finite subset such that $\varphi\in L(\Psi\times S_{0})\subseteq L(\Psi\times
S)$. Set $K_{\mathrm{a}}^{\prime}:=L(\Psi\times S_{0})\cap K_{\mathrm{a}}$.
Then $\varphi K_{\mathrm{a}}=\varphi
K_{\mathrm{a}}^{\prime}\otimes_{K_{\mathrm{a}}^{\prime}}K_{\mathrm{a}}$. We
know that the $K_{\mathrm{a}}^{\prime}\langle G\rangle$-module $L(\Psi\times
S_{0})$ embeds into $\prod_{\lambda\in\mathbb{Z}\langle
S_{0}\rangle}x^{\lambda}K_{\mathrm{a}}^{\prime}$, so $\varphi
K_{\mathrm{a}}^{\prime}\cong x^{\lambda}K_{\mathrm{a}}^{\prime}$ for some
$\lambda\in\mathbb{Z}\langle S_{0}\rangle$, and thus, $\varphi
K_{\mathrm{a}}\cong x^{\lambda}K_{\mathrm{a}}$.

As $L(\Psi\times S)$ is a cogenerator of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(G)$, to show that the
subobjects $x^{\lambda}K_{\mathrm{a}}\subset L(\Psi\times S)$ form a system of
cogenerators, it suffices to verify that $L(\Psi\times S)$ embeds into
$\prod_{\lambda\in\Xi}x^{\lambda}K_{\mathrm{a}}$. Each element of
$L(\Psi\times S)$ belongs to $L(\Psi\times S_{0})$ for a finite subset
$S_{0}\subseteq S$, while $L(\Psi\times S_{0})$ embeds into
$\prod_{\lambda\in\mathbb{Z}\langle
S_{0}\rangle}x^{\lambda}K_{\mathrm{a}}\stackrel{{\scriptstyle(-)\times\text{`0'}}}{{\hookrightarrow}}\prod_{\lambda\in\Xi}x^{\lambda}K_{\mathrm{a}}$.

If all $G$-orbits on $\Psi$ are infinite then $L(\Psi\times S)^{G}\subseteq
L$, so $(x^{\gamma}K_{\mathrm{a}})^{G}\neq 0$ only if $(x^{\gamma})^{-1}\in
K_{\mathrm{a}}$, which means that $\gamma\in\Gamma$. ∎

######  Remark 5.3.

Keeping notation of Lemma 5.2, fix a subset $\Gamma^{\perp}$ of $\Xi$
projecting bijectively onto the quotient $\Xi/\Gamma$. As any simple objects
of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
is isomorphic to $x^{\lambda}K_{\mathrm{a}}$ for a unique
$\lambda\in\Gamma^{\perp}$, and the $k$-vector space
$\mathop{\mathrm{Hom}}\nolimits_{K_{\mathrm{a}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(x^{\lambda}K_{\mathrm{a}},\prod_{\lambda\in\Xi}x^{\lambda}K_{\mathrm{a}})$
is isomorphic to $\prod_{\Gamma}k$, the maximal semisimple
$K_{\mathrm{a}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodule
(the socle) of $\prod_{\lambda\in\Xi}x^{\lambda}K_{\mathrm{a}}$ is isomorphic
to
$\bigoplus_{\lambda\in\Gamma^{\perp}}x^{\lambda}K_{\mathrm{a}}\otimes_{k}\prod_{\Gamma}k$.
Then the socle of $F_{\Psi}(\Psi\times S)$ is
$\bigoplus_{\lambda\in\Gamma^{\perp}}(x^{\lambda}K_{\mathrm{a}})^{\oplus\Gamma}$.

######  Lemma 5.4.

Let $\Psi$ be an infinite set, $F|k$ be a field extension such that $k$ is
algebraically closed in $F$. Let $S$ be a finite set, non-empty if $F=k$. Let
$F_{\Psi}(\Psi\times S)=F(S)_{\Psi}$ be the purely transcendental
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field extension 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 $(u,s)\in\Psi\times S$.

For all $u\in\Psi$ and $\gamma=\sum_{s}m_{s}[j_{s}]\in\Xi$, set
$u^{\gamma}:=\prod_{s}u_{j_{s}}^{m_{s}}$. For a subgroup
$\Gamma\subseteq\Xi:=\mathbb{Z}\langle S\rangle$, consider the following
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-subfields of $F_{\Psi}(\Psi\times S)$
and $F_{\Psi}(\Psi)=F(X)_{\Psi}$ of $F_{\Psi}$:

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

$($so $K_{\Psi,\varnothing,0}^{(F|k)}=F_{\Psi};$
$K_{\Psi,\\{\ast\\},0}^{(F|k)}=F_{\Psi}(u/v~{}|~{}u,v\in\Psi);$
$K_{\Psi,S\sqcup S^{\prime},\Gamma\oplus\mathbb{Z}\langle
S^{\prime}\rangle}^{(F|k)}=K_{\Psi,S,\Gamma}^{(F(S^{\prime})|k)});$

