cf. Proposition 2.2, the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is semisimple (in which case $K$ is a
generator of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$) if and only if $G$ is
precompact. In this note we study the case of the non-precompact group
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$ of all permutations of an infinite
set $\Psi$.

It is shown that the categories
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ 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 (Theorem 1.3) describes the Gabriel spectra (and related
objects) of the categories
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ for
some of $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant subfields $K$ of the
fraction field $F_{k,\Psi}$ of the tensor product over $k$ of the labeled by
$\Psi$ copies of $F$. In particular, the object $F_{k,\Psi}$ turns out to be
an injective cogenerator of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ for
any $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant subfield $K\subseteq
F_{k,\Psi}$.

As an application of Theorem 1.3, when transcendence degree of $F|k$ is 1, a
correspondence between the $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant
subfields of $F_{k,\Psi}$ algebraically closed in $F_{k,\Psi}$ 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
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ is trivial. However, under some mild
restriction on the field $k$, for each integer $1\leq n\leq 5$ it is possible
to find an invariant subfield $K$ of $k(\Psi)$ and an irreducible non-trivial
$n$-dimensional smooth $K$-semilinear representation of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$. It is likely that there are no such
representations for $n>5$.

In general, any smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field $K$
admits a smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field extension
$\widetilde{K}|K$ such that $\widetilde{K}$ is a cogenerator of
$\mathop{\mathrm{Sm}}\nolimits_{\widetilde{K}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
cf. Proposition 4.29.

Appendix collects some results on smooth characteristic 0 linear
representations of symmetric groups and of the automorphism group of an
infinite-dimensional vector space over a finite field.

##  1\. Introduction

###  1.1. Motivation and goals

Let $G$ be a group of permutations of a set $\Psi$. We consider $G$ as a
totally disconnected group whose base of topology is generated by the left (or
right) translates of the stabilizers of the elements of $\Psi$.

The problem of describing continuous representations of $G$ in discrete vector
spaces (i.e., with open stabilizers), called smooth in what follows, arises
naturally in many situations. E.g., some birational ‘motivic’ algebro-
geometric questions over a characteristic 0 algebraically closed field $k$ are
related to describing certain irreducible smooth representations of the
automorphism group $G$ of an algebraically closed extension $C|k$ of infinite
transcendence degree. There are some reasons to expect that the numerous
‘interesting’ representations can be embedded into the single $C$-vector space
of differential forms $\Omega^{\bullet}_{C|k}=\bigoplus_{q\geq
0}\Omega^{q}_{C|k}$. Note that the $G$-action on $\Omega^{q}_{C|k}$ is
$C$-semilinear. This leads to the study of smooth semilinear representations
of $G$.

Classical examples of the study of smooth semilinear representations are given
by (i) Hilbert’s Theorem 90 ([18, Satz 1]) stating that smooth semilinear
Galois representations are ‘trivial’, and (ii) the theory of ($q$-)difference
equations. The first case where $G$ is not locally (pre)compact (namely, $G$
is the symmetric group $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ of all
permutations of an infinite set $\Psi$) is considered in [16, 8].

In analogy with Hilbert’s Theorem 90, one may ask whether for a given
permutation group $G$ and a field $K$ endowed with a smooth $G$-action there
exists a smooth $G$-field extension $B|K$ such that all smooth semilinear
representations of $G$ are ‘as trivial as they can be’.

The purpose of this paper is (i) to describe the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ of
all smooth $K$-semilinear representations of the symmetric group
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ over other fields $K$ and (ii) to
construct, for an arbitrary field $K$ endowed with a smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action, a smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field extension $B|K$, called a ‘weak
period’ field extension, such that $B$ is a cogenerator of the category
$\mathop{\mathrm{Sm}}\nolimits_{B}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

Compared to the ‘usual’ (i.e. $k$-linear for a field $k$) representations, the
series of examples of extensions $K|k$ considered here ($K_{?}$ for
$?\in\\{\mathrm{a,b,c}\\}$, depending on a field extension $F|k$, see below)
have the feature that the structure of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
(e.g. its Gabriel spectrum) depends only on the label $?$ of $K$, but not on
$F|k$ (with a minor modification if the characteristic of $k$ is 2 and $F=k$).

Though the case of the group $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ is a toy
model of this kind of problems, a faithful and exact functor relates the
‘algebro-geometric’ category $\mathop{\mathrm{Sm}}\nolimits_{C}(G)$ of
‘quasicoherent sheaves in the smooth topology’ to the category
$\mathop{\mathrm{Sm}}\nolimits_{k(\Psi)}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
where $k(\Psi)=k(x~{}|~{}x\in\Psi)$ is the field of rational functions over
$k$ in variables labeled by $\Psi$ (and endowed with the natural
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action). In particular, at least some
of ‘interesting’ objects of $\mathop{\mathrm{Sm}}\nolimits_{C}(G)$ are
transformed to injective objects of
$\mathop{\mathrm{Sm}}\nolimits_{k(\Psi)}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

###  1.2. Basic notation

For an abelian group $A$ and a set $S$ we denote by $A\langle S\rangle$ the
abelian group which is the direct sum of copies of $A$ indexed by $S$, i.e.,
the elements of $A\langle S\rangle$ are the finite formal sums
$\sum_{i=1}^{N}a_{i}[s_{i}]$ for all integer $N\geq 0$, $a_{i}\in A$,
$s_{i}\in S$, with addition defined in the obvious way.

For a group $G$, a unital associative ring $A$ endowed with a $G$-action (by
ring automorphisms), a subgroup $H\subseteq G$, we consider $A\langle
G\rangle$ as a unital associative ring; and more generally, we consider
$A\langle G/H\rangle$ as an $A\langle G\rangle$-$A^{H}\langle
N_{G}(H)/H\rangle$-bimodule:
$(a[g])(b[g^{\prime}\pmod{H}])(c[h^{\prime}]):=ab^{g}c^{gg^{\prime}}[gg^{\prime}h^{\prime}]$
for all $g,g^{\prime}\in G$, $h^{\prime}\in N_{G}(H)/H$, $a,b\in A$ and $c\in
A^{H}$, where we write $a^{g}$ for the result of applying of $g$ to $a$.

If $S$ is a $G$-set, we consider $A\langle S\rangle$ as a left $A\langle
G\rangle$-module with the diagonal $G$-action.

The left $A\langle G\rangle$-modules are also called $A$-semilinear
representations of $G$.

For each set $S$ and an integer $s\geq 0$, denote by $\binom{S}{s}$ the set of
all subsets of $S$ of cardinality $s$. Denote by
$\mathop{\mathfrak{S}}\nolimits_{S}$ the group of all permutations of the set
$S$.

####  1.2.1. Notation for fields

For a field $L$ and a set $S$, $L(S)$ denotes the purely transcendental field
extension of $L$ with $S$ as a transcendence base (the fraction field of the
symmetric $L$-algebra of $L\langle S\rangle$). If a group acts on $L$ and on
$S$ then it acts on $L(S)$ as well. For a subfield $K$ of $L$,
$\mathop{\mathrm{Aut}}\nolimits(L|K)$ denotes the group of field automorphisms
of $L$ identical on $K$.

The fields $F_{S}=F_{k,S}$ for some field extensions $F|k$ and arbitrary sets
$S$ are defined in §1.3.

The $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-fields $K_{?}$ for a label
$?\in\\{\mathrm{a,b,c,d}\\}$ are defined in Theorem 1.3 and Lemma 5.4.

####  1.2.2. Notation for categories of smooth semilinear representations and
their Picard groups

For an associative ring $A$ and a permutation group $G$ (cf. Definition 2.1)
acting on $A$, denote by $\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ the category
of smooth left $A$-semilinear representations of $G$. E.g.,
$\mathop{\mathrm{Sm}}\nolimits_{A}(1)$ is the category of left $A$-modules.

For an object $X$ of a Grothendieck category, $E(X)$ denotes an injective
envelope of $X$.

######  Remark 1.1.

If $A$ is a division ring and there exists a non-zero object of
$\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ then the $G$-action on $A$ is smooth.
[Indeed, for any $A$-semilinear representation $V$ of $G$, any non-torsion
$v\in V$, any non-zero $a\in A$, one has
$\mathrm{Stab}_{av}\cap\mathrm{Stab}_{v}=\mathrm{Stab}_{a}\cap\mathrm{Stab}_{v}$.
If $V$ is smooth then the group on the left hand side is open, so the
stabilizer $\mathrm{Stab}_{a}$ is open, i.e. the $G$-action on $A$ is smooth.]

If $K$ is a smooth $G$-field then $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is a
tensor $k$-linear Grothendieck category, where $k:=K^{G}$. The smooth finite-
dimensional $K$-semilinear representations of $G$ form a rigid tannakian
category ($K$ is the unit, $\mathop{\mathrm{End}}\nolimits_{K\langle
G\rangle}(K)=k$; the duals are
$V^{\vee}:=\mathop{\mathrm{Hom}}\nolimits_{K}(V,K)$).

Denote by $\mathrm{Pic}_{K}(G)$ the group of isomorphism classes of invertible
objects of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, and set
$\mathrm{Pic}_{K}:=\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

###  1.3. Results

We are looking for an analogue of Hilbert’s Theorem 90 (cf. Proposition 2.2)
for the symmetric group $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ of an infinite
set $\Psi$, i.e. we study the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ of
smooth $K$-semilinear representations of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$, where $K$ is a smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field.

It is shown in Corollary 4.15 that, for any left noetherian associative ring
$A$ endowed with a smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action, the
category
$\mathop{\mathrm{Sm}}\nolimits_{A}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ is
locally noetherian.

It is possible to describe the ‘local’ structure of finitely generated objects
in $\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
(Proposition 4.6), and it turns out that the morphisms between them are split
‘locally’ (Theorem 4.17). This is related to a canonical ‘level’ filtration on
the objects, introduced in §4.6.

One more general result is that any smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field $K$ admits a smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field extension $\widetilde{K}|K$ such
that $\widetilde{K}$ is a cogenerator of
$\mathop{\mathrm{Sm}}\nolimits_{\widetilde{K}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
However, $\widetilde{K}$ constructed in Proposition 4.29 is not optimal, in
the sense that $\widetilde{K}^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}\neq
K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$.

Let $k$ be a field and $\\{A_{i}\\}_{i\in S}$ be a collection of unital
associative $k$-algebras, indexed by a set $S$. Denote by
$\bigotimes_{k,~{}i\in S}A_{i}$ the coproduct of $k$-algebras
$\\{A_{i}\\}_{i\in S}$. More explicitly, $\bigotimes_{k,~{}i\in
S}A_{i}=\mathop{\underrightarrow{\lim}}\limits_{I\subset
S}\bigotimes_{k,~{}i\in I}A_{i}$ is inductive limit of the system of the
finite tensor products $\bigotimes_{k,~{}i\in I}A_{i}$ over $k$ for all finite
subsets $I\subset S$, where the transition maps $\bigotimes_{k,~{}i\in
I}A_{i}\to\bigotimes_{k,~{}i\in I^{\prime}}A_{i}$, $I\subset I^{\prime}$, are
induced by the identifications $\bigotimes_{k,~{}i\in
I}A_{i}=\bigotimes_{k,~{}i\in I}A_{i}\otimes_{k}\bigotimes_{k,~{}i\in
I^{\prime}\smallsetminus I}k\subseteq\bigotimes_{k,~{}i\in I^{\prime}}A_{i}$.
In other words, $\bigotimes_{k,~{}i\in S}A_{i}$ consists of finite linear
combinations of tensor products of elements in $A_{i}$, almost all equal to 1.

Let $F$ be a field and $k$ be a subfield algebraically closed in $F$. For a
set $S$, denote by $F_{S}=F_{k,S}$ the field of fractions of the $k$-algebra
$\bigotimes_{k,~{}i\in S}F$, so $F_{\varnothing}:=k$. Obviously, $F_{S}=k$ if
$F=k$.

Each inclusion $S^{\prime}\subseteq S$ induces a field inclusion
$F_{S^{\prime}}\subseteq F_{S}$. In particular, each $x\in S$ determines a
field embedding $F\xrightarrow{\sim}F_{x}:=F_{\\{x\\}}\subseteq F_{S}$,
$f\mapsto f(x)$, so $F_{x}=\\{f(x)~{}|~{}f\in F\\}$. If a group $G$ acts on
$S$ and on $F$ then $G$ acts on $\bigotimes_{k,~{}i\in S}F$ by permuting the
tensor factors ($\sigma:\otimes_{i\in S}x_{i}\mapsto\otimes_{i\in
S}x_{\sigma^{-1}(i)}^{\sigma}$), and thus, $G$ acts on the field $F_{S}$. This
action is smooth if the $G$-action on $F(S)$ is smooth.

######  Definition 1.2.

The Gabriel spectrum of a Grothendieck category $\mathcal{C}$ is the
topological space whose points are isomorphism classes of indecomposable
injectives. For each object $X$ of $\mathcal{C}$ denote by $[X]$ the set of
points $E$ with $\mathop{\mathrm{Hom}}\nolimits(X,E)=0$. Then base of opens
consists of sets of the form $[X]$ as $X$ ranges over the compact (finitely
presented) objects. As $[X]\cap[Y]=[X\oplus Y]$, these sets are closed under
finite intersection, so an arbitrary open set will have the form
$\bigcup_{i}[X_{i}]$ with some compact $X_{i}$.

Some remarks on Gabriel spectra are collected in §5.4.

####  1.3.1. The fields $K_{\mathrm{a}}$

Recall that a family $\mathcal{S}$ of objects in a category is generating
(resp., cogenerating) if any pair of distinct morphisms
$g_{1},g_{2}:X\rightrightarrows Y$ admits a morphism $\theta:S\to X$ (resp.,
$\theta:Y\to S$) with $S\in\mathcal{S}$ such that $g_{1}\circ\theta\neq
g_{2}\circ\theta$ (resp., $\theta\circ g_{1}\neq\theta\circ g_{2}$). For a
Grothendieck category, a family $\mathcal{S}$ is generating (resp.,
cogenerating) if and only if any object is a quotient of a direct sum (resp.,
a subobject of a direct product) of objects in $\mathcal{S}$. An object $U$ is
called a generator (resp., cogenerator) if the family $\\{U\\}$ generating
(resp., cogenerating).

######  Theorem 1.3.

Let $\Psi$ be an infinite set, $K$ be a field endowed with a non-trivial
smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action, $\mathrm{Spec}_{K}$ be
the Gabriel spectrum of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
Fix an arbitrary element $x\in\Psi$.

Then, for all integer $s\geq 0$, the injective hulls of the objects
$K\langle\binom{\Psi}{s}\rangle$ of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
represent pairwise distinct points $P_{s}=P_{s}^{(K)}$ of $\mathrm{Spec}_{K}$.
Any set containing $P_{s}$ for infinitely many $s\geq 0$ is dense in
$\mathrm{Spec}_{K}$. The closure of $P_{s}$ in the set $\\{P_{t}~{}|~{}t\geq
0\\}$ is $\\{P_{t}~{}|~{}0\leq t\leq s\\}$ if $s\geq s_{0}(K)$, where
$[K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}:K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}]=\infty$
for any $J$ of order $\geq s_{0}(K)$.

Let $F|k$ be an extension of fields of characteristic $p\geq 0$ such that $k$
is algebraically closed in $F$.

For each label $?\in\\{\mathrm{a,b,c}\\}$, consider the following
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-fields $K_{?}:$

  * •

for a finite set $S$, non-empty if $F=k$, and a subgroup
$\Gamma\subseteq\Xi:=\mathbb{Z}\langle S\rangle$,

$K_{\mathrm{a}}=K_{\Psi,S,\Gamma}^{(F|k)}:=F_{\Psi}\left(u^{\gamma},\frac{u_{s}}{v_{s}}~{}|~{}\gamma\in\Gamma,~{}s\in
S,~{}u,v\in\Psi\right)=F_{\Psi}\left(x^{\gamma},\frac{u_{s}}{x_{s}}~{}|~{}\gamma\in\Gamma,~{}s\in
S,~{}u\in\Psi\right)$

is a subfield of the purely transcendental field extension
$F_{\Psi}(\Psi\times S)=F(S)_{\Psi}$ of $F_{\Psi}$ with a transcendence basis
consisting of the variables labeled by the
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-set $\Psi\times S$, where $u_{s}$
denotes the variable corresponding to $(u,s)\in\Psi\times S$ and
$u^{\gamma}:=\prod_{s}u_{j_{s}}^{m_{s}}$ for all $u\in\Psi$ and
$\gamma=\sum_{s}m_{s}[j_{s}]\in\Xi$ $($so $K_{\mathrm{a}}:=F_{\Psi}$ if
$S=\varnothing;$

  * •

$K_{\mathrm{b}}:=F_{\Psi}(u-v~{}|~{}u,v\in\Psi)\subset
F_{\Psi}(\Psi)=F(X)_{\Psi}\ ($here $X$ is a variable$);$

  * •

$K_{\mathrm{c}}:=F_{\Psi}\left(\frac{u-v}{u-w}~{}|~{}u,v,w\in\Psi,~{}\\#\\{u,v,w\\}=3\right)\subset
F_{\Psi}(\Psi).$

  1. (1)

For each $?\in\\{\mathrm{a,b,c}\\}$, the simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{K_{?}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
are invertible.

     * •

The invertible objects $x^{\lambda}K_{\mathrm{a}}\subseteq F_{\Psi}(\Psi\times
S)$ of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
are injective for all $\lambda\in\Xi$.

     * •

$\mathrm{Pic}_{K_{\mathrm{a}}}\cong\Xi/\Gamma$ and its elements
$P_{0,\overline{\lambda}}$ for all $\overline{\lambda}\in\Xi/\Gamma$ are
presented by $x^{\lambda}K_{\mathrm{a}}$ for all $\lambda\in\Xi$.

     * •

$\mathrm{Pic}_{K_{\mathrm{b}}}=0$.

     * •

$\mathrm{Pic}_{K_{\mathrm{c}}}\cong\mathbb{Z}$ is generated by the class of
$(x-y)K_{\mathrm{c}}\subset F_{\Psi}(\Psi)$.

  2. (2)

Any point of $\mathrm{Spec}_{K_{?}}$ of level $n\geq 0$ is contained in the
$\mathrm{Pic}_{K_{?}}$-orbit of $P_{n}$, trivial if $n>2$.

  3. (3)

The objects of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
of finite length are semisimple; for any integer $N\geq 1$, the indecomposable
objects of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
of length $N$ are isomorphic, if $p=0$.

  4. (4)

The injection $E:\mathrm{Pic}_{K}\to\mathrm{Cl}_{K}$, $\mathcal{L}\mapsto
E(\mathcal{L})$, to the set of closed points of $\mathrm{Spec}_{K}$, commutes
with the $\mathrm{Pic}_{K}$-action, it is bijective for $K=K_{\mathrm{b}}$ and
for $K=K_{\mathrm{a}}$ if $F\neq k$ or $\Gamma\neq 0;$

     * •

$\mathrm{Cl}_{K_{\mathrm{a}}}\smallsetminus\mathrm{Pic}_{K_{\mathrm{a}}}=\\{P_{1}\\}$
if $\Xi\neq\Gamma=0$ and $F=k$ $($in this case any infinite subset of
$\mathrm{Cl}_{K_{\mathrm{a}}}$ is dense$)$,

     * •

$\mathrm{Cl}_{K_{\mathrm{b}}}$ consists of the class $P_{0}$ of the $($non-
noetherian, artinian if and only if $p=0)$ object
$E(K_{\mathrm{b}})=K_{\mathrm{b}}[x]\subset F_{\Psi}(\Psi)$,

     * •

$\mathrm{Cl}_{K_{\mathrm{c}}}$ is the union of the free
$\mathrm{Pic}_{K_{\mathrm{c}}}$-orbits of $P_{0}$ and of $P_{1}$, where
$P_{0}$ is the class of
$E(K_{\mathrm{c}})=K_{\mathrm{c}}\left[\frac{x}{x-y}\right]\subset
F_{\Psi}(\Psi)$ and $P_{1}$ is the class of
$K_{\mathrm{c}}\langle\Psi\rangle$.

  5. (5)

For each $?\in\\{\mathrm{a,b,c}\\}$, the non-closed points of
$\mathrm{Spec}_{K_{?}}$ are represented by noetherian objects. The non-closed
points are $P_{2},P_{3},\dots$ and

     * •

$P_{1}$ if either $?=\mathrm{c}$ or $?=\mathrm{b}$ and $F\neq k$ or
$?=\mathrm{a}$ and $\Gamma\neq 0$,

     * •

$P_{2}^{\prime}$ of $(x-y)\cdot
K_{\mathrm{c}}\langle\binom{\Psi}{2}\rangle\subset
K_{\mathrm{b}}\langle\binom{\Psi}{2}\rangle$ if $?=\mathrm{c}$, $F=k$ and
$p\neq 2$.

If, for an integer $s\geq 1$, $K_{?}\langle\binom{\Psi}{s}\rangle$ is not
injective then $?\in\\{\mathrm{b,c}\\}$, $p=2$, $s=2$, $F=k$ and
$E(K_{?}\langle\binom{\Psi}{2}\rangle)\cong
K_{?}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$, where $\Delta_{\Psi}$
is the diagonal in $\Psi^{2}$.

  6. (6)

For $?\in\\{\mathrm{a,b}\\}$ and any $s>0$, the closure of $P_{s}$ is the set
$\mathrm{Cl}_{K_{?}}\cup\\{P_{1},P_{2},\dots,P_{s}\\}$.

For $?=\mathrm{c}$, the closure of $P_{s}$ $($for $s\geq 2)$ is

     * •

$\mathrm{Cl}_{K_{\mathrm{c}}}\cup\\{P_{1,n}~{}|~{}n\in\mathbb{Z}\\}\cup\\{P_{2},\dots,P_{s}\\}$
if $s>2$ and $p=2$,

     * •

$\mathrm{Cl}_{K_{\mathrm{c}}}\cup\\{P_{1,n}~{}|~{}n\in\mathbb{Z}\\}\cup\\{P_{2}^{\prime},P_{2},\dots,P_{s}\\}$
if $s>2$ and $p\neq 2$,

     * •

$\mathrm{Cl}_{K_{\mathrm{c}}}\cup\\{P_{1,n}~{}|~{}(-1)^{n}=1\mbox{ {\rm in}
}k\\}\cup\\{P_{2}\\}$ if $s=2;$

the closure of $P_{1,n}$ is $\mathrm{Cl}_{K_{\mathrm{c}}}\cup\\{P_{1,n}\\}$.

The fact that the points $P_{s}^{(K)}$ are well-defined and pairwise distinct
is explained in Remark 4.7. The density of infinite sets of $P_{n}$’s is shown
in Lemma 5.16 (1).

In terminology of [10, §4.20, p.295], the categories
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
for $\Xi=0$ and
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
are local, while
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
for $\Xi\neq 0$ and
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{c}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
are not.

######  Example 1.4.

If, for an $F|k$ as above, $F(X)$ is the field of rational functions over $F$
in one variable then $F(X)_{\Psi}$ is nothing but the field $F_{\Psi}(\Psi)$
of rational functions over $F_{\Psi}$ in the variables enumerated by the set
$\Psi$, while the group $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ acts on the
coefficients of rational functions and by permuting the variables. Moreover,
the points of the Gabriel spectrum are
$F(X)_{\Psi}\langle\binom{\Psi}{s}\rangle\cong\Omega^{s}_{F(X)_{\Psi}|F_{\Psi}}$
(the space of differential $s$-forms on $F(X)_{\Psi}$ over $F_{\Psi}$) for all
integer $s\geq 0$.

Theorem 1.3 provides several examples of fields $K$ and a non-precompact group
$G$ of their automorphisms such that the Gabriel spectrum (and in particular
the simple objects, cf. Lemma 5.15) of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$
admits an explicit description. In all examples
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$, though the description depends
crucially on the field $K$.

######  Remarks 1.5.

  1. (1)

If $k$ is a trivial $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field of
characteristic 0 then $k$ is an injective object of
$\mathop{\mathrm{Sm}}\nolimits_{k}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
cf. Theorem A.17. However, this is not the case if $\mathrm{char}(k)>0$, cf.
Proposition 5.29.

  2. (2)

Proposition 2.2 below implies that the pair $(G,K)$ cannot be reconstructed
from the category $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. However, for a given
set $\Psi$, one can reconstruct the field $K_{\mathrm{a}}$ with torsion free
$\mathrm{Pic}_{K_{\mathrm{a}}}$ from the category
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
as follows. The field $k$ is the endomorphism ring of any closed point of
$\mathrm{Spec}_{K_{\mathrm{a}}}$. Let $S$ be a basis of $\mathrm{Pic}_{K}$. If
there is a (unique) point $I_{1}$ of $\mathrm{Spec}_{K_{\mathrm{a}}}$ whose
closure is $\mathrm{Pic}_{K}\sqcup\\{I_{1}\\}$ (i.e. either $F\neq k$ or
$\Gamma\neq 0$) then
$L:=\mathop{\mathrm{End}}\nolimits_{\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})}(I_{1})$
is the fraction field of the group $F$-algebra of $\Gamma$, and
$K_{\mathrm{a}}\cong K_{\Psi,S,0}^{(L|k)}$. Otherwise, $K_{\mathrm{a}}\cong
K_{\Psi,S,0}^{(k|k)}$.

  3. (3)

Let $G$ be a permutation group and $L|K$ be a smooth $G$-field extension. Then

     * •

any injective object of the category $\mathop{\mathrm{Sm}}\nolimits_{L}(G)$ is
injective as an object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$;

     * •

any simple object $W$ of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ can be
embedded into any simple quotient in $\mathop{\mathrm{Sm}}\nolimits_{L}(G)$ of
the object $L\otimes_{K}W$.

######  Corollary 1.6.

Any smooth irreducible representation of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ over a field $k$ can be embedded into
$(x-y)^{n}k\left(\frac{u-v}{u-w}~{}|~{}u,v,w\in\Psi\right)\subset
x^{n}k\left(\frac{u}{v}~{}|~{}u,v\in\Psi\right)\subset k(\Psi)$ for some
$n\in\mathbb{Z}$, $x,y\in\Psi$.

This follows from Remark 1.5 (• ‣ 3) and Theorem 1.3 (1). ∎

#### 1.3.2.

Given a smooth $G$-field $K$, it is natural to study its $G$-invariant
sub-/sup- fields and to compare semilinear representations of $G$ over such
$G$-fields. In the case of $G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$ and
$K=F_{\Psi}$ for a transcendence degree 1 field extension $F|k$ of
characteristic 0, it is shown in Theorem 5.11 that if $K$ is an
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant field extension of $k$ in
$F_{\Psi}$ then the transcendence degree $d$ of $F_{\Psi}$ over $K$ is $\leq
3$. Moreover, the subfields $K$ admit an explicit description when $d\geq 1$
(and their isomorphism class depend only on $d$ if $d\geq 2$, while the case
$d=1$ admits more options). The case $d=0$ is also interesting, since it is
there where there exist smooth irreducible semilinear representations of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ of finite dimensions $>1$ (namely,
$2,\ 3,\ 4,\ 5$), cf. Examples 5.18, 5.23, 5.24. In particular, though in
Theorem 1.3 all points of a given level form a $\mathrm{Pic}_{K_{?}}$-orbit,
this is not the case if $K_{?}$ is replaced by an arbitrary $K$, even in level
0.

As for smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field extensions $L|K$
with
$L^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$,
Example 5.21 lists some of $K$ with no non-trivial finite smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field extensions, while Proposition
5.25 shows that the only ‘isotrivial’ finitely generated $L|F_{\Psi}$ is
trivial: $L=F_{\Psi}$.

#### 1.3.3.

In the appendix, for $\Psi$ being either a countable set or a vector space
over a finite field, some results on the category of smooth representations
over a field of characteristic 0 of the automorphism group of $\Psi$ are
proved. It is shown (Lemma A.8) that this category is locally of finite
length, while the principal result (Theorem A.17) is a bijective
correspondence between the simple objects and the indecomposable injectives
for this category (resulting in a description of the Gabriel spectrum of this
category).

##  2\. Hilbert’s theorem 90 and period fields

######  Definition 2.1.

A permutation group is a Hausdorff topological group $G$ admitting a base of
open subsets consisting of the left and right shifts of subgroups.

If we denote by $B$ a collection of open subgroups such that the finite
intersections of conjugates of elements of $B$ form a base of open
neighbourhoods of $1$ in $G$ (e.g., the set of all open subgroups of $G$),
then $G$ acts faithfully on the set $\Psi:=\coprod_{U\in B}G/U$, so (i) $G$
becomes a permutation group of the set $\Psi$, (ii) the shifts of the
pointwise stabilizers $G_{T}$ of the finite subsets $T\subset\Psi$ form a base
of the topology of $G$. Clearly, $G$ is totally disconnected.

The following is a version of Speiser’s generalization of Hilbert’s theorem
90, cf. [18, Satz 1], or [17, Prop. 3, p.159], or [3, Proposition 5].

######  Proposition 2.2.

Let $K$ be a field of characteristic $p\geq 0$ endowed with a smooth action of
a permutation group $G$, and $H\subseteq G$ be a closed normal subgroup. Then

  * •

the functor
$\Gamma_{H}:\mathop{\mathrm{Sm}}\nolimits_{K}(G)\to\mathop{\mathrm{Sm}}\nolimits_{K^{H}}(G/H)$,
$V\mapsto V^{H}$, is an equivalence if and only if $H$ acts on $K$ faithfully
and $H$ is precompact, i.e., any open subgroup of $H$ is of finite index;

  * •

the category $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is semisimple if and only
if $G$ is precompact and $p$ divides index of no open subgroup of the action
kernel $N:=\ker[G\to\mathop{\mathrm{Aut}}\nolimits(K)]$.

###### Proof.

As it is noted after Definition 2.1, $G$ is a permutation group of a set
$\Psi$, so $K\langle\Psi\rangle$ is a smooth faithful representation of $G$.
If $\Gamma_{H}$ is an equivalence then its left adjoint $K\otimes_{K^{H}}(-)$
should be quasi-inverse. Obviously, the adjunction map
$K\otimes_{K^{H}}K\langle\Psi\rangle^{H}\to K\langle\Psi\rangle$ is surjective
only if the $H$-action on $K$ is faithful. On the other hand, $K\langle
G/U\rangle^{H}\neq 0$ for each open subgroup $U\subseteq G$, so $U\cap H$ is
of finite index in $H$, i.e. $H$ is precompact.

If $G$ is finite then [18, Satz 1], appropriately reformulated, implies that
any $K$-semilinear representation of $G$ is a sum of copies of $K$. Namely,
with $k:=K^{G}$, the field extension $K|k$ is finite, so the multiplication
and the natural $G$-action on $K$ give rise to a $k$-algebra homomorphism from
$K\langle G\rangle$ to the algebra $\mathop{\mathrm{End}}\nolimits_{k}(K)$ of
endomorphisms of $K$ considered as $k$-vector space, which is (a) surjective
by Jacobson’s density theorem and (b) injective by the independence of
characters. Then any $K\langle G\rangle$-module is isomorphic to a direct sum
of copies of $K$.

If $H$ is precompact and faithful on $K$, we have to check that the adjunction
map $K\otimes_{K^{H}}V^{H}\xrightarrow{\xi}V$ is surjective for any object $V$
of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. For each $v\in V$, consider the
intersection $S$ of all conjugates of the stabilizer of $v$ in $H$. Thus, $v$
is contained in the $K^{S}$-semilinear representation $V^{S}$ of the group
$H/S$. As $H/S$ is finite and the action of $H/S$ on $K^{S}$ is faithful (cf.
[6, p.151]),
$V^{S}=K^{S}\otimes_{(K^{S})^{H/S}}(V^{S})^{H/S}=K^{S}\otimes_{K^{H}}V^{H}$,
i.e., $v$ is contained in the image of $\xi$.

If $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is semisimple then the projection
$\pi_{U}:K\langle G/U\rangle\xrightarrow{[g]\mapsto 1}K$, splits for any open
subgroup $U\subseteq G$, so $K\langle G/U\rangle^{G}$ contains some
$\alpha\neq 0$, and thus, $[G:U]<\infty$, i.e. $G$ is precompact. Then
$\alpha=\sum_{i}a_{i}\sum_{[g]\in O_{i}}[g]$, where $O_{i}$ are $N$-orbits on
$G/U$, so $O_{i}\cong N/(U\cap N)$, and therefore,
$\pi_{U}(\alpha)=\\#(N/(U\cap N))\sum_{i}a_{i}$. Then $\pi_{U}(\alpha)\neq 0$
only if $p\nmid[N:U\cap N]$.

Conversely, under our assumption on $N$, the categories
$\mathop{\mathrm{Sm}}\nolimits_{K}(N)$ and
$\mathop{\mathrm{Sm}}\nolimits_{K}(G/N)$ are semisimple, so for any
$V\in\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ the functors
$\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathrm{Sm}}\nolimits_{K}(N)}(V,-):\mathop{\mathrm{Sm}}\nolimits_{K}(G)\to\mathop{\mathrm{Sm}}\nolimits_{K}(G/N)$
and
$\Gamma_{G/N}:\mathop{\mathrm{Sm}}\nolimits_{K}(G/N)\to\mathop{\mathrm{Sm}}\nolimits_{K^{G}}(1)$
are exact. Then their composition
$\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathrm{Sm}}\nolimits_{K}(G)}(V,-):\mathop{\mathrm{Sm}}\nolimits_{K}(G)\to\mathop{\mathrm{Sm}}\nolimits_{K^{G}}(1)$
is exact as well, so
$\mathrm{Ext}^{>0}_{\mathop{\mathrm{Sm}}\nolimits_{K}(G)}(-,-)=0$, i.e.,
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is semisimple. ∎

######  Example 2.3.

Let $\Psi$ be an infinite-dimensional vector space over a finite field
$\mathbb{F}_{q}$, $J\subset\Psi$ be a finite-dimensional subspace, $G_{J}$ be
the pointwise stabilizer of $J$ in $\mathrm{GL}(\Psi)$,
$H\cong\mathbb{F}_{q}^{\times}\ltimes\mathop{\mathrm{Hom}}\nolimits_{\mathbb{F}_{q}}(\Psi/J,J)$
be the maximal normal proper subgroup of $G_{J}$, $K$ be a field endowed with
a smooth and faithful $H$-action. Then $H$ is compact and
$\Gamma_{H}:\mathop{\mathrm{Sm}}\nolimits_{K}(G_{J})\to\mathop{\mathrm{Sm}}\nolimits_{K^{H}}(\mathrm{PGL}(\Psi/J))$,
$V\mapsto V^{H}$, is an equivalence of categories.

######  Definition 2.4.

For a permutation group $G$ and a smooth $G$-field $F$, a smooth $G$-field
extension $K|F$ is called a $G$-period extension of $F$ if $K$ is a
cogenerator of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ and, moreover,
$K^{G}=F^{G}$.

The first natural questions are: Given a permutation group $G$ and a smooth
$G$-field $F$, does there exist a $G$-period extension of $F$? Can it be
chosen to be ‘minimal’ in some sense?

##  3\. Open subgroups, permutation modules, generators and projectives

A permutation group $G$ is called unibased if, for any open proper subgroup,
the finite intersections of its conjugates form a base of open subgroups of
$G$.

A reason to introduce this class is a ‘simplicity’ property (Proposition 3.3):
any non-zero Grothendieck subcategory of
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is equivalent to
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ itself.

######  Example 3.1.

The following groups are unibased.

(i) Simple finite groups.

(ii) Let $G$ be the automorphism group modulo centre of one of the following
structures: an infinite set $\Psi$ (i.e.
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$), an infinite-dimensional vector
space $V$ (i.e. $G=\mathrm{PGL}(V)$), an algebraically closed field extension
$F|k$ of infinite transcendence degree of an algebraically closed field $k$
(i.e. $G=\mathop{\mathrm{Aut}}\nolimits(F|k)$).

######  Lemma 3.2.

Any unibased group $G$ is topologically simple; its topology is induced by any
non-trivial continuous homomorphism $\varphi$ from $G$ to a permutation group.

###### Proof.

If $S$ is a non-trivial closed normal proper subgroup of $G$, fix an element
$g\in G\smallsetminus S$ and a neighbourhood of $g$ in $G\smallsetminus S$. We
may choose this neighbourhood of the form $gU$ for an open subgroup $U$ of
$G$. As $g\notin SU$, $SU$ is a proper open subgroup of $G$, while all
conjugates of $SU$ contain $S$, and thus, cannot generate a Hausdorff topology
on $G$.

The target of $\varphi$ admits an open subgroup $U$ that does not contain the
image of $\varphi$, so $\varphi^{-1}(U)$ is an open proper subgroup of $G$. ∎

######  Proposition 3.3.

Let $G$ be a unibased group, and $K$ be a non-trivial smooth $G$-field. Then
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is equivalent to its arbitrary non-zero
subcategory closed under direct products and subquotients.

###### Proof.

Let $V\neq 0$ be an object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. Consider
$G$ as a subgroup of $\mathop{\mathrm{Aut}}\nolimits(K)$. The objects
$K\langle G/G_{T}\rangle$ for all finite subsets $T\subseteq K$ such that
$V^{G_{T}}\neq 0$ form a system of generators of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, so it suffices to show that there is
an embedding of $K\langle G/G_{T}\rangle$ into a direct product of copies of
$V$, or equivalently, that there is a family of morphisms $K\langle
G/G_{T}\rangle\to V$ with vanishing common kernel.

Fix a nonzero $\alpha\in V^{G_{T}}$ and consider, for all $t\in K^{G_{T}}$,
the morphisms $t\alpha:K\langle G/G_{T}\rangle\to V$, $a[\sigma]\mapsto
a(t\alpha)^{\sigma}$, for all $a\in K$ and $\sigma\in G$. Let
$\xi=\sum_{i=1}^{N}a_{i}\sigma_{i}\in K\langle G/G_{T}\rangle$ be an element
in the common kernel of these morphisms. The elements of the set $G/G_{T}$ can
be considered as (pairwise distinct) $K^{\times}$-valued characters of the
group $(K^{G_{T}})^{\times}$, since $G_{K^{G_{T}}}=G_{T}$, so the element
$\xi$ can be considered as a $K$-linear relation between characters. Due to
the linear independence of such characters, one has $a_{1}=\dots=a_{N}=0$,
i.e., $\bigcap_{t\in(K^{G_{T}})^{\times}}\ker(K\langle
G/G_{T}\rangle\xrightarrow{t\alpha}V)=0$. ∎

######  Lemma 3.4.

Let $G$ be a permutation group, and $K$ be a $G$-field. Assume that for any
open subgroup $U\subset G$ there is an open subgroup $U^{\prime}\subset U$
such that any conjugate of $U^{\prime}$ meets $U$ in a subgroup of infinite
index in $U$. Then there are no non-zero projectives in
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$.