$K_{\mathrm{b}}:=F_{\Psi}(u-v~{}|~{}u,v\in\Psi),\quad
K_{\mathrm{c}}:=F_{\Psi}\left(\frac{u-v}{u-w}~{}|~{}u,v,w\in\Psi,~{}\\#\\{u,v,w\\}=3\right),$
$K_{\mathrm{d}}:=F_{\Psi}\left(\frac{(t-u)(v-w)}{(u-v)(w-t)}~{}|~{}t,u,v,w\in\Psi,~{}\\#\\{t,u,v,w\\}=4\right).$

Fix some pairwise distinct $x,y,z\in\Psi$. Then $x,y,z\in\Psi$ are
algebraically independent over $K_{\mathrm{d}}$,
$K_{\mathrm{c}}=K_{\mathrm{d}}\left(\frac{x-y}{x-z}\right)=K_{\Psi,\\{\ast\\},0}^{(F|k)}\cap
K_{\mathrm{b}}$, $x/y,x-y$ are algebraically independent over
$K_{\mathrm{c}}$, $K_{\mathrm{b}}=K_{\mathrm{c}}(x-y)$,
$K_{\Psi,\\{\ast\\},0}^{(F|k)}=K_{\mathrm{c}}(x/y)$, and
$F_{\Psi}(\Psi)=K_{\mathrm{d}}(x,y,z)=K_{\mathrm{c}}(x,y)=K_{\Psi,\\{\ast\\},0}^{(F|k)}(x)=K_{\mathrm{b}}(x);$
for all $?\in\\{\mathrm{b},\mathrm{c},\mathrm{d}\\}$, the field extensions
$K_{\mathrm{a}}|F_{\Psi}$, $K_{?}|F_{\Psi}$, $F_{\Psi}(\Psi\times
S)|K_{\Psi,S,0}^{(F|k)}$, $F_{\Psi}(\Psi)|K_{?}$ are purely transcendental.

###### Proof.

One has
$K_{\mathrm{a}}=F_{\Psi}\left(x^{\gamma},\frac{u_{s}}{x_{s}}~{}|~{}u\in\Psi\smallsetminus\\{x\\}\right)$,
since $u^{\gamma}/x^{\gamma}=(u/x)^{\gamma}$ and
$u_{s}/v_{s}=\frac{u_{s}/x_{s}}{v_{s}/x_{s}}$;
$K_{\mathrm{b}}=F_{\Psi}\left(u-x~{}|~{}u\in\Psi\smallsetminus\\{x\\}\right)$,
since $u-v=(u-x)-(v-x)$;
$K_{\mathrm{c}}=F_{\Psi}\left(\frac{x-u}{x-y}~{}|~{}u\in\Psi\smallsetminus\\{x,y\\}\right)$,
since
$\frac{u-v}{u-w}=\frac{\frac{x-u}{x-y}-\frac{x-v}{x-y}}{\frac{x-u}{x-y}-\frac{x-w}{x-y}}$;
$K_{\mathrm{d}}=F_{\Psi}\left(\frac{(x-u)(y-z)}{(u-y)(z-x)}~{}|~{}u\in\Psi\smallsetminus\\{x,y,z\\}\right)$,
since
$\frac{(x-u)(y-z)}{(u-y)(z-x)}-\frac{(x-v)(y-z)}{(v-y)(z-x)}=\frac{(x-y)(y-z)(v-u)}{(z-x)(u-y)(v-y)}$,
$\frac{(x-y)(y-z)(v-u)}{(z-x)(u-y)(v-y)}\times\frac{(z-x)(w-y)(v-y)}{(x-y)(y-z)(v-w)}=\frac{(v-u)(w-y)}{(u-y)(v-w)}$,
$\frac{(v-u)(w-y)}{(u-y)(v-w)}\times\frac{(u-y)(t-w)}{(t-u)(w-y)}=\frac{(v-u)(t-w)}{(t-u)(v-w)}$.
Thus, all these fields are purely transcendental over $F_{\Psi}$.

As
$\frac{u}{v}=\frac{1-\frac{x-u}{x-y}(1-\frac{y}{x})}{1-\frac{x-v}{x-y}(1-\frac{y}{x})}$,
$K_{\Psi,\\{\ast\\},0}^{(F|k)}=K_{\mathrm{c}}(x/y)$; as
$u-v=(x-y)(\frac{x-v}{x-y}-\frac{x-u}{x-y})$,
$K_{\mathrm{b}}=K_{\mathrm{c}}(x-y)$; as
$\frac{x-u}{x-z}\left(\frac{x-y}{x-z}+\frac{(x-u)(y-z)}{(x-z)(y-u)}-1\right)=\frac{(x-u)(y-z)}{(x-z)(y-u)}\times\frac{x-y}{x-z}$,
$K_{\mathrm{c}}=K_{\mathrm{d}}\left(\frac{x-y}{x-z}\right)$. Thus, the
extensions $F_{\Psi}(\Psi)|K_{?}$ for
$?\in\\{\mathrm{b},\mathrm{c},\mathrm{d}\\}$ and $F_{\Psi}(\Psi\times
S)|K_{\mathrm{a}}$ are purely transcendental if $\Xi/\Gamma$ is torsion free.
∎

###  5.2. Proof of Theorem 1.3 in the case $K=K_{\mathrm{a}}$

######  Proposition 5.5.

Let $\Psi$ be a set, $F$ be a field and $k$ be a proper subfield algebraically
closed in $F$, $K=F_{\Psi}$. Then $K$ is an injective object of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

###### Proof.

By Lemma 4.12, it suffices to construct, for any finite $J\subset\Psi$, a
morphism of
$K^{(J)}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-modules
$\xi:K\to K^{(J)}$ identical on $K^{(J)}$. We consider $K$ as the fraction
field of the algebra $K^{(J)}\otimes_{k}F_{J}$ and embed both the algebra and
the field into a field of series with coefficients in
$K^{(J)}\otimes_{k}\overline{k}$ for an algebraic closure $\overline{k}$ of
$k$.

For each ordered group $\Gamma$, let $\overline{k}((\Gamma))$ be the field of
Hahn power series over $\overline{k}$, i.e. the set of formal expressions of
the form $\sum_{s\in\Gamma}a_{s}\cdot s$, where $a_{s}\in\overline{k}$ and the
set $\\{s\in\Gamma~{}|~{}a_{s}\neq 0\\}$ is well-ordered. Fix a totally
ordered divisible group $\Gamma$, such that transcendence degree of the field
extension $k((\Gamma))|k$ is at least that of $F|k$. By [7], there is a field
embedding $F_{J}\hookrightarrow\overline{k}((\Gamma))$ over $k$, so the
$\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$-field $K$ becomes a subfield of
$(K^{(J)}\otimes_{k}\overline{k})((\Gamma))$ with
$\mathop{\mathfrak{S}}\nolimits_{\Psi|J}$ acting on the coefficients. Define
$\widetilde{\xi}:K\to K^{(J)}\otimes_{k}\overline{k}$ as the ‘constant term’
of the Hahn power series expression: $\sum_{s\in\Gamma}a_{s}\cdot s\mapsto
a_{0}$. Fix a $k$-linear functional $\nu:\overline{k}\to k$ identical on $k$.
Finally, we define $\xi:K\to K^{(J)}$ as $(id\cdot\nu)\circ\widetilde{\xi}$. ∎

######  Lemma 5.6.

If any non-zero object in a Grothendieck category $\mathcal{A}$ admits a
simple subquotient, e.g. $\mathcal{A}=\mathop{\mathrm{Sm}}\nolimits_{A}(G)$
for a permutation group $G$ and a unital $G$-ring $A$, then injective hulls of
the simple objects form a system of cogenerators.

###### Proof.

Assume that, for any non-zero morphism $X\to Y$, its image admits a subobject
$Z$ with a simple quotient $Q$. Then the composition $Z\to Q\hookrightarrow
E(Q)$ extends to a morphism $\varphi:Y\to E(Q)$ non-zero on $Z$, and
therefore, with a non-zero composition $X\to
Y\xrightarrow{\varphi}Q\hookrightarrow E(Q)$. Thus, all such $E(Q)$ form a
system of cogenerators. ∎

######  Theorem 5.7.

Let $\Psi$ be a set, $F$ be a field and $k\subsetneqq F$ be a subfield
algebraically closed in $F$.

Let $K\subseteq F_{\Psi}$ be an
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant subfield. Then the object
$F_{\Psi}$ is an injective cogenerator of the category
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$. In
particular, (i) any smooth $K$-semilinear representation of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ can be embedded into a direct product
of copies of $F_{\Psi}$; (ii) any smooth $F_{\Psi}$-semilinear representation
of $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ of finite length is isomorphic to a
direct sum of copies of $F_{\Psi}$.

###### Proof.

By Proposition 4.6, for any smooth simple
$F_{\Psi}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $M$ there
is a finite subset $J\subset\Psi$ and an isomorphism of
$F_{\Psi}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-modules
$\bigoplus_{s=0}^{N}F_{\Psi}\langle\binom{\Psi\smallsetminus
J}{s}\rangle^{\kappa_{s}}\xrightarrow{\sim}M$ for some integer
$N,\kappa_{0},\dots,\kappa_{N}\geq 0$. By Lemma 4.4, the
$F_{\Psi}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $M$
admits a simple $F_{\Psi\smallsetminus
J}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-submodule
$M^{\prime}$. By Lemma 3.13, Remark 3.14 (1) and Corollary 3.12, there are no
simple $F_{\Psi\smallsetminus
J}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-submodules in
$F_{\Psi}\langle\binom{\Psi\smallsetminus J}{s}\rangle$ for $s>0$, so
$M^{\prime}$ is isomorphic to $F_{\Psi\smallsetminus J}$, again by Corollary
3.12, and thus, $M$ is isomorphic to $F_{\Psi}$. As $F_{\Psi}$ is injective
(Proposition 5.5), Lemma 5.6 shows that $F_{\Psi}$ is an injective cogenerator
of $\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
∎

######  Corollary 5.8.

Let $F|k$, $\Psi$ and $K\subseteq F_{\Psi}$ be as in Theorem 5.7. Then any
smooth $K$-semilinear irreducible representation of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ can be embedded into $F_{\Psi}$.