###### Proof.

Let $P\neq 0$ be an object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. Fix a
non-zero $e\in P$. For each $u\in P$, fix an open subgroup
$U_{u}\subset\mathrm{Stab}_{e}\cap\mathrm{Stab}_{u}$ such that any conjugate
of $U_{u}$ meets $\mathrm{Stab}_{e}$ in a subgroup of infinite index. Consider
the surjection $\bigoplus_{u\in P}K\langle G/U_{u}\rangle\xrightarrow{\pi}P$,
$[g]_{u}\mapsto gu$. If the support of an element in $K\langle
G/U_{u}\rangle^{\mathrm{Stab}_{e}}$ contains $[g]\in G/U_{u}$ then it contains
$\mathrm{Stab}_{e}gU_{u}/U_{u}$. This set is of the same cardinality as
$g^{-1}\mathrm{Stab}_{e}gU_{u}/U_{u}=g^{-1}\mathrm{Stab}_{e}g/(g^{-1}\mathrm{Stab}_{e}g\cap
U_{u})\cong\mathrm{Stab}_{e}/(\mathrm{Stab}_{e}\cap gU_{u}g^{-1})$. As the
latter set is infinite, $K\langle G/U_{u}\rangle^{\mathrm{Stab}_{e}}=0$, so
$\pi$ does not split, and thus, $P$ is not projective. ∎

######  Example 3.5.

The automorphism groups of infinite sets, infinite-dimensional vector spaces,
algebraically closed field extensions $F|k$ of infinite transcendence degree
satisfy assumptions of Lemma 3.4.

For any set $\Psi$ and a subset $T\subseteq\Psi$, we denote by
$\mathop{\mathfrak{S}}\nolimits_{\Psi|T}$ the pointwise stabilizer of $T$ in
the group $\mathop{\mathfrak{S}}\nolimits_{\Psi}$. Let
$\mathop{\mathfrak{S}}\nolimits_{\Psi,T}:=\mathop{\mathfrak{S}}\nolimits_{\Psi\smallsetminus
T}\times\mathop{\mathfrak{S}}\nolimits_{T}$ denote the group of all
permutations of $\Psi$ preserving $T$ (in other words, the setwise stabilizer
of $T$ in the group $\mathop{\mathfrak{S}}\nolimits_{\Psi}$, or equivalently,
the normalizer of $\mathop{\mathfrak{S}}\nolimits_{\Psi|T}$ in
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$).

###  3.1. Substructures

Let $G\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi}$ be a permutation group
of a set $\Psi$.

For a subset $S\subset\Psi$, (i) we denote by $G_{S}$ the pointwise stabilizer
of the set $S$; (ii) we call the fixed set $\Psi^{G_{S}}$ the $G$-closure of
$S$. We say that a subset $S\subset\Psi$ is $G$-closed if $S=\Psi^{G_{S}}$.

Any intersection $\bigcap_{i}S_{i}$ of $G$-closed sets $S_{i}$ is $G$-closed:
as $G_{S_{i}}\subseteq G_{\bigcap_{j}S_{j}}$, one has $G_{S_{i}}s=s$ for any
$s\in\Psi^{G_{\bigcap_{j}S_{j}}}$, so $s\in\Psi^{G_{S_{i}}}=S_{i}$ for any
$i$, and thus, $s\in\bigcap_{i}S_{i}$. This implies that the subgroup
generated by $G_{S_{i}}$’s is dense in $G_{\bigcap_{i}S_{i}}$ (and coincides
with $G_{\bigcap_{i}S_{i}}$ if at least one of $G_{S_{i}}$’s is open).

The $G$-closed subsets of $\Psi$ form a small category with the morphisms
being all those embeddings that are induced by elements of $G$:
$\mathop{\mathrm{Hom}}\nolimits(X,Y):=\\{g\in G~{}|~{}g(X)\subseteq
Y\\}/G_{X}$.

For a $G$-closed subset $T\subset\Psi$, (hiding $G$ and $\Psi$ from notation)
set $\mathop{\mathrm{Aut}}\nolimits(T):=N_{G}(G_{T})/G_{T}$.

######  Lemma 3.6.

Let $\Psi$ be either (i) an infinite set, or (ii) an infinite-dimensional
vector space over a field, or (iii) an algebraically closed field extension of
a field $k$ of infinite transcendence degree. Let $G$ be the automorphism
group of $\Psi$, i.e. either $\mathop{\mathfrak{S}}\nolimits_{\Psi}$, or
$\mathrm{GL}(\Psi)$, or the field automorphism group of $\Psi$ over $k$. For
an integer $n$, let $T\subset\Psi$ be a subset of order $n$, resp. a subspace
of dimension $n$, resp. a subfield of transcendence degree $n$ over $k$. Then
there exist elements $g_{0},\dots,g_{n}\in G$ such that
$G=\bigcup_{i=0}^{n}G_{T}g_{i}G_{T}g_{i}^{-1}$.

Let, in the cases (i) and (ii), $T^{\prime}\subset\Psi$ be a finite $G$-closed
subset, while $T,T^{\prime}$ be algebraically closed subfields in $\Psi|k$ of
finite transcendence degree in the case (iii). Then the subgroups $G_{T}$ and
$G_{T^{\prime}}$ generate the subgroup $G_{T\cap T^{\prime}}$. If
$T\subsetneqq T^{\prime}$ and $V^{G_{T}}=V^{G_{T^{\prime}}}$ for a $G$-set $V$
then $V^{G_{T}}=V^{G}$.

###### Proof.

Choose $n+1$ subsets (resp. subspaces, resp. subfields) in general position
$T_{0},\dots,T_{n}$ isomorphic to $T$, i.e.
$\\#(T\cup\bigcup_{i=0}^{n}T_{i})=(n+2)n$ (resp.
$\dim(T+\sum_{i=0}^{n}T_{i})=(n+2)n$, resp. $\mathrm{tr.deg}(TT_{0}\cdots
T_{n}|k)=(n+2)n$). Looking at the $G_{T}$-orbit of the identical embedding
$T_{i}\hookrightarrow\Psi$, we see that $G_{T}G_{T_{i}}=\\{g\in
G~{}|~{}g(T)\cap T_{i}=\varnothing\\}$ (resp. $G_{T}G_{T_{i}}=\\{g\in
G~{}|~{}\dim(g(T)+T_{i})=2n\\}$, resp. $G_{T}G_{T_{i}}=\\{g\in
G~{}|~{}\mathrm{tr.deg}(g(T)T_{i}|k)=2n\\}$), so
$G=\bigcup_{i=0}^{n}G_{T}g_{i}G_{T}g_{i}^{-1}$ for elements
$g_{0},\dots,g_{n}\in G$ such that $g_{i}(T)=T_{i}$ for all $i$.

More generally, let us show that $G_{T}G_{T^{\prime}}=\\{g\in G_{T\cap
T^{\prime}}~{}|~{}g(T^{\prime})\cap T=T\cap T^{\prime}\\}=:\Xi$. The inclusion
$\subseteq$ is trivial. On the other hand, the set
$\Xi/G_{T^{\prime}}=G_{T}/(G_{T}\cap G_{T^{\prime}})$ consists of all
embeddings of $T\cup T^{\prime}$ (or rather of its $G$-closure) into $\Psi$
identical on $T$ that are induced by elements of $G$, while such embeddings
form a $G_{T}$-orbit.

Let $T_{0}\subseteq T$ be a minimal $G$-closed subset such that
$V^{G_{T}}=V^{G_{T_{0}}}$. Assuming $T_{0}$ is not initial (i.e.
$\neq\varnothing$ or 0), let $g\in N_{G}(G_{T^{\prime}})$ be an element such
that $g(T_{0})\neq T_{0}$. Then

$V^{G_{T^{\prime}}}=V^{G_{T^{\prime}}}\cap
g(V^{G_{T^{\prime}}})=V^{G_{T_{0}}}\cap g(V^{G_{T_{0}}})=V^{G_{T_{0}}}\cap
V^{G_{g(T_{0})}}=V^{\langle G_{T_{0}},G_{g(T_{0})}\rangle}=V^{G_{T_{0}\cap
g(T_{0})}},$

contradicting the minimality of $T_{0}$. ∎

######  Lemma 3.7.

Let $G$ be either $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ for an infinite set
$\Psi$, or $\mathrm{GL}_{\mathbb{F}_{q}}(\Psi)$ for an infinite-dimensional
vector space $\Psi$ over a finite field $\mathbb{F}_{q}$. For any open
subgroup $U$ of $G$ there exists a unique $G$-closed subset $T\subset\Psi$
such that $G_{T}\subseteq U$ and the following equivalent conditions hold: (a)
$T$ is minimal; (b) $G_{T}$ is normal in $U$; (c) $G_{T}$ is of finite index
in $U$. In particular, (i) such $T$ is finite, (ii) the open subgroups of $G$
correspond bijectively to the pairs $(T,H)$ consisting of a finite $G$-closed
subset $T\subset\Psi$ and a subgroup
$H\subseteq\mathop{\mathrm{Aut}}\nolimits(T)$ under $(T,H)\mapsto\\{g\in
N_{G}(G_{T})~{}|~{}\text{{\rm restriction of $g$ to $T$ belongs to $H$}}\\}$.

###### Proof.

Any open subgroup $U$ in $G$ contains the subgroup $G_{T}$ for a finite
$G$-closed subset $T\subset\Psi$. Assume that $T$ is chosen to be minimal. If
$\sigma\in U$ then $U\supseteq\sigma G_{T}\sigma^{-1}=G_{\sigma(T)}$, and
therefore, (i) $\sigma(T)$ is also minimal, (ii) $U$ contains the subgroup
generated by $G_{\sigma(T)}$ and $G_{T}$. By Lemma 3.6, the subgroup generated
by $G_{\sigma(T)}$ and $G_{T}$ is $G_{T\cap\sigma(T)}$ and thus, $U$ contains
the subgroup $G_{T\cap\sigma(T)}$. The minimality of $T$ means that
$T=\sigma(T)$, i.e., $U\subseteq N_{G}(G_{T})$. If $T^{\prime}\subset\Psi$ is
another minimal subset such that $G_{T^{\prime}}\subseteq U$ then, by Lemma
3.6, $G_{T\cap T^{\prime}}\subseteq U$, so $T=T^{\prime}$, which proves (b)
and (the uniqueness in the case) (a). It follows from (b) that $G_{T}\subseteq
U\subseteq N_{G}(G_{T})$, so $G_{T}$ is of finite index in $U$. As the
subgroups $G_{T}$ and $G_{T^{\prime}}$ are not commensurable for
$T^{\prime}\neq T$, we get the uniqueness in the case (c). ∎

######  Lemma 3.8.

Let $G$ be a group acting on a field $K$. Let $U\subset G$ be a subgroup of
infinite index. Let $K^{\prime}\subseteq K$ be a $G$-invariant subfield. Then
there are no non-zero $K^{\prime}\langle G\rangle$-submodules in $K\langle
G/U\rangle$ finite-dimensional over $K^{\prime}$.

###### Proof.

Let $V\subseteq K\langle G/U\rangle$ be a $K^{\prime}\langle
G\rangle$-submodule finite-dimensional over $K^{\prime}$, and $S\subset G/U$
be the union of the supports in $G/U$ of all elements of $V$. Then $S$ is
finite and $G$-stable. However, the only finite and $G$-stable subset of $G/U$
is empty, so $V=0$. ∎

######  Lemma 3.9.

Let $G$ be a group acting on a field $K$. Then the following conditions on a
subgroup $U\subseteq G$ are equivalent:

  * •

an element $g\in G$ acts identically on $K^{U}$ if and only if $g\in U$;

  * •

$K\langle G/U\rangle$ embeds into a product of copies of $K$.

Assume that $U$ satisfies these conditions, and $V\xrightarrow{\lambda}K$ is a
morphism of $K\langle G\rangle$-modules inducing a non-zero map $V^{U}\to
K^{U}$. Then $K\langle G/U\rangle$ embeds into a product of copies of $V$.

###### Proof.

For any $K\langle G\rangle$-module $V$, define a morphism of $K\langle
G\rangle$-modules $K\langle
G/U\rangle\xrightarrow{\varphi_{V}}\mathop{\mathrm{Hom}}\nolimits_{K^{G}}(V^{U},V)$
by $\sum_{g}b_{g}[g]\mapsto[v\mapsto\sum_{g}b_{g}v^{g}]$. The first condition
means that the elements $[g]\in G/U$ can be considered as certain pairwise
distinct one-dimensional characters $\chi_{[g]}:(K^{U})^{\times}\to
K^{\times}$. By Artin’s independence of characters theorem, the characters
$\chi_{[g]}$ are linearly independent in the $K$-vector space of all functions
$(K^{U})^{\times}\to K$, so $\varphi_{K}$ is injective. Conversely, if
$\varphi_{K}$ is injective then $\chi_{[g]}\neq\chi_{[1]}$ for any $[g]\in
G/U\smallsetminus\\{[1]\\}$, i.e. $g$ acts non-trivially on $K^{U}$.

Fix some $v\in V^{U}$ with $\lambda(v)=1$. Then the composition

$K\langle
G/U\rangle\xrightarrow{\varphi_{V}}\mathop{\mathrm{Hom}}\nolimits_{K^{G}}(V^{U},V)\xrightarrow{\lambda}\mathop{\mathrm{Hom}}\nolimits_{K^{G}}(V^{U},K)\xrightarrow{|_{K^{U}\cdot
v}}\mathop{\mathrm{Hom}}\nolimits_{K^{G}}(K^{U}\cdot
v,K)\cong\mathop{\mathrm{Hom}}\nolimits_{K^{G}}(K^{U},K)$

coincides with $\varphi_{K}$, which is injective. ∎

######  Lemma 3.10.

Let $G$ be a group, $K$ be a $G$-field, $k\subseteq K^{G}$ be a subfield,
$F|k$ be a field extension. Assume that $k$ is algebraically closed in $K$.
Let $L$ be the fraction field of $K\otimes_{k}F$.

Then any simple $K\langle G\rangle$-submodule $M$ of $L$ coincides with
$K\cdot a$ for some non-zero $a\in L^{G}$.

###### Proof.

Let $Q\in L^{\times}$ be a non-zero element of $M$, so $Q=\alpha/\beta$ is a
ratio of a pair of elements $\alpha,\beta\in K\otimes_{k}A$ for a finitely
generated $k$-subalgebra $A$ of $F$. There is a finite field extension
$k^{\prime}|k$ and a $k$-algebra homomorphism $\varphi:A\to k^{\prime}$ with a
non-zero image of $\alpha\beta$ under the $G$-equivariant $K$-algebra
homomorphism $\Phi:K\otimes_{k}A\to K\otimes_{k}k^{\prime}$,
$\sum_{i}b_{i}\otimes a_{i}\mapsto\sum_{i}b_{i}\varphi(a_{i})$. The
homomorphism $\Phi$ extends to $(K\otimes_{k}A)[g\beta^{-1}~{}|~{}g\in G]$
with a non-zero restriction to a morphism of left $K\langle G\rangle$-modules
$M\to K\otimes_{k}k^{\prime}\cong\bigoplus_{\text{a $k$-basis of
$[k^{\prime}:k]$}}K$, so $M\cong K$, and thus, $M=K\cdot a$ for some non-zero
$a\in M^{G}\subseteq L^{G}$. ∎

######  Remark 3.11.

If $G$ has no open proper subgroups of finite index (e.g. if
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$ for an infinite set $\Psi$) and $K$
is smooth then $K^{G}$ is algebraically closed in $K$: any finite extension of
$K^{G}$ in $K$ is fixed by an open subgroup of $G$ of finite index, i.e. by
$G$.

If $K^{G}$ is algebraically closed in $K$ then all $G$-fields $L$ in Lemma
3.10 come from the particular case $k=K^{G}$, since $L$ is the fraction field
of $K\otimes_{K^{G}}\tilde{F}$, where $\tilde{F}$ is the fraction field of
$K^{G}\otimes_{k}F$.

######  Corollary 3.12.

Let $\Psi$ be an infinite set, $J\subset\Psi$ be a subset with infinite
complement. Let $F$ be a field and $k\subsetneqq F$ be a subfield
algebraically closed in $F$. Then any simple $F_{\Psi\smallsetminus
J}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-submodule $M$ of
$F_{\Psi}$ coincides with $aF_{\Psi\smallsetminus J}$ for some $a\in
F_{J}^{\times}$. In particular, $M$ is isomorphic to $F_{\Psi\smallsetminus
J}$.

###### Proof.

This is Lemma 3.10 for $G=\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$,
$K=F_{\Psi\smallsetminus J}$, $k=K^{G}$, with $F$ replaced by $F_{J}$ (so
$L=F_{\Psi}$). ∎

######  Lemma 3.13.

Let $G$ be a group acting on a field $K$. Let $U\subset G$ be a subgroup of
infinite index such that an element $g\in G$ acts identically on $K^{U}$ if
and only if $g\in U$. Set $k:=K^{G}$. Let $F|k$ be a field extension such that
$k$ is algebraically closed either in $F$ or in $K$. Let $L$ be the fraction
field of $K\otimes_{k}F$. Then

  * •

$L\langle G/U\rangle$ contains no simple $K\langle G\rangle$-submodules;

  * •

for each integer $s\geq 0$, there is a natural bijection between
$s$-dimensional $k$-vector subspaces $\Xi$ in $K^{U}$ and $K\langle
G\rangle$-submodules $V_{\Xi}$ in $K\langle G/U\rangle$ such that $K\langle
G/U\rangle/V_{\Xi}\cong K^{s}$ in $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$.

###### Proof.

In notation of Lemma 3.9, $\varphi_{L}$ is injective. Then, for any non-zero
element $\alpha\in L\langle G/U\rangle$, there exists an element $Q\in L^{U}$
such that the composition $L\langle
G/U\rangle\xrightarrow{\varphi_{L}}\mathop{\mathrm{Hom}}\nolimits_{F}(L^{U},L)\xrightarrow{\text{evaluation
at $Q$}}L$ does not vanish at $\alpha$. Then $\alpha$ generates a
$K$-semilinear subrepresentation $V$ admitting a non-zero morphism to $L$. By
Lemma 3.10, $V$ is isomorphic to $K$, contradicting Lemma 3.8.

The image of $\varphi_{K}$ is dense, which means that for any finite-
dimensional $k$-vector subspace $\Xi\subset K^{U}$ the composition $K\langle
G/U\rangle\xrightarrow{\varphi_{K}}\mathop{\mathrm{Hom}}\nolimits_{k}(K^{U},K)\xrightarrow{\text{restriction
to $\Xi$}}\mathop{\mathrm{Hom}}\nolimits_{k}(\Xi,K)$ is surjective: otherwise
there was a non-zero element $\sum_{i}a_{i}\otimes f_{i}\in K\otimes_{k}\Xi$
vanishing on the image, i.e., $\sum_{i}a_{i}\otimes f_{i}^{g}=0$ for all $g\in
G/U$, contradicting linear independence of characters $(K^{U})^{\times}\to
K^{\times}$.

To each $k$-vector subspace $\Xi\subset K^{U}$ we associate the $K\langle
G\rangle$-submodule $V_{\Xi}:=\bigcap_{\xi\in\Xi}\ker\xi$, where all $\xi$’s
are considered as morphisms $K\langle G/U\rangle\to K$. Conversely, to each
$K\langle G\rangle$-submodule $V\subset K\langle G/U\rangle$ we associate the
$k$-vector subspace $\Xi_{V}:=\bigcap_{v\in V}\ker v\subset K^{U}$, where all
$v$’s are considered as $k$-linear maps $K^{U}\to K$. ∎

Example. Let $G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$ and
$U\subset\mathop{\mathfrak{S}}\nolimits_{\Psi}$ be a maximal proper subgroup,
i.e., $U=\mathop{\mathfrak{S}}\nolimits_{\Psi,I}$ for a finite non-empty
subset $I\subset\Psi$ (so $\mathop{\mathfrak{S}}\nolimits_{\Psi}/U$ can be
identified with the set $\binom{\Psi}{\\#I}$). Suppose that
$K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,I}}\neq k$. Then we are under
assumptions of Lemma 3.13, so there are no irreducible $K$-semilinear
subrepresentations in $K\langle\binom{\Psi}{\\#I}\rangle$.