###### Proof.

For any smooth simple
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $V$ the
$F_{\Psi}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module
$V\otimes_{K}F_{\Psi}$ admits a simple quotient isomorphic, by Theorem 5.7, to
$F_{\Psi}$. This means that $V$ can be embedded into $F_{\Psi}$. ∎

As a consequence, we get a description of level 1 ‘rank 1’ objects of
$\mathop{\mathrm{Sm}}\nolimits_{F_{\Psi}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$:

######  Corollary 5.9.

Let $\Psi$ be an infinite set, $F$ be a field and $k\subsetneqq F$ be a
subfield algebraically closed in $F$. Then there are natural bijections

  1. (1)

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

  2. (2)

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

###### Proof.

By Theorem 5.7, this follows from Lemma 4.8. ∎

######  Corollary 5.10.

The indecomposable objects $K_{\mathrm{a}}\langle\binom{\Psi}{s}\rangle$ of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
are injective for all integer $s\geq 0$.

###### Proof.

This is Lemma 4.13, since by Theorem 5.7 and Lemmas 5.2 and 4.11 (1),
$K_{\mathrm{a}}$ and $K_{\mathrm{a}}(\Psi)$ are injective objects of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
∎

####  5.2.1. Invariant subfields of $F_{\Psi}$ and one-dimensional groups in
characteristic 0

Let $G$ be a group, $\widetilde{K}$ be a smooth $G$-field of characteristic
$p\geq 0$, $L\subset\widetilde{K}$ be a $G$-invariant subfield. Set
$k:=\widetilde{K}^{G}$. Denote by
$\mathrm{Der}(\widetilde{K}|L)=\mathop{\mathrm{Hom}}\nolimits_{\widetilde{K}}(\Omega_{\widetilde{K}|L},\widetilde{K})$
the Lie algebra of derivations of $\widetilde{K}$ over $L$, and by
$\mathbb{D}$ the Lie algebra
$\mathrm{Der}(\widetilde{K}|L)^{G}=\mathop{\mathrm{Hom}}\nolimits_{\widetilde{K}\langle
G\rangle}(\Omega_{\widetilde{K}|L},\widetilde{K})$ of $G$-equivariant
derivations of $\widetilde{K}$ over $L$ (so $\mathbb{D}$ is a left $k$-vector
space).

Consider the following squarewise commutative diagram of natural maps:

$\begin{array}[]{ccccc}\left\\{\begin{array}[]{c}\mbox{$G$-invariant}\\\
\mbox{subfields of $\widetilde{K}$}\\\ \mbox{containing
$L$}\end{array}\right\\}&\xrightarrow{\mathrm{Ann}^{G}_{\widetilde{K}}}&\left\\{\begin{array}[]{c}\mbox{restricted
Lie}\\\ \mbox{$k$-subalgebras of
$\mathbb{D}$}\end{array}\right\\}&\xrightarrow{\alpha}&\left\\{\begin{array}[]{c}\mbox{$G$-invariant
subfields of}\\\ \mbox{$\widetilde{K}$ containing $L\widetilde{K}^{p}$ and}\\\
\mbox{separably closed in $\widetilde{K}$}\end{array}\right\\}\\\
\|&\circlearrowleft&H^{0}(G,-)\uparrow\downarrow\hbox
to0.0pt{$\displaystyle\widetilde{K}\otimes_{k}(-)$\hss}\phantom{H^{0}(G,-)}&\circlearrowleft&\|\\\
\left\\{\begin{array}[]{c}\mbox{$G$-invariant}\\\ \mbox{subfields of
$\widetilde{K}$}\\\ \mbox{containing
$L$}\end{array}\right\\}&\xrightarrow{\mathrm{Ann}_{\widetilde{K}}}&\left\\{\begin{array}[]{c}\mbox{$G$-invariant}\\\
\mbox{restricted Lie $\widetilde{K}$-sub-}\\\ \mbox{algebras of
$\mathrm{Der}(\widetilde{K}|L)$}\end{array}\right\\}&\xrightarrow{\beta}&\left\\{\begin{array}[]{c}\mbox{$G$-invariant
subfields of}\\\ \mbox{$\widetilde{K}$ containing $L\widetilde{K}^{p}$ and}\\\
\mbox{separably closed in $\widetilde{K}$}\end{array}\right\\}\\\
\gamma\downarrow\hbox to0.0pt{$\displaystyle K\mapsto\widetilde{K}\mbox{-span
of }\Omega_{K|L}$\hss}\phantom{\gamma}&&\bigcap\\\
\left\\{\begin{array}[]{c}\mbox{$\widetilde{K}\langle G\rangle$-sub-}\\\
\mbox{modules of
$\Omega_{\widetilde{K}|L}$}\end{array}\right\\}&\xrightarrow{\mathrm{Ann_{\Omega}}}&\left\\{\begin{array}[]{c}\mbox{$\widetilde{K}\langle
G\rangle$-submodules}\\\ \mbox{of
$\mathrm{Der}(\widetilde{K}|L)$}\end{array}\right\\},\end{array}$