######  Remarks 3.14.

  1. (1)

Let $G$ be as in Lemma 3.7. It follow from the explicit description of open
subgroups in Lemma 3.7 that index of any proper open subgroup $U$ in $G$ is
infinite.

One cannot claim that, for an arbitrary smooth $K$, there are no elements in
$\mathop{\mathfrak{S}}\nolimits_{\Psi}\smallsetminus U$ acting identically on
$K^{U}$, even if $K^{U}\neq K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$. For
instance, if $K=F_{\binom{\Psi}{s}}$ for $s>1$,
$U=\mathop{\mathfrak{S}}\nolimits_{\Psi|S}$ for a finite $S$, $\\#S=s$, then
$K^{U}\cong F$, but $H=\mathop{\mathfrak{S}}\nolimits_{\Psi,S}$ acts trivially
on $K^{U}$.

However, this is the case if $K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|x}}\neq
K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$, or equivalently,
$K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}(\Psi)$ embeds into $K$.

  2. (2)

One cannot claim that $K\langle G/U\rangle$ is trivial, when $U$ is of finite
index in $G$, even if $G$ acts faithfully on $K$. E.g., let
$U=\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x,y\\}}$,
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x,y\\}}=\mathop{\mathfrak{S}}\nolimits_{2}\times
U$, $K=K^{\prime}=k(\frac{z-y}{x-y}~{}|~{}z\in\Psi\smallsetminus\\{x,y\\})$.
Then $K^{U}=k$ and $K\langle G/U\rangle^{G}=(K\langle
G/U\rangle^{U})^{G}=(k\langle G/U\rangle)^{G}=k([1]+[(12)])$.

######  Lemma 3.15.

Let $K$ be a field, $G$ be a group of automorphisms of the field $K$. Let
$U\subseteq H\subseteq G$ be open subgroups of $G$, where the index of $U$ in
$H$ is finite. Suppose that $[K^{U}:K^{H}]=\\#(H/U)$. Then there is a natural
isomorphism $\rho_{G}:K\langle
G/H\rangle\otimes_{K^{H}}K^{U}\xrightarrow{\sim}K\langle G/U\rangle$, given by
$\sum_{i}a_{i}[g_{i}]\otimes f_{i}\mapsto\sum_{i}\sum_{[\xi]\in
G/U,~{}[\xi]\bmod H=[g_{i}]}a_{i}f_{i}^{\xi}[\xi]$.

###### Proof.

Any element $\alpha$ of $K\langle G/H\rangle\otimes_{K^{H}}K^{U}$ can be
presented as $\sum_{i}a_{i}[g_{i}]\otimes f_{i}$, where $[g_{i}]$ are
determined uniquely if they are pairwise distinct. Then $\rho_{G}(\alpha)=0$
only if $\sum_{i:~{}[\xi]\bmod H=[g_{i}]}a_{i}f_{i}^{\xi}=0$ for all $\xi$,
i.e. only if $\alpha=0$. As $[K^{U}:K^{H}]=\\#(H/U)$, $K$-dimensions of the
source and the target of the embedding $\rho_{H}:K\langle
H/H\rangle\otimes_{K^{H}}K^{U}=K\otimes_{K^{H}}K^{U}\to K\langle H/U\rangle$
coincide, so $\rho_{H}$ is bijective. Then the induction functor $K\langle
G\rangle\otimes_{K\langle H\rangle}(-)$ transforms the bijection $\rho_{H}$ to
the isomorphism $\rho_{G}$. ∎

######  Example 3.16.

If $G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$ for an infinite set $\Psi$ then,
by Lemma 3.7, the assumption of Lemma 3.15 for all $U\subseteq H$ is
equivalent to the faithfulness of the
$\mathop{\mathfrak{S}}\nolimits_{J}$-action on
$K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}$ for all finite subsets
$J\subset\Psi$. This is the case if, for a field $k$, $K$ contains one of the
following fields:

  * •

$k(x/y~{}|~{}x,y\in\Psi)$;

  * •

$k(x-y~{}|~{}x,y\in\Psi)$ and either $k$ is not of characteristic 2, or $U$
coincides with $\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x,y\\}}$ for no $x\neq
y$ in $\Psi$;

  * •

$F_{\Psi}\left(\frac{x-y}{x-z}~{}|~{}x,y,z\in\Psi\right)$,
$F_{\Psi}\left(\frac{(w-x)(y-z)}{(x-y)(z-w)}~{}|~{}w,x,y,z\in\Psi\right)$,
where $F\neq k$;

  * •

$k\left(\frac{x-y}{x-z}~{}|~{}x,y,z\in\Psi\right)$, where $U$ coincides with
$\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x,y\\}}$ for no $x\neq y$ in $\Psi$,

  * •

$k\left(\frac{(w-x)(y-z)}{(x-y)(z-w)}~{}|~{}w,x,y,z\in\Psi\right)$, where $U$
coincides with $\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x,y\\}}$ for no $x\neq
y$ in $\Psi$ and $U$ admits a subgroup
$\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x,y,z\\}}$ of index $\leq 3$ for no
pairwise distinct $x,y,z\in\Psi$.

Then Lemma 3.15 asserts that, for any such open subgroup $U\subseteq G$, one
has $K\langle G/U\rangle\cong K\langle\binom{\Psi}{s}\rangle^{r}$ for some
integer $s\geq 0$ and $r\geq 1$.

######  Definition 3.17.

A pair $(K,G)$ consisting of a permutation group $G$ and a $G$-field $K$ is
called a Galois pair if there is an open subgroup $U_{0}$ such that the
following equivalent conditions hold:

  1. (1)

for any open subgroup $U\subseteq U_{0}$ and any subgroup
$\widetilde{U}\subseteq G$ containing $U$ as a proper subgroup of finite index
one has $K^{U}\neq K^{\widetilde{U}}$;

  2. (2)

for any open subgroup $U\subseteq U_{0}$ and any subgroup
$\widetilde{U}\subseteq G$ containing $U$ as a proper normal subgroup of
finite index the $\widetilde{U}/U$-action on $K^{U}$ is faithful.

###### Proof.

(1)$\Rightarrow$(2). If $U\subseteq U_{0}$ is normal in $\widetilde{U}$ and
$h\in\widetilde{U}/U$ acts trivially on $K^{U}$ then $K^{U}=K^{\langle
g,U\rangle}$, where $g$ is a preimage of $h$ in $\widetilde{U}$, so $\langle
g,U\rangle=U$, i.e. $g\in U$, so the $\widetilde{U}/U$-action on $K^{U}$ is
faithful.

(2)$\Rightarrow$(1). If $U\subseteq U_{0}$ and $\widetilde{U}\subseteq G$
contains $U$ as a subgroup of finite index, set
$U^{\prime}:=\bigcap_{g\in\widetilde{U}}gUg^{-1}\subseteq U$. Then
$\widetilde{U}/U^{\prime}$ acts faithfully on $K^{U^{\prime}}$, i.e.
$\widetilde{U}/U^{\prime}$ is the Galois group of
$K^{U^{\prime}}|K^{\widetilde{U}}$, and therefore,
$K^{U}:=(K^{U^{\prime}})^{U/U^{\prime}}=(K^{U^{\prime}})^{\widetilde{U}/U^{\prime}}=:K^{\widetilde{U}}$
if and only if $U=\widetilde{U}$. ∎

######  Lemma 3.18.

Let $K$ be a field endowed with a faithful smooth $G$-action, where $G$ is
either $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ for an infinite set $\Psi$, or
a quotient by a central subgroup of the automorphism group of an infinite-
dimensional vector space $\Psi$ over a field. Then $(K,G)$ is a Galois pair.

###### Proof.

Let $J\subset\Psi$ be a maximal $G$-closed subset such that $K^{G_{J}}=K^{G}$,
and $x_{1},x_{2},\dots$ be a sequence of elements of $\Psi$ in general
position with respect to $J$, i.e. each $x_{i+1}$ is not in the $G$-closure
$J^{(i)}$ of $J\cup\\{x_{1},\dots,x_{i}\\}$. As $G_{J\cup\\{x_{i}\\}}$ and
$G_{J\cup\\{x_{j}\\}}$ generate $G_{J}$ for all $i\neq j$,
$K^{G_{J\cup\\{x_{i}\\}}}\cap K^{G_{J\cup\\{x_{j}\\}}}=K^{G_{J}}$, so the
group
$\mathop{\mathfrak{S}}\nolimits_{\\{x_{1},\dots,x_{N}\\}}\subset\mathop{\mathrm{Aut}}\nolimits(J^{(N)}|J)$
acts faithfully on $\sum_{i=1}^{N}K^{G_{J\cup\\{x_{i}\\}}}$. Then, for any
$N\geq 5$, the group $\mathop{\mathrm{Aut}}\nolimits(J^{(N)})$ acts faithfully
on $K^{G_{J^{(N)}}}\supseteq\sum_{i=1}^{N}K^{G_{J\cup\\{x_{i}\\}}}$, since all
non-central normal subgroups of $\mathop{\mathrm{Aut}}\nolimits(J^{(N)})$
(i.e. $\mathop{\mathfrak{S}}\nolimits_{J^{(N)}}$ and $\mathfrak{A}_{J^{(N)}}$
if $G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$;
$\mathrm{SL}(J^{(N)})\cdot\langle\mbox{central subgroup}\rangle$ if
$G=\mathrm{GL}(\Psi)$) meet the subgroup
$\mathop{\mathfrak{S}}\nolimits_{\\{x_{1},\dots,x_{N}\\}}\subset\mathop{\mathrm{Aut}}\nolimits(J^{(N)}|J)$
non-trivially. ∎

######  Lemma 3.19.

Let $K$ be a field and $G$ be a group of automorphisms of the field $K$. Let
$B$ be such a system of open subgroups of $G$ that any open subgroup contains
a subgroup conjugated, for some $H\in B$, to an open subgroup of finite index
in $H$. Suppose that $(K,G)$ is a Galois pair. Then the objects $K\langle
G/H\rangle$ for all $H\in B$ form a system of generators of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ of smooth $K$-semilinear
representations of $G$.

###### Proof.

Let $V$ be a smooth semilinear representation of $G$. Then the stabilizer of
any vector $v\in V$ is open, i.e., the stabilizer of some vector $v^{\prime}$
in the $G$-orbit of $v$ admits an open subgroup $U\subseteq G$ which is of
finite index in some $H\in B$. The $K$-linear envelope $W$ of the (finite)
$H$-orbit of $v^{\prime}$ is a smooth $K$-semilinear representation of $H$. We
may assume that $U$ is normal in $H$. Then the elements of the $H$-orbit of
$v^{\prime}$ are fixed by $U$, so $W=K\otimes_{K^{U}}W^{U}$.

If $(K,G)$ is a Galois pair then we may assume that $U\subseteq U_{0}$ as in
Definition 3.17, so the $H/U$-action on $K^{U}$ is faithful. By Proposition
2.2, this implies that
$W^{U}=K^{U}\otimes_{(K^{U})^{H/U}}(W^{U})^{H/U}=K^{U}\otimes_{K^{H}}W^{H}$.
Then $W=K\otimes_{K^{H}}W^{H}$, i.e., $v^{\prime}$ belongs to the $K$-linear
envelope of the $K^{H}$-vector subspace fixed by $H$. As a consequence, there
is a morphism from a finite cartesian power of $K\langle G/H\rangle$ to $V$,
containing $v^{\prime}$ (and therefore, containing $v$ as well) in the image.
∎

######  Example 3.20.

Let $K$ be a field endowed with a smooth faithful
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action. Then (i) the assumptions of
Lemma 3.19 hold if $B$ is the set of subgroups
$\mathop{\mathfrak{S}}\nolimits_{\Psi,T}$ for a collection of finite subsets
$T\subset\Psi$ of unbounded cardinality, (ii) $K\langle\binom{\Psi}{N}\rangle$
is isomorphic to
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathop{\mathfrak{S}}\nolimits_{\Psi,T}\rangle$
for any $T$ of order $N$.

Thus, any collection of the objects $K\langle\binom{\Psi}{N}\rangle$ for
infinitely many $N$ form a system of generators of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ of
smooth $K$-semilinear representations of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$.

Let $K$ admit an $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-equivariant field
embedding $\iota:k\left(\frac{x-y}{x-z}~{}|~{}x,y,z\in\Psi\right)\to K$ for a
prime field $k$. For each $n\in\mathbb{Z}$ and some $x\neq y$ in $\Psi$,
denote by $(x-y)^{n}K$ the one-dimensional $K$-vector space spanned by
$(x-y)^{n}$ endowed with the semilinear
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action
$\sigma:(x-y)^{n}\mapsto(x-y)^{n}\cdot\left(\frac{x^{\sigma}-y^{\sigma}}{x-y}\right)^{n}$.
Obviously, the isomorphism class of $(x-y)^{n}K$ is independent of
$x,y\in\Psi$. Then $n\mapsto(x-y)^{n}K$ gives rise to homomorphism from
$\mathbb{Z}$ to the Picard group of the isomorphism classes of invertible
objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
which is trivial if and only if $\iota$ extends to an
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-equivariant field embedding of
$k(u-v~{}|~{}u,v\in\Psi)$.

For any object $V$ of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
set $(x-y)^{n}V:=(x-y)^{n}K\otimes_{K}V$. Then

$\bigwedge\nolimits_{K}^{N}K\langle\Psi\rangle\xrightarrow[\sim]{[s_{1}]\wedge\dots\wedge[s_{N}]\mapsto\prod_{1\leq
i<j\leq
N}(s_{i}-s_{j})\\{s_{1},\dots,s_{N}\\}}(x-y)^{\binom{N}{2}}K\langle\binom{\Psi}{N}\rangle$

is an isomorphism. If $K=k(\Psi)$ for a
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-subfield $k$ then
$\bigwedge_{K}^{N}K\langle\Psi\rangle\cong\Omega^{N}_{K|k}$ (differential
$N$-forms on $K$ over $k$), $[s_{1}]\wedge\dots\wedge[s_{N}]\leftrightarrow
ds_{1}\wedge\dots\wedge ds_{N}$. ∎

######  Lemma 3.21.

Let $G$ be a group, $U\subseteq G$ be a subgroup such that
$[U:U\cap(gUg^{-1})]=\infty$ unless $g\in U$, $K$ be a $G$-field, and $V$ be
an invertible $K\langle G\rangle$-module $($i.e. $\dim_{K}V=1)$.

Then (i) $V\langle G/U\rangle$ is indecomposable; (ii) $V\langle
G/U\rangle\cong K\langle G/U\rangle$ if and only if $V^{U}\neq 0$.

###### Proof.

As $UgU=g(g^{-1}Ug)U$ consists of $[g^{-1}Ug:(g^{-1}Ug)\cap
U]=[U:U\cap(gUg^{-1})]$ classes in $G/U$, the only finite $U$-orbit on $G/U$
is $\\{[U]\\}$, so $V\langle G/U\rangle^{U}\subseteq V\cdot[U]$. Then
$\mathop{\mathrm{End}}\nolimits_{K\langle G\rangle}(V\langle
G/U\rangle)=\mathop{\mathrm{End}}\nolimits_{K\langle G\rangle}(K\langle
G/U\rangle)\cong K\langle G/U\rangle^{U}=K^{U}\cdot[U]$ is a field, so
$V\langle G/U\rangle$ is indecomposable. If $0\neq v\in V^{U}$ then $K\langle
G/U\rangle\to V\langle G/U\rangle$, $\sum_{[g]\in
G/U}a_{g}[g]\mapsto\sum_{[g]\in G/U}a_{g}v^{g}[g]$, is an isomorphism. ∎

For a wide class of categories, one can define the length of an object. In
particular, the length of a $G$-closed subset $X$ is the minimal cardinality
of the subsets $S\subset\Psi$ such that $X$ is the $G$-closure of $S$.

Examples. 1\. Let $\Psi$ be an infinite set, possibly endowed with a structure
of a projective space. Let $G$ be the group of automorphisms of $\Psi$,
respecting the structure, if any. Let $J$ be the $G$-closure of a finite
subset in $\Psi$, i.e., a finite subset or a finite-dimensional subspace. Let
$U$ be the stabilizer of $J$ in $G$. Then $G/U$ is identified with the set of
all $G$-closed subsets in $\Psi$ of the same length as $J$.

2\. By Lemma 3.21, the object $K\langle G/U\rangle$ of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is indecomposable in the following
examples:

  1. (1)

$G$ is the group of projective automorphisms of an infinite projective space
$\Psi$ (i.e., either $\Psi$ is infinite-dimensional, or $\Psi$ is defined over
an infinite field), $U$ is the setwise stabilizer in $G$ of a finite-
dimensional subspace $J\subseteq\Psi$. Then $G/U$ is identified with the
Grassmannian of all subspaces in $\Psi$ of the same dimension as $J$.

  2. (2)

$G$ is the group of permutations of an infinite set $\Psi$, $U$ is the
stabilizer in $G$ of a finite subset $J\subset\Psi$. Then $G/U$ is identified
with the set $\binom{\Psi}{\\#J}$ of all subsets in $\Psi$ of order $\\#J$.

  3. (3)

$G$ is the automorphism group of an algebraically closed extension $F$ of a
field $k$, $U$ is the stabilizer in $G$ of an algebraically closed
subextension $L|k$ of finite transcendence degree. Then $G/U$ is identified
with the set of all subextensions in $F|k$ isomorphic to $L|k$.

##  4\. Structure of smooth semilinear representations of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$

The following result will be used in the particular case of the trivial
$G$-action on the $A$-module $V$ (i.e., $\chi\equiv id_{V}$), claiming the
injectivity of the natural map $A\otimes_{A^{G}}V^{G}\to V$ (since
$V_{id_{V}}=V^{G}$).

######  Lemma 4.1.

Let $G$ be a group, $A$ be a division ring endowed with a $G$-action
$G\to\mathop{\mathrm{Aut}}\nolimits_{\mathrm{ring}}(A)$, $V$ be an $A\langle
G\rangle$-module and $\chi:G\to\mathop{\mathrm{Aut}}\nolimits_{A}(V)$ be a
$G$-action on the $A$-module $V$.

Set $V_{\chi}:=\\{w\in V~{}|~{}\sigma w=\chi(\sigma)w\text{ {\rm for all}
$\sigma\in G$}\\}$.

Then $V_{\chi}$ is an $A^{G}$-module and the natural map
$A\otimes_{A^{G}}V_{\chi}\to V$ is injective.

###### Proof.

This is well-known: Suppose that some elements $w_{1},\dots,w_{m}\in V_{\chi}$
are $A^{G}$-linearly independent, but $A$-linearly dependent for a minimal
$m\geq 2$. Then $w_{1}=\sum_{j=2}^{m}\lambda_{j}w_{j}$ for some
$\lambda_{j}\in A^{\times}$.

Applying $\sigma-\chi(\sigma)$ for each $\sigma\in G$ to both sides of the
latter equality, we get
$\sum_{j=2}^{m}(\lambda_{j}^{\sigma}-\lambda_{j})\chi(\sigma)w_{j}=0$, and
therefore, $\sum_{j=2}^{m}(\lambda_{j}^{\sigma}-\lambda_{j})w_{j}=0$. By the
minimality of $m$, one has $\lambda_{j}^{\sigma}-\lambda_{j}=0$ for each
$\sigma\in G$, so $\lambda_{j}\in A^{G}$ for any $j$, contradicting the
$A^{G}$-linear independence of $w_{1},\dots,w_{m}$. ∎

###  4.1. Growth estimates

For each integer $N\geq 0$, fix a $G$-closed subset $\Psi_{N}\subset\Psi$ of
length $N$. For a division ring endowed with a $G$-action and an $A\langle
G\rangle$-module $M$, define a function $d_{M}:{\mathbb{Z}}_{\geq
0}\to{\mathbb{Z}}_{\geq 0}\coprod\\{\infty\\}$ by
$d_{M}(N):=\dim_{A^{G_{\Psi_{N}}}}(M^{G_{\Psi_{N}}})$.

######  Lemma 4.2.

Let $G$ be either $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ $($and then $q:=1)$
or the group of automorphisms of an $\mathbb{F}_{q}$-vector space $\Psi$
identical on a fixed subspace of finite dimension $v\geq 0$. Then, for any
integer $N\geq 0$, the $G$-closed subsets of length $N$ form a non-empty
$G$-orbit. Let $A$ be a division ring endowed with a $G$-action. If $M$ is a
non-zero $A\langle G\rangle$-submodule of $A\langle G/G_{\Psi_{n}}\rangle$ for
some $n\geq 0$ then $d_{M}$ grows as a $q$-polynomial of degree $n$:
$\frac{d_{m+n}(N)}{d_{m}(N)d_{n}(n)}\leq d_{M}(N)\leq d_{n}(N)$ for some
$m\geq 0$, where $d_{n}(N)$ is the number of embeddings
$\Psi_{n}\hookrightarrow\Psi_{N}$ induced by elements of $G$, and

$d_{n}(N)=\begin{cases}q^{(N+v)n}\prod_{i=0}^{n-1}(1-q^{i-N})&\mbox{if $q>1$,
so
$\frac{d_{m+n}(N)}{d_{m}(N)d_{n}(n)}=\frac{q^{Nn}}{\prod_{i=0}^{n-1}(q^{n}-q^{i})}\prod_{i=0}^{n-1}(1-q^{i+m-N})$}\\\
\prod_{i=0}^{n-1}(N-i)&\mbox{if $q=1$, so
$\frac{d_{m+n}(N)}{d_{m}(N)d_{n}(n)}=\frac{1}{n!}\prod_{i=0}^{n-1}(N-m-i)$.}\end{cases}$

###### Proof.

As $M^{G_{\Psi_{N}}}\subseteq A\langle
N_{G}(G_{\Psi_{N}})/(N_{G}(G_{\Psi_{N}})\cap G_{\Psi_{n}})\rangle$ and (by
Lemma 4.1) $A\otimes_{A^{G_{\Psi_{N}}}}M^{G_{\Psi_{N}}}\to M\subseteq A\langle
G/G_{\Psi_{n}}\rangle$ is injective, there is a natural inclusion

$A\otimes_{A^{G_{\Psi_{N}}}}M^{G_{\Psi_{N}}}\hookrightarrow A\langle
N_{G}(G_{\Psi_{N}})/(N_{G}(G_{\Psi_{N}})\cap
G_{\Psi_{n}})\rangle=A\langle\mathop{\mathrm{Aut}}\nolimits(\Psi_{N})/\mathop{\mathrm{Aut}}\nolimits(\Psi_{N}|\Psi_{n})\rangle,$

if $n\leq N$. (Here $\mathop{\mathrm{Aut}}\nolimits(\Psi_{N}|\Psi_{n})$
denotes the automorphisms of $\Psi_{N}$ identical on $\Psi_{n}$.) Then one has
$d_{M}(N)\leq\\#(\mathop{\mathrm{Aut}}\nolimits(\Psi_{N})/\mathop{\mathrm{Aut}}\nolimits(\Psi_{N}|\Psi_{n}))=d_{n}(N)$.

To obtain a lower bound of $d_{M}(N)$, fix a non-zero element $\alpha\in
M\subseteq A\langle G/G_{\Psi_{n}}\rangle$. Let $\Psi_{m}$ be the $G$-closure
of the union of the images of $\Psi_{n}$ under the elements of
$G/G_{\Psi_{n}}$ in the support of $\alpha$. Fix a $G$-closed subset
$\Psi_{m-n}\subset\Psi_{m}$ of length $m-n$. For each length-$m$ $G$-closed
subset $\Psi_{m-n}\subset P\subseteq\Psi_{N}$, fix an element $\sigma_{P}\in
G_{\Psi_{m-n}}$ such that $\sigma_{P}(\Psi_{m})=P$. Then the supports of
$\sigma_{P}\alpha$ for all such $P$ are pairwise disjoint, apart from those
embeddings of $\Psi_{n}$ whose images meet $\Psi_{m-n}$ non-trivially. This
implies that $\sigma_{P}\alpha$ are linearly independent, which gives
$d_{M}(N)\geq\binom{N-m}{n}_{q}$.

The lower bound of $d_{M}(N)$ is given by the number of $G$-closed subsets in
$\Psi_{N}$ with length-0 intersection with $\Psi_{m}$. Indeed, for any non-
zero element $\alpha\in M\subseteq A\langle G/G_{\Psi_{n}}\rangle$ there exist
an integer $m\geq 0$ and elements $\xi,\eta\in G$ such that $\xi\alpha$ is
congruent to
$\sum_{\sigma\in\mathop{\mathrm{Aut}}\nolimits(\Psi_{n})}b_{\sigma}\eta\sigma$
for some non-zero collection $\\{b_{\sigma}\in
A\\}_{\sigma\in\mathop{\mathrm{Aut}}\nolimits(\Psi_{n})}$ modulo monomorphisms
whose images have intersection of positive length with a fixed finite
$\Psi_{m}$. ∎

Let $q$ be either 1 or a primary integer. Let $S$ be a plain set if $q=1$ and
an $\mathbb{F}_{q}$-vector space if $q>1$. For each integer $s\geq 0$, we
denote by $\binom{S}{s}_{q}$ the set of subobjects of $S$ ($G$-closed subsets
of $\Psi$, if $S=\Psi$, where $G=\mathop{\mathfrak{S}}\nolimits_{\Psi}$ if
$q=1$ and $G=\mathrm{GL}_{\mathbb{F}_{q}}(\Psi)$ if $q>1$) of length $s$. In
other words, $\binom{S}{s}_{1}:=\binom{S}{s}$, while $\binom{S}{s}_{q}$ is the
Grassmannian of the $s$-dimensional subspaces in $S$ if $q>1$.

Recall, that an injection $M\hookrightarrow N$ in an abelian category is
called an essential extension if any non-zero subobject of $N$ has a non-zero
intersection with the image of $M$, cf. [2, Ch. 6, §2].

######  Corollary 4.3.

Let $G$ be either $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ $($and then $q:=1)$
or the group of automorphisms of an $\mathbb{F}_{q}$-vector space $\Psi$
fixing a finite-dimensional subspace of $\Psi$. Let $A$ be a division ring
endowed with a $G$-action. Let $\Xi$ be a finite subset in
$\mathop{\mathrm{Hom}}\nolimits_{A\langle G\rangle}(A\langle
G/G_{T}\rangle,A\langle G/G_{T^{\prime}}\rangle)$ for some finite $G$-closed
$T^{\prime}\subsetneqq T\subset\Psi$. Then

  1. (1)

any non-zero $A\langle G\rangle$-submodule of
$A\langle\binom{\Psi}{m}_{q}\rangle$ is essential for any integer $m\geq 0$;

  2. (2)

there are no non-zero isomorphic $A\langle G\rangle$-submodules in $A\langle
G/G_{T}\rangle$ and $A\langle G/G_{T^{\prime}}\rangle$;

  3. (3)

the common kernel $V_{\Xi}$ of all elements of $\Xi$ is an essential $A\langle
G\rangle$-submodule in $A\langle G/G_{T}\rangle$.

###### Proof.

(1) It follows from the lower growth estimate of Lemma 4.2 that, for any non-
zero $A\langle G\rangle$-submodule $M$ of
$A\langle\binom{\Psi}{m}_{q}\rangle$, $d_{M}(N)\sim
d_{A\langle\binom{\Psi}{m}_{q}\rangle}(N)$ when $N\to\infty$, so the growth of
intersection of any pair of non-zero submodules is the same.

(2) follows immediately from Lemma 4.2, since for any non-zero $A\langle
G\rangle$-submodule $M$ of $A\langle G/G_{T}\rangle$, there exist constants
$\beta>\alpha>0$ such that $\alpha<d_{M}(N)/d_{A\langle
G/G_{T}\rangle}(N)<\beta$ for all $N\gg 0$.

(3) Suppose that there exists a non-zero submodule $M\subseteq A\langle
G/G_{T}\rangle$ such that $M\cap V_{\Xi}=0$. Then restriction of some
$\xi\in\Xi$ to $M$ is non-zero. If $\xi|_{M}$ is not injective, replacing $M$
with $\ker\xi\cap M$, we can assume that $\xi|_{M}=0$. In other words, we can
assume that restriction to $M$ of any $\xi\in\Xi$ is either injective or zero.
In particular, restriction to $M$ of some $\xi\in\Xi$ is injective, i.e. $\xi$
embeds $M$ into $A\langle G/G_{T^{\prime}}\rangle$, contradicting (2). ∎

###  4.2. Smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-sets and
$F_{\Psi}$-semilinear representation of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ as sheaves

Let $\mathrm{FI}$ be the category of all finite sets, where the morphisms are
the injections.

This category does not admit coproducts, since, for any non-empty $J_{1}$ and
$J_{2}$ and their elements $x_{1}$ and $x_{2}$, there are diagrams
$J_{1}\hookrightarrow J\hookleftarrow J_{2}$ gluing together and separating
$x_{1}$ and $x_{2}$.

Consider the opposite category $\mathrm{FI}^{\mathrm{op}}$ as a site, where
any non-empty sieve is covering.

A presheaf $\mathcal{F}$ on $\mathrm{FI}^{\mathrm{op}}$ is a sheaf if and only
if, for any diagram $J_{1}\hookleftarrow J_{0}\hookrightarrow J_{2}$ in
$\mathrm{FI}$, the diagram
$\mathcal{F}(J_{0})\to\mathcal{F}(J_{1})\times\mathcal{F}(J_{2})\rightrightarrows\mathcal{F}(J_{1}\sqcup_{J_{0}}J_{2})$
is an equalizer, where $\sqcup_{J_{0}}$ is the colimit of $J_{1}\hookleftarrow
J_{0}\hookrightarrow J_{2}$ in the category of sets.

For each ‘reasonable’ category $\mathcal{C}$, each $J\in\mathrm{FI}$ and each
$\mathcal{C}$-valued sheaf $\mathcal{F}$ on $\mathrm{FI}^{\mathrm{op}}$,
denote by $\mathcal{F}_{+J}$ the sheaf $I\mapsto\mathcal{F}(I\sqcup J)$. Thus,

  * •

$(-)_{+J}:\mathcal{F}\mapsto\mathcal{F}_{+J}$ is an endofunctor on the
category of $\mathcal{C}$-valued sheaves on $\mathrm{FI}^{\mathrm{op}}$;

  * •

$(-)_{+\varnothing}$ is the identity functor;

  * •

any embedding of finite sets $J_{1}\hookrightarrow J_{2}$ induces a morphism
of functors $(-)_{+J_{1}}\to(-)_{+J_{2}}$;

  * •

for each pair of finite $J,J^{\prime}$, there is a natural isomorphism
$(-)_{+J}\circ(-)_{+J^{\prime}}\cong(-)_{+(J\sqcup J^{\prime})}$.

For a sheaf of rings $\mathcal{A}$, we say ‘an $\mathcal{A}$-module’ instead
of ‘a sheaf of $\mathcal{A}$-modules’.

If $\mathcal{A}$ is a sheaf of rings and $\mathcal{F}$ is an
$\mathcal{A}$-module then $\mathcal{F}_{+J}$ is an $\mathcal{A}_{+J}$-module,
so there is a natural morphism of functors
$\mathcal{A}_{+J}\otimes_{{\mathcal{A}}}(-)\to(-)_{+J}$ on the category of
$\mathcal{A}$-modules.

######  Lemma 4.4.

Let $\Psi$ be an infinite set. Let $F$ be a field and $k$ be a subfield
algebraically closed in $F$. To each sheaf of sets $\mathcal{F}$ on
$\mathrm{FI}^{\mathrm{op}}$ we associate the
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-set
$\mathcal{F}(\Psi):=\mathop{\underrightarrow{\lim}}\limits_{J\subset\Psi}\mathcal{F}(J)$,
where $J$ runs over the finite subsets of $\Psi$. This gives rise to the
following equivalences of categories:

$\displaystyle\nu_{\Psi}:\\{\text{{\rm sheaves of sets on
$\mathrm{FI}^{\mathrm{op}}$}}\\}\xrightarrow{\sim}\\{\text{{\rm smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-sets}}\\},\quad\mathcal{F}\mapsto\mathcal{F}(\Psi);$
$\displaystyle\\{\text{{\rm sheaves of $k$-vector spaces on
$\mathrm{FI}^{\mathrm{op}}$}}\\}\xrightarrow{\sim}\mathop{\mathrm{Sm}}\nolimits_{k}(\mathop{\mathfrak{S}}\nolimits_{\Psi});$
$\displaystyle\\{\text{{\rm$\mathcal{O}$-modules}}\\}\xrightarrow{\sim}\mathop{\mathrm{Sm}}\nolimits_{F_{\Psi}}(\mathop{\mathfrak{S}}\nolimits_{\Psi}),\quad\text{where
$\mathcal{O}$ is the sheaf of fields $S\mapsto F_{S}$ on
$\mathrm{FI}^{\mathrm{op}}$}.$

The functor $\nu_{\Psi}$ admits a quasi-inverse $\nu_{\Psi}^{-1}$ such that
for any infinite subset $\Psi^{\prime}\subseteq\Psi$ the functor
$\nu_{\Psi^{\prime}}\circ\nu_{\Psi}^{-1}$ is given by $M\mapsto
M^{(\Psi^{\prime})}_{(\Psi)}=M^{\prime}:=\mathop{\underrightarrow{\lim}}\limits_{J\subset\Psi^{\prime}}M^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}\subseteq
M^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\Psi^{\prime}}}$, where $J$ runs over
the finite subsets of $\Psi^{\prime}$, and gives rise to the following
equivalences of categories:

$\\{\text{{\rm smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-sets}}\\}\xrightarrow{\sim}\\{\text{{\rm
smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}}$-sets}}\\};\
\mathop{\mathrm{Sm}}\nolimits_{k}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\xrightarrow{\sim}\mathop{\mathrm{Sm}}\nolimits_{k}(\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}});\
\mathop{\mathrm{Sm}}\nolimits_{F_{\Psi}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\xrightarrow{\sim}\mathop{\mathrm{Sm}}\nolimits_{F_{\Psi^{\prime}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}}).$

For infinite subsets $\Psi^{\prime\prime}\subseteq\Psi^{\prime}\subseteq\Psi$
and
$V\in\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
one has

$\Psi^{(\Psi^{\prime\prime})}_{(\Psi^{\prime})}=\Psi^{\prime\prime}\cup(\Psi\smallsetminus\Psi^{\prime});\quad(F_{\Psi})^{(\Psi^{\prime\prime})}_{(\Psi^{\prime})}=F_{\Psi^{\prime\prime}\cup(\Psi\smallsetminus\Psi^{\prime})};\quad(V^{(\Psi^{\prime})}_{(\Psi)})^{(\Psi^{\prime\prime})}_{(\Psi^{\prime})}=V^{(\Psi^{\prime\prime})}_{(\Psi)}.$

Fix an embedding of a finite set $J$ into $\Psi$. Then the functor
$\nu_{\Psi\smallsetminus J}\circ(-)_{+J}\circ\nu_{\Psi}^{-1}$ is isomorphic to
the functor $\\{\text{{\rm smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-sets}}\\}\xrightarrow{(-)^{(\Psi\smallsetminus
J)}_{(\Psi\smallsetminus
J)}:=\mathrm{Res}_{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}}\\{\text{{\rm
smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$-sets}}\\}=\\{\text{{\rm
smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi\smallsetminus J}$-sets}}\\}$.

###### Proof.

For any pair of sheaves $\mathcal{F}$ and $\mathcal{G}$, a map of sets
$\alpha:\mathcal{F}(\Psi)\to\mathcal{G}(\Psi)$ and any embedding
$\iota:S\hookrightarrow\Psi$ such that
$\alpha(\mathcal{F}(\Psi)^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota(S)}})\subseteq\mathcal{G}(\Psi)^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota(S)}}$
there is a unique map $\alpha_{S}:\mathcal{F}(S)\to\mathcal{G}(S)$ making
commutative the square

$\begin{array}[]{ccc}\mathcal{F}(S)&\stackrel{{\scriptstyle\alpha_{S}}}{{\longrightarrow}}&\mathcal{G}(S)\\\
\downarrow\hbox
to0.0pt{$\displaystyle\iota_{\mathcal{F}}$\hss}&&\downarrow\hbox
to0.0pt{$\displaystyle\iota_{\mathcal{G}}$\hss}\\\
\mathcal{F}(\Psi)&\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}&\mathcal{G}(\Psi).\end{array}$

If $\alpha$ is a morphism of $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-sets then
$\alpha_{S}$ is independent of $\iota$, since all embeddings
$S\hookrightarrow\Psi$ form a single
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-orbit. This gives gives rise to a
natural bijection
$\mathop{\mathrm{Hom}}\nolimits_{\mathop{\mathfrak{S}}\nolimits_{\Psi}}(\mathcal{F}(\Psi),\mathcal{G}(\Psi))\xrightarrow{\sim}\mathop{\mathrm{Hom}}\nolimits(\mathcal{F},\mathcal{G})$,
the inverse map is given by restriction to finite subsets of $\Psi$.

To construct a functor $\nu_{\Psi}^{-1}$ quasi-inverse to $\nu_{\Psi}$, for
each finite set $S$ fix an embedding $\iota_{S}:S\hookrightarrow\Psi$, which
is identical if $S$ is a subset of $\Psi$. Then to a smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-set $M$ we associate the presheaf
$S\mapsto M^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota_{S}(S)}}$ and to each
embedding $j:S\hookrightarrow T$ we associate a unique map
$M^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota_{S}(S)}}\hookrightarrow
M^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota_{T}(T)}}$ induced by the element
$\iota_{T}j\iota_{S}^{-1}\in\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota_{T}(T)}\backslash\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathop{\mathfrak{S}}\nolimits_{\Psi|\iota_{S}(S)}$.
It follows from Lemma 3.6 that this presheaf is a sheaf. ∎

######  Example 4.5.

Given a sheaf of $\mathbb{Z}$-modules $\mathcal{F}$, an integer $s\geq 0$ and
$J\in\mathrm{FI}$, define the presheaves of abelian groups
$h^{J}_{\mathcal{F}}:I\mapsto\mathcal{F}(I)\langle\\{J\hookrightarrow
I\\}\rangle$ and
$\bar{h}^{s}_{\mathcal{F}}:I\mapsto\mathcal{F}(I)\langle\binom{I}{s}\rangle$.
E.g., $h^{\varnothing}_{\mathcal{F}}=\bar{h}^{0}_{\mathcal{F}}=\mathcal{F}$
and
$h^{\\{\ast\\}}_{\mathcal{F}}=\bar{h}^{1}_{\mathcal{F}}:I\mapsto\mathcal{F}(I)\langle
I\rangle$. Then $h^{J}_{\mathcal{F}}$ and $\bar{h}^{s}_{\mathcal{F}}$ are
sheaves of abelian groups, faithfully flat for
$\mathcal{F}=\underline{\mathbb{Z}}$.