where $\mathrm{Ann}_{\widetilde{K}}(K):=\mathrm{Der}(\widetilde{K}|K)$;
$\mathrm{Ann}_{\Omega}(V):=\mathop{\mathrm{Hom}}\nolimits_{\widetilde{K}}(\Omega_{\widetilde{K}|L}/V,\widetilde{K})$;
$\beta(\mathfrak{a}):=\bigcap_{\xi\in\mathfrak{a}}\ker\xi$;
$\alpha(\Lambda):=\beta(\widetilde{K}\otimes_{k}\Lambda)=\bigcap_{\xi\in\Lambda}\ker\xi$.

Then $\beta$ is surjective and the composition
$\beta\circ\mathrm{Ann}_{\widetilde{K}}(-)$ is idempotent, so it is identical
on the target of $\beta$. The composition with
$\mathrm{Ann}_{\Omega}(-)\circ\gamma$ factors through
$\mathrm{Ann}_{\widetilde{K}}(-)$.

The restriction of $\mathrm{Ann}_{\Omega}(-)$ (resp. of
$\mathrm{Ann}_{\widetilde{K}}(-)$) to the subset of $\widetilde{K}\langle
G\rangle$-submodules of finite codimension (resp. of subfields of finite
transcendence codegree) factors through the subset of smooth finite-
dimensional $\widetilde{K}\langle G\rangle$-submodules of
$\mathrm{Der}(\widetilde{K}|L)$.

We are going to restrict our attention to the case of those $\widetilde{K}$
for which any smooth finite-dimensional $\widetilde{K}\langle
G\rangle$-modules is a direct sum of copies of $\widetilde{K}$. Then the
images under $\mathrm{Ann}_{\Omega}(-)$ (resp. under
$\mathrm{Ann}_{\widetilde{K}}(-)$) of $\widetilde{K}\langle
G\rangle$-submodules of finite codimension (resp. of subfields of finite
transcendence codegree) are contained in
$\widetilde{K}\otimes_{k}\mathbb{D}\subseteq\mathrm{Der}(\widetilde{K}|L)$.
This implies that the image of $\alpha$ contains all subfields of finite
transcendence codegree, and the composition $\alpha\circ
H^{0}(G,-)\circ\mathrm{Ann}_{\widetilde{K}}(-)$ is identical on such
subfields.

Recall (p.1.3), that each $x\in\Psi$ determines a field embedding
$(x):F\xrightarrow{\sim}F_{x}=\\{f(x)~{}|~{}f\in F\\}\subseteq F_{\Psi}$.

######  Theorem 5.11.

Let $\Psi$ be an infinite set, $F|k$ be a transcendence degree 1 field
extension of characteristic $0$ such that $k$ is algebraically closed in $F$,
$K\neq k$ be an $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant field
extension of $k$ in $F_{\Psi}$ of transcendence degree $d$ algebraically
closed in $F_{\Psi}$. Then $d\leq 3$. If $d=3$ then there exists a unique
$R\in\mathrm{PGL}_{2}(k)\backslash(F\smallsetminus k)$ such that
$K=k\left(\frac{(R(w)-R(x))(R(y)-R(z))}{(R(w)-R(z))(R(x)-R(y))}~{}|~{}w,x,y,z\in\Psi\right)$.
If $d=2$ then there exists a unique $R\in\mathbb{P}_{k}(F/k)$ such that
$K=k\left(\frac{R(u)-R(w)}{R(u)-R(v)}~{}|~{}u,v,w\in\Psi\right)$.

If $d=1$ then there is a system $(\pi_{ij}:W_{i}\to W_{j})_{ij}$ of isogenies
between torsors $W_{i}$ over absolutely irreducible one-dimensional algebraic
$k$-groups $E_{i}$ endowed with a compatible system of $k$-field embeddings
$\sigma_{i}:k(W_{i})\hookrightarrow F$ (i.e.
$\sigma_{i}\pi_{ij}^{*}=\sigma_{j}$ for all $i,j$) such that
$K=\bigcup_{i}(\overline{k}(W_{i})_{\overline{k},\Psi}^{E_{i}(\overline{k})})^{\mathrm{Gal}(\overline{k}|k)}\subset\bigcup_{i}k(W_{i})_{\Psi}\subseteq
F_{\Psi}$, where $E_{i}(\overline{k})$ acts on $\overline{k}(W_{i})_{\Psi}$
diagonally.