###  4.3. Local structure of finitely generated objects in
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$

######  Proposition 4.6.

Let $\mathcal{O}$ be a non-constant sheaf of fields on
$\mathrm{FI}^{\mathrm{op}}$, and $V$ be a finitely generated
$\mathcal{O}$-module. Then there is an integer $n\geq 0$ such that for any
finite set $J$ of order $\geq n$

  * •

the $\mathcal{O}_{+J}$-module $V_{+J}$ is isomorphic to
$\bigoplus_{s=0}^{N}(\bar{h}^{s}_{\mathcal{O}_{+J}})^{\kappa_{s}}$ for some
integer $N,\kappa_{0},\dots,\kappa_{N}\geq 0$;

  * •

$V$ can be embedded into a product of copies of $\mathcal{O}_{+J}$.

###### Proof.

By Lemma 4.21, for each finite set $J^{\prime}$, the
$\mathcal{O}_{+J^{\prime}}$-module $V_{+J^{\prime}}$ is finitely generated,
while by Lemma 3.19, there is a surjection of
$\mathcal{O}_{+J^{\prime}}$-modules
$\bigoplus_{s=0}^{N(J^{\prime})}\bar{h}^{s}_{\mathcal{O}_{+J^{\prime}}}{}^{m_{s,J^{\prime}}}\to
V_{+J^{\prime}}$ for some $N(J^{\prime})\geq 0$ and $m_{s,J^{\prime}}\geq 0$.
Choose such a surjection so that $N(J^{\prime})$ is minimal and
$m_{N(J^{\prime}),J^{\prime}}\geq 1$ is minimal as well. It is clear, that for
each finite set $J^{\prime\prime}$ with
$\\#J^{\prime\prime}\geq\\#J^{\prime}$, one has $N(J^{\prime\prime})\leq
N(J^{\prime})$ and $m_{N(J^{\prime\prime}),J^{\prime\prime}}\leq
m_{N(J^{\prime}),J^{\prime}}$ if $N(J^{\prime\prime})=N(J^{\prime})$.

Fix some $J^{\prime}$ with minimal possible $N(J^{\prime})$ and
$m_{N,J^{\prime}}\geq 1$. The pair $(N,m)$ depends only on $V$, so denote it
$(N_{V},m_{V})$. We replace $\mathcal{O}$ by $\mathcal{O}_{+J^{\prime}}$ and
$V$ by $V_{+J^{\prime}}$. Obviously, $(N_{V},m_{V})=(N_{V_{+J}},m_{V_{+J}})$
for any finite $J$.

By induction on $N$, we prove the statement for all $V$ with $N_{V}=N$, the
case $N=0$ being trivial.

Let $\alpha:(\bar{h}^{N}_{\mathcal{O}})^{m}\to V$ and
$\beta:\bigoplus_{s=0}^{N-1}(\bar{h}^{s}_{\mathcal{O}})^{m_{s}}\to V$ be two
morphisms such that
$\alpha+\beta:(\bar{h}^{N}_{\mathcal{O}})^{m}\oplus\bigoplus_{s=0}^{N-1}(\bar{h}^{s}_{\mathcal{O}})^{m_{s}}\to
V$ is surjective.

Suppose that $\alpha$ is not injective, and
$0\neq(\xi_{1},\dots,\xi_{m})\in(\ker\alpha)(I)$ for a finite $I$. Without
loss of generality, we may assume that $\xi_{m}=\sum_{i=1}^{b}a_{i}I_{i}\neq
0$ for some $I_{i}\subseteq I$ of order $N$ and non-zero $a_{i}$. Set
$J:=I\smallsetminus I_{1}$. Then $\alpha$ factors through a quotient
$(\bar{h}^{N}_{\mathcal{O}})^{m}/\langle(\xi_{1},\dots,\xi_{m})\rangle$. On
the other hand, the inclusion
$(\bar{h}^{N}_{\mathcal{O}})^{m-1}\stackrel{{\scriptstyle(-,0)}}{{\hookrightarrow}}(\bar{h}^{N}_{\mathcal{O}})^{m}$
induces a surjection of $\mathcal{O}_{+J}$-modules

$(\bar{h}^{N}_{\mathcal{O}})_{+J}^{m-1}\oplus\bigoplus_{\varnothing\neq\Lambda\subseteq
J}\bar{h}^{N-\\#\Lambda}_{\mathcal{O}_{+J}}\xrightarrow{(-,0)+\sum_{\Lambda}(\underbrace{0,\dots,0}_{m-1},-\cup\Lambda)}((\bar{h}^{N}_{\mathcal{O}})^{m}/\langle(\xi_{1},\dots,\xi_{m})\rangle)_{+J}$

giving rise to a surjection of $\mathcal{O}_{+J}$-modules
$(\bar{h}^{N}_{\mathcal{O}})_{+J}^{m-1}\oplus\bigoplus_{s=0}^{N-1}\bar{h}^{s}_{\mathcal{O}_{+J}}{}^{\binom{\\#J}{N-s}+m_{s}}\to
V_{+J}$. This contradicts the minimality assumption on $m$, thus showing that
$\alpha$ is injective.

By Lemma 4.2 and the induction hypothesis, for any subsheaf
$\mathcal{F}\subseteq\mathrm{Im}(\beta)$ there is some $C>0$ such that
$\dim_{{\mathcal{O}}(J)}\mathcal{F}(J)\leq C\cdot\\#J^{N-1}$ for all
sufficiently big $J$. For any non-zero subsheaf
$\mathcal{F}\subseteq(\bar{h}^{N}_{\mathcal{O}})^{m}$, Lemma 4.2 gives an
estimate $\dim_{{\mathcal{O}}(J)}\mathcal{F}(J)\geq C^{\prime}\cdot\\#J^{N}$
for some $C^{\prime}>0$ and all sufficiently big $J$. This implies that
$\mathrm{Im}(\alpha)\cap\mathrm{Im}(\beta)=0$. Therefore,
$V\cong(\bar{h}^{N}_{\mathcal{O}})^{m}\oplus\mathrm{Im}(\beta)$. By induction
hypothesis, $\mathrm{Im}(\beta)_{+J}$ is of required type for some $J$, thus
completing the induction step.

Fix a finite set $I$ such that $\mathcal{O}(I)\neq\mathcal{O}(\varnothing)$.
By Remark 3.11, $\mathcal{O}(\varnothing)$ is algebraically closed in
$\mathcal{O}(J)$ for any $J$, so $\mathcal{O}(I)$ is transcendental over
$\mathcal{O}(\varnothing)$, and therefore,
$\mathcal{O}(I)^{\mathop{\mathfrak{S}}\nolimits_{I}}$ is transcendental over
$\mathcal{O}(\varnothing)$ as well. By Lemma 3.6,
$\mathcal{O}(J)\neq\mathcal{O}(I)$ for any $I\subsetneqq J$. Then Lemma 3.9
implies that, for each integer $s>0$, the sheaf
$\bar{h}^{s}_{\mathcal{O}_{+I}}$ can be embedded into a product of copies of
$\mathcal{O}_{+I}$. Now any $\mathcal{O}$-module $V$ embeds into $V_{+J}$ for
any finite $J$, so if $V$ is finitely generated we can use the above
description of $V_{+J}$ for sufficiently big $J$ to conclude that, $V$ can be
embedded into a product of copies of $\mathcal{O}_{+I}$ for any sufficiently
large finite $I$. ∎

######  Remark 4.7.

By Krull–Schmidt–Remak–Azumaya Theorem the integers
$N,\kappa_{0},\dots,\kappa_{N}\geq 0$ in Proposition 4.6 are uniquely
determined. Clearly, $N$ and $\kappa_{N}$ are independent of $J$. We call $N$
the level of $V$. It is easy to show that any non-zero submodule of
$K\langle\binom{\Psi}{N}\rangle$ is of level $N$, while the key point of the
proof of the Proposition is that any quotient of
$K\langle\binom{\Psi}{N}\rangle$ by a non-zero submodule is of level $<N$.

In particular, the objects $K\langle\binom{\Psi}{N}\rangle$ of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ are
monoform, so their injective hulls are indecomposable, and they correspond to
pairwise distinct points in the Gabriel spectrum.

######  Lemma 4.8.

Let $\Psi$ be an infinite set, and $K$ be an
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field. Then the cokernel of any non-
zero morphism of
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules
$\varphi:K\langle\binom{\Psi}{s}\rangle\to K\langle\Psi\rangle$ is at most
$(s-1)$-dimensional. In particular, any non-zero submodule of
$K\langle\Psi\rangle$ is of finite codimension.

Suppose that $K$ is a non-trivial smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field, $K$ is a cogenerator of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ and
$K\langle\Psi\rangle$ is injective. Set
$k:=K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$ and
$F:=K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|x}}$ for some $x\in\Psi$. Then
there are natural bijections

  1. (1)

between the $K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodules
of $K\langle\Psi\rangle$ of codimension $s$ and the $s$-dimensional $k$-vector
subspaces in $F$;

  2. (2)

between the isomorphism classes of
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodules of
$K\langle\Psi\rangle$ of codimension $s$ and the $F^{\times}$-orbits of the
$s$-dimensional $k$-vector subspaces in the space $F$.

###### Proof.

Fix a subset $S\subset\Psi$ of order $s-1$. For any element
$b\in\Psi\smallsetminus S$, the set $\\{b\\}\cup S$ is sent by $\varphi$ to a
linear combination of its elements with non-zero coefficients, and therefore,
the image of $b$ in the cokernel of $\varphi$ is a linear combination of
images of the elements of $S$.

We already know that the $K$-vector space $K\langle\Psi\rangle/M$ is finite-
dimensional for any non-zero
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodule $M$ of
$K\langle\Psi\rangle$, so by our assumption the
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module
$K\langle\Psi\rangle/M$ is isomorphic to a sum of copies of $K$, and
therefore, $M$ is the common kernel of the elements of a finite-dimensional
$k$-vector subspace of
$\mathop{\mathrm{Hom}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\Psi\rangle,K)=F$.
Clearly, the
$\mathop{\mathrm{End}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\Psi\rangle)^{\times}$-action
preserves the isomorphism classes of the common kernels (since
$\mathop{\mathrm{End}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\Psi\rangle)=F$
is a field). On the other hand,
$\mathop{\mathrm{Hom}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\Psi\rangle/M,K\langle\Psi\rangle)=0$
and, as $K\langle\Psi\rangle$ is injective, the restriction morphism
$\mathop{\mathrm{End}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\Psi\rangle)\to\mathop{\mathrm{Hom}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(M,K\langle\Psi\rangle)$
is an isomorphism, i.e., any morphism of
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules $M\to
K\langle\Psi\rangle$ is induced by an endomorphism of $K\langle\Psi\rangle$. ∎

######  Corollary 4.9.

Let $K$ be a field endowed with a non-trivial smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action.

Then, for any smooth finitely generated
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $V$, there is a
unique polynomial $P_{V}(X)$ over $\mathbb{Q}$ such that
$P_{V}(\\#T)=\dim_{K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T}}}V^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T}}$
for any sufficiently large finite subset $T\subset\Psi$.

In particular, $V$ is admissible, i.e., $\dim_{K^{U}}V^{U}<\infty$ for any
open subgroup $U\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi}$.

###### Proof.

By Proposition 4.6, there is a finite subset $J\subset\Psi$ and an isomorphism
of $K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-modules
$\bigoplus_{s=0}^{N}K\langle\binom{\Psi\smallsetminus
J}{s}\rangle^{\kappa_{s}}\xrightarrow{\sim}V$ for some integer
$N,\kappa_{0},\dots,\kappa_{N}\geq 0$. Then
$\dim_{K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T}}}V^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T}}=\sum_{s=0}^{N}\kappa_{s}\binom{\\#(T\smallsetminus
J)}{s}$ for any finite subset $T\subset\Psi$ containing $J$, i.e.
$P_{V}(X)=\sum_{s=0}^{N}\kappa_{s}\binom{X-\\#J}{s}$ is the unique polynomial
whose value at $\\#T$ is
$\dim_{K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T}}}V^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T}}$
for such $T$. ∎

###  4.4. Injective objects and resolutions

######  Proposition 4.10.

Let $\mathcal{O}$ be a non-constant sheaf of fields on
$\mathrm{FI}^{\mathrm{op}}$. Then, for any finitely generated
$\mathcal{O}$-module $W$, there is a finite set $J$ and a resolution of the
form

(1)
$0\to\mathcal{O}_{+J}\otimes_{\mathcal{O}}W\to\bigoplus_{s=0}^{N}\bar{h}^{s}_{\mathcal{O}_{+J}}{}^{\kappa_{0,s}}\to\bigoplus_{s=0}^{N-1}\bar{h}^{s}_{\mathcal{O}_{+J}}{}^{\kappa_{1,s}}\to\cdots\to\bigoplus_{s=0}^{1}\bar{h}^{s}_{\mathcal{O}_{+J}}{}^{\kappa_{N-1,s}}\to\mathcal{O}_{+J}^{\kappa_{N}}\to
0$

for some integer $\kappa_{ij}\geq 0$, where $N$ is the level of $W$.

###### Proof.

We construct recursively an ascending sequence of finite sets
$\varnothing=:J_{0}\subseteq J_{1}\subseteq\cdots\subseteq J_{N}$ and a
sequence of $\mathcal{O}_{+J_{i}}$-modules $W_{i}$ of level $\leq N-i$, where
$W_{0}:=W$. Choose $J_{i+1}$ so that $(W_{i})_{+(J_{i+1}\smallsetminus
J_{i})}$ is as in Proposition 4.6. Define $W_{i+1}$ as the cokernel of the
morphism
$\mathcal{O}_{+J_{i+1}}\otimes_{\mathcal{O}_{+J_{i}}}W_{i}\hookrightarrow(W_{i})_{+(J_{i+1}\smallsetminus
J_{i})}$. Then $W_{i+1}$ is of level $<N-i$ and the sequence

$0\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{i})}}(W_{i})^{(\Psi_{N})}_{(\Psi_{i})}\hookrightarrow
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{i+1})}}(W_{i})^{(\Psi_{N})}_{(\Psi_{i+1})}\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{i+1})}}(W_{i+1})^{(\Psi_{N})}_{(\Psi_{i+1})}\to
0$

is exact. Combining all these short exact sequences we get a resolution

$0\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{0})}}(W_{0})^{(\Psi_{N})}_{(\Psi_{0})}\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{1})}}(W_{0})^{(\Psi_{N})}_{(\Psi_{1})}\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{2})}}(W_{1})^{(\Psi_{N})}_{(\Psi_{2})}\to\cdots\\\
\cdots\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{N-1})}}(W_{N-2})^{(\Psi_{N})}_{(\Psi_{N-1})}\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{N})}}(W_{N-1})^{(\Psi_{N})}_{(\Psi_{N})}\to
K\otimes_{K^{(\Psi_{N})}_{(\Psi_{N})}}(W_{N})^{(\Psi_{N})}_{(\Psi_{N})}\to 0$

of required type (1). ∎

######  Lemma 4.11.

Let $C$ be a collection of objects in an abelian category $\mathcal{A}$ such
that (i) $\mathop{\mathrm{End}}\nolimits(P)$ is a division ring for all $P\in
C$, (ii) $\mathop{\mathrm{Hom}}\nolimits(P,P^{\prime})=0$ for all $P\neq
P^{\prime}$ in $C$.

  1. (1)

Suppose in addition that $C$ is cogenerating. Then all objects in $C$ are
injective.

  2. (2)

Suppose in addition that $\mathcal{A}$ is a locally finitely generated
Grothendieck category, while any finitely generated object of $\mathcal{A}$
admits an embedding into a product of objects in $C$. Then each finitely
generated object in $C$ is injective. If all objects in $C$ are finitely
generated then $C$ cogenerates $\mathcal{A}$.

###### Proof.

Assuming that $C$ is cogenerating, if $P_{0}\in C$ and $i:P_{0}\to V$ is a
monomorphism then there exists a morphism $f:V\to P$ for some $P\in C$ such
that $f\circ i:P_{0}\to P$ is non-zero, so $P=P_{0}$ and $(f\circ i)^{-1}\circ
f:V\to P_{0}$ is a splitting of $i$, i.e., $P_{0}$ is injective.

Assuming that $\mathcal{A}$ is locally finitely generated, if $P_{0}\in C$ is
finitely generated and admits a non-trivial essential extension $E$ then $E$
can be chosen to be finitely generated as well. Any morphism $E\to P$ for
$P\neq P_{0}$ in $C$ vanishes on $P_{0}$, so there exists a morphism $f:E\to
P_{0}$ non-vanishing on $P_{0}$. But $\ker f\cap P_{0}=0$, so $\ker f=0$, i.e.
$E=P_{0}$, and thus, $P_{0}$ is injective. Now, if all objects in $C$ are
finitely generated then they are injective, so for any object $W$ and each
finitely generated $W_{0}\subseteq W$ there is an embedding
$W_{0}\hookrightarrow\prod_{i\in I(W_{0})}P_{i}$ into a product of objects in
$C$, while this embedding extends to a morphism $f_{W_{0}}:W\to\prod_{i\in
I(W_{0})}P_{i}$, thus giving an embedding
$(f_{W_{0}}):W\to\prod_{W_{0}}\prod_{i\in I(W_{0})}P_{i}$. ∎

######  Lemma 4.12.

Let $\mathcal{O}$ be a non-constant sheaf of fields, and $\mathcal{V}$ be a
level 0 simple $\mathcal{O}$-module. Then $\mathcal{V}$ is injective if and
only if, for any finite set $J$, any $\mathcal{O}$-module embedding
$\iota:\mathcal{V}\hookrightarrow\mathcal{O}_{+J}$ is split.

###### Proof.

The ‘only if’ direction is trivial. In the opposite direction, let an
$\mathcal{O}$-module $\mathcal{E}$ be an essential extension of $\mathcal{V}$.
We are going to figure out whether $\mathcal{E}$ can be non-trivial, so we may
assume that $\mathcal{E}$ is finitely generated. By Proposition 4.6, there is
a finite set $J$ and an isomorphism of $\mathcal{O}_{+J}$-modules
$\mathcal{E}_{+J}\xrightarrow{\sim}\bigoplus_{s=0}^{N}(\bar{h}^{s}_{\mathcal{O}_{+J}})^{\kappa_{s}}$
for some integer $N,\kappa_{0},\dots,\kappa_{N}\geq 0$. Then $\mathcal{E}$ is
a finitely generated $\mathcal{O}$-submodule of
$\bigoplus_{s=0}^{N}(\bar{h}^{s}_{\mathcal{O}_{+J}})^{\kappa_{s}}$. Fix a non-
zero $\mathcal{O}_{+J}$-morphism $\lambda:\mathcal{E}_{+J}\to\mathcal{O}_{+J}$
(e.g. $\lambda$ can be chosen to be the composition of the natural projection
$\mathcal{E}_{+J}\xrightarrow{\sim}\bigoplus_{s=0}^{N}(\bar{h}^{s}_{\mathcal{O}_{+J}})^{\kappa_{s}}\to\mathcal{O}_{+J}^{\kappa_{0}}$
with any $\mathcal{O}_{+J}$-morphism
$\mathcal{O}_{+J}^{\kappa_{0}}\to\mathcal{O}_{+J}$, which is non-zero on the
image of $\mathcal{V}\subset\mathcal{E}$; by Lemma 3.8,
$\bar{h}^{s}_{\mathcal{O}_{+J}}$ does not contain level 0
$\mathcal{O}$-submodules if $s>0$). Then $\lambda$ embeds $\mathcal{E}$ into
$\mathcal{O}_{+J}$. By our assumption, we can choose a morphism of
$\mathcal{O}$-modules $\xi:\mathcal{O}_{+J}\to\mathcal{V}$ such that the
composition $\xi\circ\lambda\circ\iota$ splits the embedding $\iota$. ∎

######  Lemma 4.13.

Let $G$ be a permutation group, $\Psi$ be a smooth $G$-set, $K$ be a smooth
$G$-field. Let $V$ be an object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ such
that $V\otimes_{K}K(\Psi)$ is injective and either $K$ or $V$ is injective as
well. Then the object $V\langle\binom{\Psi}{s}\rangle$ is injective for any
integer $s\geq 0$.