###### Proof.

Suppose that $K$ contains an element of $F_{\\{x_{1},\dots,x_{n}\\}}\setminus
k$. Then $d\leq n-1$. For each $x\in\Psi$, the elements of
$\mathrm{Der}(F_{\Psi})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}|\\{x\\}}$
preserve $F_{x}$, so the restrictions
$\mathrm{Der}(F_{\Psi}|F_{\Psi\smallsetminus\\{x\\}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x\\}}}\xrightarrow{\rho_{x}}\mathrm{Der}(F_{x}|k)$
and
$\mathrm{Der}(F_{\Psi}|k)^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}\xrightarrow{r_{x}}\mathrm{Der}(F_{x}|k)$
are well-defined. In fact, $\rho_{x}$ and $r_{x}$ are bijective. Denote by
$\eta\mapsto\eta_{x}$ the inverse of the composition of $\rho_{x}$ with the
isomorphism
$\mathrm{Der}(F_{x}|k)\xrightarrow[\sim]{\iota_{x}}\mathrm{Der}(F|k)$ (induced
by $(x):F\xrightarrow{\sim}F_{x}$). The composition $\iota_{x}\circ r_{x}$ is
independent of $x$, and thus, gives a natural Lie $k$-algebra isomorphism
$\mathrm{Der}(F|k)\xrightarrow{\sim}\mathrm{Der}(F_{\Psi}|k)^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$,
$\eta\mapsto\sum_{u\in\Psi}\eta_{u}$.

It follows from the discussion preceeding the proposition, that $K$ is the
common kernel of the elements of a finite-dimensional Lie $k$-subalgebra
$\Lambda$ of $\mathrm{Der}(F|k)$:
$K=\bigcap_{\eta\in\Lambda}\ker(\sum_{x\in\Psi}\eta_{x})$.

If $d=2$ then let $\eta_{1}$ be a non-zero element of $[\Lambda,\Lambda]$, and
$\eta_{2}\in\Lambda$ be such that $[\eta_{1},\eta_{2}]=\eta_{1}$. Then
$\eta_{1}(\eta_{2}/\eta_{1})=1$. Set $R:=\eta_{2}/\eta_{1}\in F\smallsetminus
k$, so $\eta_{1}=\frac{\mathrm{d}}{\mathrm{d}R}$ and
$\eta_{2}=R\frac{\mathrm{d}}{\mathrm{d}R}$. As $\eta_{1}$ is defined uniquely
up to a $k^{\times}$-multiple, and $\eta_{2}$ is defined uniquely modulo
$[\Lambda,\Lambda]$, the class of $R$ in $\mathbb{P}_{k}(F/k)$ is well-
defined. Then $K$ is contained in the common kernel of
$\sum_{x\in\Psi}\frac{\partial}{\partial R(x)}$ and
$\sum_{x\in\Psi}R(x)\frac{\partial}{\partial R(x)}$, which means that it
contains the intersection of the subfield generated over $k$ by $R(x)-R(y)$
for all $x,y\in\Psi$ and the subfield generated over $k$ by $R(x)/R(y)$ for
all $x,y\in\Psi$.

Fix a rank one valuation $v:F^{\times}/k^{\times}\to\mathbb{Q}$ of $F$ (e.g.
by embedding of $F\subseteq F\otimes_{k}\overline{k}$ into the field of
Puiseux series
$\overline{k}((X^{\mathbb{Q}})):=\mathop{\underrightarrow{\lim}}\limits\overline{k}((X^{1/N}))$).
Let us show that if $\Lambda$ is a $d$-dimensional Lie $k$-subalgebra of
$k((X^{\mathbb{Q}}))$ then $d\leq 3$. Let
$S=\\{m_{1},\dots,m_{d}\\}\subset\mathbb{Q}$ be the set of values of $v$ on
$(\frac{\mathrm{d}}{\mathrm{d}X})^{-1}\Lambda\smallsetminus\\{0\\}$, and
$m_{1}<\cdots<m_{d}$. Choose a basis
$(Q_{1}\frac{\mathrm{d}}{\mathrm{d}X},\dots,Q_{d}\frac{\mathrm{d}}{\mathrm{d}X})$
of $\Lambda$ such that $Q_{i}=X^{m_{i}}(1+\dots)$. Then
$[Q_{i}\frac{\mathrm{d}}{\mathrm{d}X},Q_{j}\frac{\mathrm{d}}{\mathrm{d}X}]=(m_{j}-m_{i})X^{m_{i}+m_{j}-1}(1+\dots)\frac{\mathrm{d}}{\mathrm{d}X}$,
so $m_{i}+m_{j}-1\in S$ for all $i<j$. As $m_{1}+m_{2}-1\geq m_{1}$, one has
$m_{2}\geq 1$. Then $m_{d}>1$ (if $d\geq 3$), so $m_{d-1}+m_{d}-1>m_{d-1}$,
and thus, $m_{d-1}+m_{d}-1=m_{d}$, which implies $m_{d-1}=1$. This means that
$d=3$ (and $m_{2}=1$, so $m_{1}=1-m$ and $m_{3}=m+1$ for some $m>0$). Then
$Q_{2}\frac{\mathrm{d}}{\mathrm{d}X}$ and
$Q_{3}\frac{\mathrm{d}}{\mathrm{d}X}$ generate a two-dimensional Lie
subalgebra $\Lambda^{\prime}\subset\Lambda$.

If $d=3$ then rescale $\eta_{2}\in\Lambda^{\prime}$ so that
$[\eta_{3},\eta_{2}]=\eta_{3}$. Then, as we have seen in the case $d=2$,
$\eta_{2}=R\frac{\mathrm{d}}{\mathrm{d}R}$ and
$\eta_{3}=\frac{\mathrm{d}}{\mathrm{d}R}$ for some $R\in F\smallsetminus k$.
On the other hand, $[\eta_{1},\eta_{3}]=a\eta_{2}+b\eta_{3}$ for some $a,b\in
k$, i.e.
$-\frac{\mathrm{d}}{\mathrm{d}R}(\eta_{1}/\frac{\mathrm{d}}{\mathrm{d}R})\frac{\mathrm{d}}{\mathrm{d}R}=aR\frac{\mathrm{d}}{\mathrm{d}R}+b\frac{\mathrm{d}}{\mathrm{d}R}$,
or
$-\mathrm{d}(\eta_{1}/\frac{\mathrm{d}}{\mathrm{d}R})=\mathrm{d}(aR^{2}/2+bR)$,
and thus, $\eta_{1}=-(aR^{2}/2+bR+c)\frac{\mathrm{d}}{\mathrm{d}R}$ for some
$c\in k$, so $\Lambda/\Lambda^{\prime}$ is spanned by
$\frac{\mathrm{d}}{\mathrm{d}R^{-1}}$. Then $K$ is the common kernel of
$\sum_{x\in\Psi}\frac{\partial}{\partial R(x)}$,
$\sum_{x\in\Psi}R(x)\frac{\partial}{\partial R(x)}$ and
$\sum_{x\in\Psi}R(x)^{2}\frac{\partial}{\partial R(x)}$, which means that it
is the intersection of the subfields
$k\left(R(x)-R(y)~{}|~{}x,y\in\Psi\right)$,
$k\left(R(x)/R(y)~{}|~{}x,y\in\Psi\right)$, and
$k\left(1/R(x)-1/R(y)~{}|~{}x,y\in\Psi\right)$.

If $d=1$, we employ the Lefschetz principle, i.e. reduce the problem to the
case of a countable $k$ and embed $k$ into $\mathbb{C}$. Let
$0\neq\eta\in\mathrm{Der}(F|k)$ be such that $\sum_{x\in\Psi}\eta_{x}$
annihilates $K$. For each $f\in K$, there is a smooth projective curve
$\overline{C}$ over $k$ with an embedding $k(\overline{C})\hookrightarrow F$
such that $f\in k(\overline{C})_{\Psi}$ and
$\eta\in\mathrm{Der}(k(\overline{C})|k)$. Let $C\subseteq\overline{C}$ be the
complement to the set of poles of the 1-form $1/\eta$. If considered locally
in the analytic topology on $C$, $1/\eta=du$ for an analytic function
$u(x)=\int_{*}^{x}1/\eta$. Then $f=\varphi(u_{i}-v~{}|~{}i\in S)$ for a finite
subset $S\subset\Psi$, a fixed $v\in\Psi$, and an analytic function $\varphi$.
As the path from $\ast$ to $x$ varies, the values of the branches of $u(x)$
are modified by the integral linear combinations of the periods of the 1-form
$1/\eta$. As $f$ is non-constant, there are 3 options for the periods: (1)
they are zero; (2) they form a cyclic group $\mathbb{Z}\cdot\omega$ for some
$\omega\in\mathbb{C}^{\times}$; (3) they form a lattice $L$ in $\mathbb{C}$.
In the case (1) $u(x)$ is rational and
$f\in\mathbb{C}(u(x)-u(y)~{}|~{}x,y\in\Psi)$. In the case (2)
$Q(x):=\exp(\frac{2\pi i}{\omega}u(x))$ is rational and
$f\in\mathbb{C}(Q(x)/Q(y)~{}|~{}x,y\in\Psi)$. In the case (3), let
$\wp_{L}(z):=\frac{1}{z^{2}}+\sum_{\lambda\in
L\smallsetminus\\{0\\}}\left(\frac{1}{(z-\lambda)^{2}}-\frac{1}{\lambda^{2}}\right)$
be the Weierstraß $\wp$-function, so
$\wp^{\prime}_{L}(z)^{2}=4\wp_{L}(z)^{3}+2a\wp_{L}(z)+b$ for some
$a,b\in\mathbb{C}$. Then $g(x):=\wp_{L}(u(x))$,
$h(x):=\wp^{\prime}_{L}(u(x))$,
$\wp_{L}(u(x)-u(y))=\left[\frac{1}{2}\frac{h(x)+h(y)}{g(x)-g(y)}\right]^{2}-g(x)-g(y)$
and
$\wp^{\prime}_{L}(u(x)-u(y))=\frac{1}{2}(h(x)+h(y))\left[\frac{(g(x)-g(y))(6g(x)^{2}+a)-(h(x)+h(y))h(x)}{(g(x)-g(y))^{3}}\right]-h(x)$,
are rational and
$f\in\mathbb{C}(\wp_{L}(u(x)-u(y)),\wp^{\prime}_{L}(u(x)-u(y))~{}|~{}x,y\in\Psi)$.
∎