###### Proof.

Let $\widetilde{K}\subset K(\Psi)$ be the subfield generated over $K$ by
squares of the elements of $\Psi$. There is an isomorphism $\bigoplus_{s\geq
0}\widetilde{K}\langle\binom{\Psi}{s}\rangle\xrightarrow{\sim}K(\Psi)$,
$[S]\mapsto\prod_{t\in S}t\cdot\widetilde{K}$, so each
$\widetilde{K}\langle\binom{\Psi}{s}\rangle$ is isomorphic to a direct summand
of the object $K(\Psi)$ of $\mathop{\mathrm{Sm}}\nolimits_{\widetilde{K}}(G)$.

As $\widetilde{K}$ and $K(\Psi)$ are isomorphic $G$-field extensions of $K$
(under $t^{2}\mapsto t$ for all $t\in\Psi$), they are isomorphic as objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, as well as $V\otimes_{K}\widetilde{K}$
and $V\otimes_{K}K(\Psi)$ are.

We assume that $V\otimes_{K}K(\Psi)$ is injective in
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, so each
$V\otimes_{K}\widetilde{K}\langle\binom{\Psi}{s}\rangle$ is an injective
object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. As we also assume that
either $K$ or $V$ is injective, the inclusion $V\hookrightarrow
V\otimes_{K}\widetilde{K}$ admits a splitting
$V\otimes_{K}\widetilde{K}\xrightarrow{\pi}V$. Then the inclusion
$V\otimes_{K}K\langle\binom{\Psi}{s}\rangle\hookrightarrow
V\otimes_{K}\widetilde{K}\langle\binom{\Psi}{s}\rangle$ splits as well:
$\sum_{i}a_{i}[S_{i}]\mapsto\sum_{i}\pi(a_{i})[S_{i}]$, and thus,
$V\langle\binom{\Psi}{s}\rangle=V\otimes_{K}K\langle\binom{\Psi}{s}\rangle$ is
an injective object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. ∎

###  4.5.
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ is
locally noetherian; local structure of morphisms between compact objects in
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$

In the following result, our principal examples of the ring $A$ will be
division rings endowed with an $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action,
though localization of $\mathbb{Z}[x~{}|~{}x\in\Psi]$ at all non-constant
indecomposable polynomials gives one more example.

######  Proposition 4.14.

Let $A$ be an associative left noetherian ring endowed with an arbitrary
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action, and $W$ be a left
$A\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module generated by
finitely many smooth vectors. Then the left $A\langle U\rangle$-module $W$ is
noetherian for any open subgroup
$U\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi}$. $($Example:
$W=A\langle\Psi^{s}\rangle$ for $s\geq 0.)$

###### Proof.

The module $W$ is a quotient of a finite direct sum of
$A\langle\Psi^{s}\rangle$ for some integer $s\geq 0$, so it suffices to show
that the left $A\langle U\rangle$-module $A\langle\Psi^{s}\rangle$ is
noetherian for any integer $s\geq 0$, i.e. that any $A\langle
U\rangle$-submodule $M\subset A\langle\Psi^{s}\rangle$ is finitely generated
for all $U=\mathop{\mathfrak{S}}\nolimits_{\Psi|S}$ with finite
$S\subset\Psi$. We proceed by induction on $s\geq 0$, the case $s=0$ being
trivial. Assume that $s>0$ and the $A\langle U\rangle$-modules
$A\langle\Psi^{j}\rangle$ are noetherian for all $j<s$. Fix a subset
$I_{0}\subset\Psi\smallsetminus S$ of order $s$.

Let $M_{0}$ be the image of $M$ under the $A$-linear projector
$\pi_{0}:A\langle\Psi^{s}\rangle\to A\langle I_{0}^{s}\rangle\subset
A\langle\Psi^{s}\rangle$ omitting all $s$-tuples containing elements other
than those of $I_{0}$. As $A$ is noetherian and $I_{0}^{s}$ is finite, the
$A$-module $M_{0}$ is finitely generated. Let the $A$-module $M_{0}$ be
generated by the images of some elements $\alpha_{1},\dots,\alpha_{N}\in
M\subseteq A\langle\Psi^{s}\rangle$. Then $\alpha_{1},\dots,\alpha_{N}$ belong
to the $A$-submodule $A\langle I^{s}\rangle$ of $A\langle\Psi^{s}\rangle$ for
some finite subset $I\subset\Psi$.

Let $J\subset I\cup S$ be the complement to $I_{0}$. For each pair
$\gamma=(j,x)$, where $1\leq j\leq s$ and $x\in J$, set
$\Psi^{s}_{\gamma}:=\\{(x_{1},\dots,x_{s})\in\Psi^{s}~{}|~{}x_{j}=x\\}$. This
is a smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$-set. Then the set
$\Psi^{s}$ is the union of the $\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$-orbit
consisting of $s$-tuples of pairwise distinct elements of $\Psi\smallsetminus
J$ and of a finite union of $\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$-orbits
embeddable into $\Psi^{s-1}$:
$\bigcup_{\gamma}\Psi^{s}_{\gamma}\cup\bigcup_{1\leq i<j\leq s}\Delta_{ij}$,
where $\Delta_{ij}:=\\{(x_{1},\dots,x_{s})\in\Psi^{s}~{}|~{}x_{i}=x_{j}\\}$
are diagonals.

As (i)
$M_{0}\subseteq\sum_{j=1}^{N}A\alpha_{j}+\sum_{\gamma\in\\{1,\dots,s\\}\times
J}A\langle\Psi^{s}_{\gamma}\rangle$, (ii) $g(M_{0})\subset
A\langle\Psi^{s}\rangle$ is determined by $g(I_{0})$, (iii) for any $g\in U$
such that $g(I_{0})\cap J=\varnothing$ there exists $g^{\prime}\in U_{J}$ with
$g(I_{0})=g^{\prime}(I_{0})$ ($U_{J}$ acts transitively on the
$s$-configurations in $\Psi\smallsetminus J$), one has inclusions of $A\langle
U_{J}\rangle$-modules

$\sum_{j=1}^{N}A\langle U\rangle\alpha_{j}\subseteq M\subseteq\sum_{g\in
U}g(M_{0})\subseteq\sum_{g\in
U_{J}}g(M_{0})+\sum_{\gamma\in\\{1,\dots,s\\}\times
J}A\langle\Psi^{s}_{\gamma}\rangle.$

On the other hand, $g(M_{0})\subseteq
g(\sum_{j=1}^{N}A\alpha_{j})+\sum_{\gamma\in\\{1,\dots,s\\}\times
J}A\langle\Psi^{s}_{\gamma}\rangle$ for $g\in U_{J}$, and therefore, the
$A\langle U_{J}\rangle$-module $M/\sum_{j=1}^{N}A\langle U\rangle\alpha_{j}$
becomes a subquotient of the noetherian, by the induction assumption,
$A\langle U_{J}\rangle$-module $\sum_{\gamma\in\\{1,\dots,s\\}\times
J}A\langle\Psi^{s}_{\gamma}\rangle$, so the $A\langle U_{J}\rangle$-module
$M/\sum_{j=1}^{N}A\langle U\rangle\alpha_{j}$ is finitely generated, and thus,
$M$ is finitely generated as well. ∎

######  Corollary 4.15.

Let $A$ be a left noetherian associative ring endowed with a smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action. Then any smooth finitely
generated left $A\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module
$W$ is noetherian if considered as a left $A\langle U\rangle$-module for any
open subgroup $U\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi}$.

###### Proof.

The module $W$ is a quotient of a finite direct sum of
$A\langle\Psi^{m}\rangle$ for some integer $m\geq 0$, while
$A\langle\Psi^{m}\rangle$ are noetherian by Proposition 4.14. ∎

In particular, the category of smooth
$A\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules is locally
noetherian, i.e., any smooth finitely generated left
$A\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module is noetherian.

######  Lemma 4.16.

Let $K$ be a field endowed with an
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action;
$k:=K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$; $Q_{s}$ and $\Lambda_{s}$ be
$k$-vector spaces. Then any embedding of
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules
$\iota:\bigoplus_{s=0}^{N}K\langle\binom{\Psi}{s}\rangle\otimes_{k}Q_{s}\hookrightarrow
M:=\bigoplus_{s\geq 0}K\langle\binom{\Psi}{s}\rangle\otimes_{k}\Lambda_{s}$
splits, and $\mathrm{Coker}(\iota)\cong\bigoplus_{s\geq
0}K\langle\binom{\Psi}{s}\rangle\otimes_{k}\Lambda_{s}^{\prime}$ for subspaces
$\Lambda_{s}^{\prime}\subseteq\Lambda_{s}$ of codimension $\dim Q_{s}$.

###### Proof.

We proceed by induction on $N\geq 0$. As
$\mathop{\mathrm{Hom}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\binom{\Psi}{s}\rangle,K\langle\binom{\Psi}{t}\rangle)=0$
for $s<t$, $\iota$ factors through
$\bigoplus_{s=0}^{N}K\langle\binom{\Psi}{s}\rangle\otimes_{k}\Lambda_{s}\subseteq
M$. Fix a subset $S\subset\Psi$ of order $N$. Let $\iota_{N}$ be the
restriction of $\iota$ to the summand
$K\langle\binom{\Psi}{N}\rangle\otimes_{k}Q_{N}$. As
$\mathop{\mathrm{End}}\nolimits_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K\langle\binom{\Psi}{N}\rangle)\cong
K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,S}}$, it follows from Lemma 4.2 that
the composition of $\iota_{N}$ with the projection $M\to
K\langle\binom{\Psi}{N}\rangle\otimes_{k}\Lambda_{N}$ is given on $[S]\otimes
Q_{N}$ by an injective element $\alpha$ of
$\mathop{\mathrm{Hom}}\nolimits_{k}(Q_{N},\Lambda_{N})\otimes_{k}K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,S}}$.
Choose a $k$-vector subspace $\Lambda_{N}^{\prime}\subseteq\Lambda_{N}$ such
that
$\Lambda_{N}\otimes_{k}K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,S}}=\mathrm{Im}(\alpha)\dotplus\Lambda_{N}^{\prime}\otimes_{k}K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,S}}$.
Then
$M=\bigoplus_{s=0}^{N-1}K\langle\binom{\Psi}{s}\rangle\otimes_{k}\Lambda_{s}\oplus\mathrm{Im}(\iota_{N})\oplus
K\langle\binom{\Psi}{N}\rangle\otimes_{k}\Lambda_{N}^{\prime}\oplus\bigoplus_{s>N}K\langle\binom{\Psi}{s}\rangle\otimes_{k}\Lambda_{s}$,
which reduces the level. ∎

######  Theorem 4.17.

Let $K$ be a field endowed with a faithful smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action. Then for any morphism
$\varphi:M^{\prime}\to M$ between smooth finitely generated
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules there exist an
open subgroup $U\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi}$ and
decompositions $M^{\prime}=\ker\varphi\oplus M^{\prime}_{0}$ and
$M=\mathrm{Im}(\varphi)\oplus M_{0}$ in
$\mathop{\mathrm{Sm}}\nolimits_{K}(U)$.

###### Proof.

By Corollary 4.15, $\ker\varphi$ is finitely generated. By Proposition 4.6,
there exists an open subgroup
$U\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi}$ such that restrictions to
$U$ of the inclusions $\iota_{1}:\ker\varphi\hookrightarrow M^{\prime}$ and
$\iota_{2}:\mathrm{Im}(\varphi)\hookrightarrow M$ are as in Lemma 4.16. Then
by Lemma 4.16, $\iota_{1}$ and $\iota_{2}$ are split in
$\mathop{\mathrm{Sm}}\nolimits_{K}(U)$, i.e. $M^{\prime}=\ker\varphi\oplus
M^{\prime}_{0}$ and $M=\mathrm{Im}(\varphi)\oplus M_{0}$. ∎

######  Remark 4.18.

Obviously, a surjection
$\bigoplus_{s=0}^{N}K\langle\binom{\Psi}{s}\rangle^{\lambda_{s}}\to\bigoplus_{s=0}^{N}K\langle\binom{\Psi}{s}\rangle^{\kappa_{s}}$
in $\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
need not be split: $K\langle\Psi\rangle\to K$,
$\sum_{i}a_{i}[x_{i}]\mapsto\sum_{i}a_{i}$.

######  Corollary 4.19.

In notation of Corollary 4.9, for any exact sequence $0\to V_{1}\to
V_{2}\to\cdots\to V_{n}\to 0$ of finitely generated objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
one has $\sum_{i=1}^{n}(-1)^{i}P_{V_{i}}=0$.

###### Proof.

By Theorem 4.17, this sequence splits after restriction to an open subgroup of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$, thus giving the claim. ∎

###  4.6. The level filtration

######  Definition 4.20.

Let $G$ be a permutation group of a set $\Psi$, $A$ be an associative ring
endowed with a smooth $G$-action. For each integer $s\geq 0$ and each object
$V$ of $\mathop{\mathrm{Sm}}\nolimits_{A}(G)$, define

  * •

$\widetilde{N}_{s}^{G}V=\widetilde{N}_{s}^{\Psi,G}V$ as the image of the
natural morphism $\bigoplus_{J\in\binom{\Psi}{s}}A\langle
G/G_{J}\rangle\otimes_{A^{G_{J}}}V^{G_{J}}\to V;$

  * •

$N_{s}V=N_{s}^{G}V=N_{s}^{\Psi,G}V$ as the sum of all subobjects
$V^{\prime}\subseteq V$ such that there exists an open subgroup $U\subseteq G$
with the property that $\widetilde{N}_{s}^{U}V^{\prime}=V^{\prime}$.

We say that an object $V$ of $\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ is of
level $\leq s$ with respect to $\Psi$ if $V=N_{s}^{\Psi,G}V$.

######  Lemma 4.21.

Let $G$ be a permutation group of a set $\Psi$, $A$ be an associative ring
endowed with a smooth $G$-action, $V$ be an object of
$\mathop{\mathrm{Sm}}\nolimits_{A}(G)$. Then

  * •

$\widetilde{N}_{0}^{G}$ and $N_{0}^{G}$ are independent of $\Psi$;

  * •

$\widetilde{N}_{s}^{G}\subseteq N_{s}^{G}$,
$\widetilde{N}_{s}^{G}\widetilde{N}_{t}^{G}=\widetilde{N}_{t}^{G}\widetilde{N}_{s}^{G}=\widetilde{N}_{s}^{G}$
and $N_{s}^{G}N_{t}^{G}=N_{t}^{G}N_{s}^{G}=N_{s}^{G}$ for $s\leq t$;

  * •

$\widetilde{N}_{0}^{G}V\subseteq\widetilde{N}_{1}^{G}V\subseteq\widetilde{N}_{2}^{G}V\subseteq\cdots$
and $N_{0}^{G}V\subseteq N_{1}^{G}V\subseteq N_{2}^{G}V\subseteq\cdots$ are
functorial exhausting filtrations. Both filtrations stabilize for finitely
generated $V$.

Let $U\subseteq G$ be an open subgroup. Then any smooth $A\langle
G\rangle$-module $V$ is also smooth when considered as an $A\langle
U\rangle$-module, and $N_{s}^{G}V\subseteq N_{s}^{U}V$. If $V\neq N_{0}^{G}V$
then $N_{0}^{G}V\neq N_{0}^{U}V$.

Suppose that the set $U\backslash G/U^{\prime}$ is finite for any open
subgroup $U^{\prime}\subseteq G$. Then any smooth finitely generated $A\langle
G\rangle$-module is also finitely generated as an $A\langle U\rangle$-module.

###### Proof.

The equality
$\widetilde{N}_{s}^{G}\widetilde{N}_{t}^{G}=\widetilde{N}_{t}^{G}\widetilde{N}_{s}^{G}=\widetilde{N}_{s}^{G}$
for $s\leq t$ is evident. Then $V^{\prime}=\widetilde{N}_{s}^{G}$ satisfies
$\widetilde{N}_{s}^{G}V^{\prime}=V^{\prime}$, so
$V^{\prime}=\widetilde{N}_{s}^{G}V\subseteq N_{s}^{G}V$.

Let us show that $N_{s}^{G}V\subseteq N_{s}^{U}V$, or equivalently, that
$V^{\prime}\subseteq N_{s}^{U}V$ for any subobject $V^{\prime}\subseteq V$ in
$\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ such that there exists an open subgroup
$U^{\prime}\subseteq G$ with the property that
$\widetilde{N}_{s}^{U^{\prime}}V^{\prime}=V^{\prime}$. By definition, the
natural morphism $\bigoplus_{J\in\binom{\Psi}{s}}A\langle
U^{\prime}/U^{\prime}_{J}\rangle\otimes_{A^{U^{\prime}_{J}}}(V^{\prime})^{U^{\prime}_{J}}\to
V^{\prime}$ in $\mathop{\mathrm{Sm}}\nolimits_{A}(U^{\prime})$ is surjective.
Set $U^{\prime\prime}:=U\cap U^{\prime}$. The left $U^{\prime\prime}$-action
on $U^{\prime}/U^{\prime}_{J}$ gives a splitting $A\langle
U^{\prime}/U^{\prime}_{J}\rangle=\bigoplus_{[g]\in U^{\prime\prime}\backslash
U^{\prime}/U^{\prime}_{J}}A\langle
U^{\prime\prime}gU^{\prime}_{J}/U^{\prime}_{J}\rangle.$ Using the equality
$U^{\prime\prime}\cap gU^{\prime}_{J}g^{-1}=U^{\prime\prime}_{g(J)}$ and the
$U^{\prime\prime}$-equivariant bijection
$U^{\prime\prime}gU^{\prime}_{J}/U^{\prime}_{J}\xrightarrow{\sim}U^{\prime\prime}/(U^{\prime\prime}\cap
gU^{\prime}_{J}g^{-1})$ ($[g]\mapsto[1]$), we see that the image of $A\langle
U^{\prime\prime}gU^{\prime}_{J}/U^{\prime}_{J}\rangle\otimes_{A^{U^{\prime}_{J}}}(V^{\prime})^{U^{\prime}_{J}}$
in $V$ ($[ug]\otimes v\mapsto ugv$) coincides with the image of $A\langle
U^{\prime\prime}/U^{\prime\prime}_{g(J)}\rangle\otimes_{A^{U^{\prime}_{g(J)}}}(V^{\prime})^{U^{\prime}_{g(J)}}$
($[u]\otimes gv\mapsto ugv$), i.e. the natural morphism
$\bigoplus_{J\in\binom{\Psi}{s}}A\langle
U^{\prime\prime}/U^{\prime\prime}_{J}\rangle\otimes_{A^{U^{\prime}_{J}}}(V^{\prime})^{U^{\prime}_{J}}\to
V^{\prime}$ is surjective, and therefore, the natural morphism
$\bigoplus_{J\in\binom{\Psi}{s}}A\langle
U^{\prime\prime}/U^{\prime\prime}_{J}\rangle\otimes_{A^{U^{\prime}_{J}}}(V^{\prime})^{U^{\prime\prime}_{J}}\to
V^{\prime}$ is surjective as well, i.e.
$\widetilde{N}_{s}^{U^{\prime\prime}}V^{\prime}=V^{\prime}$.

The $A\langle G\rangle$-modules $A\langle G/U^{\prime}\rangle$ for all open
subgroups $U^{\prime}$ of $G$ form a generating family of the category of
smooth $A\langle G\rangle$-modules. It suffices, thus, to check that $A\langle
G/U^{\prime}\rangle$ is a finitely generated $A\langle U\rangle$-module for
all open subgroups $U^{\prime}$ of $G$. Choose representatives $\alpha_{i}\in
G/U^{\prime}$ of the elements of $U\backslash G/U^{\prime}$. Then
$G/U^{\prime}=\coprod_{i}U\alpha_{i}$, so $A\langle
G/U^{\prime}\rangle\cong\bigoplus_{i}A\langle
U/(U\cap\alpha_{i}U^{\prime}\alpha_{i}^{-1})\rangle$ is a finitely generated
$A\langle U\rangle$-module. ∎

Examples. 1\. The finiteness assumption of Lemma 4.21 is valid for any Roelcke
precompact group $G$, i.e., such that the set $U^{\prime}\backslash
G/U^{\prime}$ is finite for any open subgroup $U^{\prime}\subseteq G$.
Examples of such groups include the profinite groups, the symmetric groups
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$, the automorphism group of a vector
space over a finite field [this is shown in Lemma 3.6]; the open subgroups of
any Roelcke precompact group, the products of Roelcke precompact groups.

2\. For $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ or $\mathrm{GL}(\Psi)$, the
restriction functor splits the generators $A\langle\binom{\Psi}{t}_{q}\rangle$
into finite direct sums of indecomposable generators via canonical
isomorphisms of $A\langle G_{J}\rangle$-modules
$A\langle\binom{\Psi}{t}_{q}\rangle=\bigoplus_{\Lambda\subseteq
J}M_{\Lambda}$, where $M_{\Lambda}$ is the free $A$-module on the set of all
subobjects of $\Psi$ of length $t$ and meeting $J$ along $\Lambda$.

######  Example 4.22.

1\. If an object $V$ of $\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ is a sum of
$G$-invariant finitely generated $A$-submodules then $V=N_{0}^{\Psi}V$.
[Indeed, any finitely generated $A$-module is fixed by an open subgroup of
$G$.]

2\. If $G$ is locally precompact and an open subgroup of $G$ acts faithfully
on a field $K$ then any object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ is of
level 0. [Indeed, let $U$ be an open precompact subgroup of $G$ acting
faithfully on $K$. Then $V=K\otimes_{K^{U}}V^{U}$, so
$V=\widetilde{N}_{0}^{U}V$, and therefore, $V=N_{0}^{G}V$.]

For each $j\geq 0$, let $\mathop{\mathrm{Sm}}\nolimits_{K}^{\leq j}(G)$ be the
full subcategory of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ of the objects of
level y $\leq j$. Then the inclusion functor
$\mathop{\mathrm{Sm}}\nolimits_{K}^{\leq
j}(G)\hookrightarrow\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ admits a right
adjoint
$N_{j}:\mathop{\mathrm{Sm}}\nolimits_{K}(G)\to\mathop{\mathrm{Sm}}\nolimits_{K}^{\leq
j}(G)$, $V\mapsto N_{j}V$, the maximal subobject of $V$ of level $\leq j$.

######  Corollary 4.23.

In notation of Corollary 4.9, for any object $V$ of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
$N_{s}V$ consists of the elements $v\in V$ such that $\deg
P_{K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle v}\leq s$. In
particular, the level filtration

  * •

on $\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
is strictly compatible with injections, i.e. $N_{s}V=V\cap N_{s}W$ for any
inclusion $V\subset W$ in
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\
($so that $N_{s}E(V)=0$ whenever $N_{s}V=0);$

  * •

is strictly compatible with $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field
extensions $\widetilde{K}|K$:
$N_{s}(\widetilde{K}\otimes_{K}V)=\widetilde{K}\otimes_{K}N_{s}V$.

$\mathop{\mathrm{Sm}}\nolimits_{K}^{\leq
s}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ is a cocomplete Serre subcategory
of $\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
non-complete if $K$ is non-trivial.

###### Proof.

We may assume that $V$ is finitely generated, so, for a finite $J\subset\Psi$,
its restriction to $\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$ splits as
$\bigoplus_{j}K\langle\binom{\Psi\smallsetminus J}{j}\rangle^{\kappa_{j}}$. By
Lemma 4.2, any element outside of $\bigoplus_{j\leq
s}K\langle\binom{\Psi\smallsetminus J}{j}\rangle^{\kappa_{j}}$ generates a
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi\smallsetminus
J}\rangle$-submodule that grows as a polynomial of degree $>s$.

By Corollary 4.19, the polynomial $P$ is additive on exact sequences, so (i)
the level of a subquotient of $V$ does not exceed the level of $V$; (ii) if
$V$ is an extension of $V_{1}$ by $V_{2}$ then the level of $V$ coincides with
the maximum of the levels of $V_{1}$ and $V_{2}$. ∎

######  Definition 4.24.

Let $G$ be a group and $U\subseteq G$ be a subgroup. We say that $U$ is an
f-subgroup if there exist integers $N,m\geq 1$ and elements $g_{ij}\in G$ for
$1\leq i\leq m$ and $1\leq j\leq N$ such that
$\bigcup_{j=1}^{N}g_{1j}Ug_{2j}U\cdots Ug_{mj}=G$.

######  Example 4.25.

In the following examples $U$ is an f-subgroup of a group $G$.

  * •

$U$ is a subgroup of $G$ of finite index;

  * •

$U$ is an f-subgroup of an f-subgroup of $G$;

  * •

$U$ contains an f-subgroup of $G$;

  * •

$U$ is an open subgroup of a Roelcke precompact group $G$;

  * •

$U$ is a parabolic subgroup of a linear algebraic group $G$ over an
algebraically closed field;

  * •

$U$ is an open subgroup of $G$, which is either $\mathrm{GL}(V)$ for an
infinite-dimensional vector space $V$ or $\mathop{\mathrm{Aut}}\nolimits(F|k)$
for an algebraically closed field extension $F|k$ of infinite transcendence
degree.

###### Proof.

The first 4 cases are evident. The last one follows from Lemma 3.6. The 5-th
case follows from the Bruhat decomposition. ∎

######  Lemma 4.26.

Let $G$ be a group acting on a left noetherian ring $A$, $U\subseteq G$ be an
f-subgroup, $V$ be an $A\langle G\rangle$-module generated as an $A$-module by
$V^{U}$. Then $V$ is a sum of $A\langle G\rangle$-submodules noetherian as
$A$-modules.

###### Proof.

For each finitely generated $A$-submodule $B\subseteq V$, denote by
$B^{\prime}$ an $A$-submodule of $V$ containing $B$ and generated by finitely
many elements of $V^{U}$. Then $B^{\prime}$ is an $A\langle U\rangle$-module.
For any $v\in V$ and any finite collection $g_{1},g_{2},\dots,g_{m}\in G$, the
$A$-module $A\langle g_{1}Ug_{2}U\cdots Ug_{m}\rangle v$ is contained in the
finitely generated $A$-module
$(g_{1}((g_{2}(\cdots(g_{m-2}((g_{m-1}((Ag_{m}v)^{\prime}))^{\prime}))^{\prime}\cdots))^{\prime}))^{\prime}$.
As $U$ is an f-subgroup of $G$, $A\langle G\rangle v=\sum_{j=1}^{N}A\langle
g_{1j}Ug_{2j}U\cdots Ug_{mj}\rangle v$, so this shows that any cyclic
$A\langle G\rangle$-submodule of $V$ is contained in a finitely generated
$A$-module. As $A$ is noetherian, any cyclic $A\langle G\rangle$-submodule of
$V$ is a noetherian $A$-module. ∎

######  Corollary 4.27.

Let $G$ be a permutation group such that any open subgroup is an f-subgroup,
$A$ be a left noetherian associative ring endowed with a smooth $G$-action,
$V$ be an object of $\mathop{\mathrm{Sm}}\nolimits_{A}(G)$. Then $N_{0}^{G}V$
is the sum of all subobjects of $V$ noetherian as $A$-modules.

###### Proof.

If an object $V$ of $\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ is finitely
generated as $A$-module then its generators are fixed by an open subgroup of
$G$, and thus, $V$ is of level 0. Conversely, a level 0 object $V$ of
$\mathop{\mathrm{Sm}}\nolimits_{A}(G)$ is a sum of subobjects $V_{i}$
generated as $A$-modules by some elements fixed by an open subgroup $U_{i}$ of
$G$. By Lemma 4.26, $V_{i}$ is a sum of $A\langle G\rangle$-submodules
noetherian as $A$-modules. ∎

###  4.7. Existence of weak period extensions

It is not clear so far, whether there exist $G$-period extensions (Definition
2.4). However, we may weaken Definition 2.4 by omitting the condition on the
fixed subfield:

######  Definition 4.28.

For a permutation group $G$ and a smooth $G$-field $F$, a smooth $G$-field
extension $K|F$ is called a weak $G$-period extension of $F$ if $K$ is a
cogenerator of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$.

######  Proposition 4.29.

Let $\mathcal{O}$ be a non-constant sheaf of fields on
$\mathrm{FI}^{\mathrm{op}}$. For a fixed infinite set, say $\mathbb{N}$, let
$\widetilde{\mathcal{O}}:=\mathop{\underrightarrow{\lim}}\limits_{J\subset\mathbb{N}}\mathcal{O}_{+J}$
be the colimit over the finite subsets $J$. Then $\widetilde{\mathcal{O}}$ is
a weak period sheaf over $\mathcal{O}$.

###### Proof.

By Lemma 4.11, it suffices to check that any finitely generated
$\widetilde{\mathcal{O}}$-module $W$ can be embedded into a product of copies
of $\widetilde{\mathcal{O}}$.

The $\widetilde{\mathcal{O}}$-module $W$ is a quotient of a finite direct sum
of $\widetilde{\mathcal{O}}$-modules
$h^{I}_{\widetilde{\mathcal{O}}}=\widetilde{\mathcal{O}}\otimes
h^{I}_{\mathbb{Z}}$ by a submodule. By Proposition 4.14, this submodule is
finitely generated, so there is a finite set $J$ and an
$\mathcal{O}_{+J}$-module $W_{0}$ such that
$W=\widetilde{\mathcal{O}}\otimes_{\mathcal{O}_{+J}}W_{0}$.

By Proposition 4.6, there is a finite set $J^{\prime}\supseteq J$ such that
the $\mathcal{O}_{+J^{\prime}}$-module
$W_{1}:=(W_{0})_{+(J^{\prime}\smallsetminus J)}$ is a sum of sheaves
$\bar{h}^{s}_{\mathcal{O}_{+J^{\prime}}}$. By Lemmas 3.9 and 3.7,
$\bar{h}^{s}_{\mathcal{O}_{+J^{\prime}}}$ embeds into a product of copies of
$\mathcal{O}_{+J^{\prime}}$ for sufficiently big $J^{\prime}$. As
$\mathcal{O}_{+J^{\prime}}\otimes_{\mathcal{O}_{+J}}W_{0}$ is embedded into
$W_{1}$,
$W=\widetilde{\mathcal{O}}\otimes_{\mathcal{O}_{+J^{\prime}}}(\mathcal{O}_{+J^{\prime}}\otimes_{\mathcal{O}_{+J}}W_{0})$
is embedded into
$\widetilde{\mathcal{O}}\otimes_{\mathcal{O}_{+J^{\prime}}}W_{1}$, which is a
sum of sheaves $\bar{h}^{s}_{\widetilde{\mathcal{O}}}$. ∎

##  5\. The spectrum of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ for
$K=K_{\mathrm{a}},K_{\mathrm{b}},K_{\mathrm{c}}$

###  5.1. A construction of $G$-extensions with Picard group containing a
given abelian group

For a permutation group $G$ and a smooth $G$-field $K$, denote by
$\mathrm{Pic}_{K}(G)$ the group of isomorphism classes of invertible objects
of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, called the Picard group of
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, i.e.
$\mathrm{Pic}_{K}(G)=H^{1}_{\text{{\rm cont}}}(G,K^{\times})$.

The group $\mathrm{Pic}_{K}(G)$ acts continuously on the Gabriel spectrum of
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ by
$\mathcal{L}:I\mapsto\mathcal{L}\otimes_{K}I$.

######  Example 5.1 (The fields $K_{\mathrm{a}}=K_{\Psi,S,\Gamma}^{L}$).

Fix a quadruple $\mathrm{a}=(\Psi,S,\Gamma,L)$, where $\Psi$ and $S$ are sets,
$L$ is a field, $\Gamma$ is a subgroup of the group $\Xi:=\mathbb{Z}\langle
S\rangle$.

Let $L(\Psi\times S)=L(S)_{L,\Psi}$ be the purely transcendental field
extension of $L$ with a transcendence basis consisting of the variables
labeled by the set $\Psi\times S$. Denote by $u_{s}$ the variable
corresponding to $(u,s)\in\Psi\times S$. For each $s\in S$, sending $u\in\Psi$
to $u_{s}$ gives rise to a field embedding $L(\Psi)\hookrightarrow
L(\Psi\times S)$, $f\mapsto f_{s}=f^{[s]}$. For all $f\in L(\Psi)^{\times}$
and $\gamma=\sum_{s}m_{s}[j_{s}]\in\Xi$, set
$f^{\gamma}:=\prod_{s}f_{j_{s}}^{m_{s}}$. Then the embeddings $u\mapsto u_{s}$
give rise to the homomorphism $\Xi\otimes L(\Psi)^{\times}\to L(\Psi\times
S)^{\times}$, $\gamma\otimes f\mapsto f^{\gamma}$.

Define a subfield $K_{\mathrm{a}}=K_{\Psi,S,\Gamma}^{L}$ of $L(\Psi\times S)$
by

$K_{\mathrm{a}}:=L\left(u^{\gamma},\frac{u_{s}}{v_{s}}~{}|~{}\gamma\in\Gamma,~{}s\in
S,~{}u,v\in\Psi\right)=L\left(x^{\gamma},\frac{u_{s}}{x_{s}}~{}|~{}\gamma\in\Gamma,~{}s\in
S,~{}u\in\Psi\right)$

for an arbitrary fixed $x\in\Psi$ (so $K_{\Psi,\varnothing,0}^{L}:=L$,
$K_{\Psi,\\{\ast\\},0}^{L}=L(u/v~{}|~{}u,v\in\Psi)$).

If $G$ is a group acting on the set $\Psi$ and on the field $L$ then

  * •

$G$ acts on the fields $L(\Psi)$ and $L(\Psi\times S)$, while the above
homomorphisms
$L(\Psi)\stackrel{{\scriptstyle(-)_{s}}}{{\hookrightarrow}}L(\Psi\times S)$
and $\Xi\otimes L(\Psi)^{\times}\to L(\Psi\times S)^{\times}$ are
$G$-equivariant,

  * •

$G$ preserves the one-dimensional $K_{\mathrm{a}}$-vector subspaces
$x^{\lambda}K_{\mathrm{a}}\subseteq L(\Psi\times S)$ for all $\lambda\in\Xi$,

  * •

$L(\Psi\times S)|K_{\mathrm{a}}$ is a $G$-field extension of degree equal to
the index of $\Gamma$.

This shows that, for any non-precompact permutation group $G$, any smooth
$G$-field $L$, any abelian group $\Lambda$, there exists a smooth $G$-field
extension $K_{\mathrm{a}}|L$ and a group embedding
$\Lambda\hookrightarrow\mathrm{Pic}_{K_{\mathrm{a}}}(G)$.

######  Lemma 5.2.

We keep notation of Example 5.1.

  1. (1)

The $K_{\mathrm{a}}\langle G\rangle$-module $L(\Psi\times S)$ embeds into
$\prod_{\lambda\in\Xi}x^{\lambda}K_{\mathrm{a}}$ if $\Gamma$ is finitely
generated.

  2. (2)

Assume that the $G$-action on $L(\Psi)$ is smooth. Then sending $\lambda$ to
the class of $x^{\lambda}K_{\mathrm{a}}$ gives rise to a homomorphism
$\Xi\to\mathrm{Pic}_{K_{\mathrm{a}}}(G)$ that factors through
$\iota:\Xi/\Gamma\to\mathrm{Pic}_{K_{\mathrm{a}}}(G)$. If, moreover, all
$G$-orbits on $\Psi$ are infinite then the homomorphism $\iota$ is injective.

  3. (3)

Assume that the $G$-action on $L(\Psi)$ is smooth, all $G$-orbits on $\Psi$
are infinite, $\Gamma$ is finitely generated, $\mathrm{Pic}_{L(\Psi\times
S)}(G)=0$. Then

     * •

the map $\iota:\Xi/\Gamma\to\mathrm{Pic}_{K_{\mathrm{a}}}(G)$,
$\lambda\mapsto[x^{\lambda}K_{\mathrm{a}}]$, is an isomorphism, and

     * •

the invertible subobjects $x^{\lambda}K_{\mathrm{a}}\subset L(\Psi\times S)$
form a system of cogenerators of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(G)$ if $L(\Psi\times S)$ is a
cogenerator of $\mathop{\mathrm{Sm}}\nolimits_{L(\Psi\times S)}(G)$.

###### Proof.

Suppose that $\Xi$ admits a basis $S^{\prime}$ such that $\Gamma$ is generated
by the elements $m(s)s$ for all $s\in S^{\prime}$. (E.g. if $\Gamma$ is
finitely generated then such an $S^{\prime}$ is provided by the Smith normal
form.) Let $S_{0}:=\\{s\in S^{\prime}~{}|~{}m(s)=0\\}$ and
$S_{+}:=S^{\prime}\smallsetminus S_{0}$. Fix a total order on $S_{0}$, and
order $\mathbb{Z}\langle S_{0}\rangle$ lexicographically. Then the Hahn
$\mathbb{Z}\langle S_{0}\rangle$-power series over the field
$\widetilde{K_{\mathrm{a}}}:=L(x_{j},u_{s}/v_{s}~{}|~{}u,v\in\Psi,~{}s\in
S,~{}j\in S_{+})$ form a $G$-field
$\widetilde{K_{\mathrm{a}}}((x^{\mathbb{Z}\langle
S_{0}\rangle})):=\\{\sum_{\lambda\in
J}a_{\lambda}x^{\lambda}~{}|~{}J\subset\mathbb{Z}\langle S_{0}\rangle\mbox{ is
well-ordered, }a_{\lambda}\in\widetilde{K_{\mathrm{a}}}\\}$ containing
naturally $L(\Psi\times S)$. On the other hand, the
$\widetilde{K_{\mathrm{a}}}\langle G\rangle$-module
$\widetilde{K_{\mathrm{a}}}((x^{\mathbb{Z}\langle S_{0}\rangle}))$ is
naturally embedded into $\prod_{\lambda\in\mathbb{Z}\langle
S_{0}\rangle}x^{\lambda}\widetilde{K_{\mathrm{a}}}$, while
$\widetilde{K_{\mathrm{a}}}=\bigoplus_{\xi\in\Gamma^{\perp}}x^{\xi}K_{\mathrm{a}}$,
where $\Gamma^{\perp}=\\{\sum_{s\in S_{+}}m_{s}[s]\in\mathbb{Z}\langle
S_{+}\rangle~{}|~{}0\leq m_{s}<|m(s)|\\}$. This means that the
$K_{\mathrm{a}}\langle G\rangle$-module $L(\Psi\times S)$ embeds into
$\prod_{\lambda\in\Xi}x^{\lambda}K_{\mathrm{a}}$.

If $\mathrm{Pic}_{L(\Psi\times S)}(G)=0$ then any element of
$\mathrm{Pic}_{K_{\mathrm{a}}}(G)$ can be considered as a subobject of
$L(\Psi\times S)$, i.e. a one-dimensional $K_{\mathrm{a}}$-vector subspace
$\varphi K_{\mathrm{a}}$ for a rational function $\varphi$. Let $S_{0}$ be a