###### Lemma 5.2 and Corollary 5.10 list all indecomposable injectives.

Let $K$ be $K_{\mathrm{a}}$. Any indecomposable injective object is an
injective envelope of a non-zero cyclic subobject, so we only have to show
that any smooth finitely generated
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $V$ can be
embedded into a direct sum of $x^{\lambda}K$ for some
$\lambda\in\mathbb{Z}\langle S\rangle$ and of $K\langle\binom{\Psi}{s}\rangle$
for several integer $s\geq 1$.

By Proposition 4.6, there is a subset $\Psi^{\prime}\subset\Psi$ with finite
complement $J$ and an isomorphism of
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-modules
$\bigoplus_{s=0}^{N}K\langle\binom{\Psi^{\prime}}{s}\rangle^{\kappa_{s}}\xrightarrow{\sim}V$
for some integer $N,\kappa_{0},\dots,\kappa_{N}\geq 0$. In particular,
$V^{\prime}:=\mathop{\underrightarrow{\lim}}\limits_{I\subset\Psi^{\prime}}V^{\mathop{\mathfrak{S}}\nolimits_{\Psi|I}}$,
where $I$ runs over the finite subsets of $\Psi^{\prime}$, can be embedded
into
$\bigoplus_{s=0}^{N}K\langle\binom{\Psi^{\prime}}{s}\rangle^{\kappa_{s}}$.

By Lemma 4.4, it suffices to show that the
$K^{\prime}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-module
$K\langle\binom{\Psi^{\prime}}{s}\rangle$ is isomorphic to a direct sum of
copies of modules
$x^{\lambda}K^{\prime},K^{\prime}\langle\Psi^{\prime}\rangle,K^{\prime}\langle\binom{\Psi^{\prime}}{2}\rangle,\dots$.
(Obviously,
$K_{\mathrm{a}}^{\prime}=F_{\Psi^{\prime}}(u^{\gamma},u_{s}/v_{s}~{}|~{}\gamma\in\Gamma,~{}s\in
S,~{}u,v\in\Psi^{\prime})$.)

As
$K\langle\binom{\Psi^{\prime}}{s}\rangle=K\otimes_{K^{\prime}}K^{\prime}\langle\binom{\Psi^{\prime}}{s}\rangle$,
this can be reduced to the case $s=0$ as follows. Assuming that $K$ is a
direct sum of the required type, we only have to (i) note that
$K^{\prime}\langle\binom{\Psi^{\prime}}{s}\rangle\to
x^{\lambda}K^{\prime}\otimes_{K^{\prime}}K^{\prime}\langle\binom{\Psi^{\prime}}{s}\rangle$,
$[J]\mapsto(\sum_{t\in J}t)^{\lambda}\cdot[J]$, is an isomorphism for any
$s\geq 1$, and (ii) to check that
$K^{\prime}\langle\binom{\Psi^{\prime}}{n}\times\binom{\Psi^{\prime}}{s}\rangle\cong\bigoplus_{j=0}^{n+s}K^{\prime}\langle\binom{\Psi^{\prime}}{j}\rangle^{\oplus
N_{j}}$. It is clear that
$K^{\prime}\langle\binom{\Psi^{\prime}}{n}\times\binom{\Psi^{\prime}}{s}\rangle\xrightarrow{\sim}\bigoplus_{j=0}^{\min(n,s)}K^{\prime}\langle\binom{\Psi^{\prime}}{j,n-j,s-j}\rangle$,
$[(I,J)]\mapsto[(I\cap J,I\smallsetminus I\cap J,J\smallsetminus I\cap J)]$,
where $\binom{\Psi^{\prime}}{j,n-j,s-j}$ denotes the set of triples of
disjoint subsets of $\Psi^{\prime}$ of orders $j,n-j,s-j$. According to
Example 3.16, $K^{\prime}\langle\binom{\Psi^{\prime}}{j,n-j,s-j}\rangle$ is
isomorphic to a direct sum of copies of
$K^{\prime}\langle\binom{\Psi^{\prime}}{n+s-j}\rangle$.

Now we treat the case of $s=0$. By Lemma 4.13, the objects
$x^{\lambda}K^{\prime},K^{\prime}\langle\Psi^{\prime}\rangle,K^{\prime}\langle\binom{\Psi^{\prime}}{2}\rangle,\dots$
are injective, while by Corollary 4.15, the category
$\mathop{\mathrm{Sm}}\nolimits_{K^{\prime}}(\mathop{\mathfrak{S}}\nolimits_{\Psi|J})$
is locally noetherian, so all direct sums of their copies are injective as
well. This means that we can split off any direct sum of isomorphic copies of
the above objects from $K$. It is thus sufficient to show that the
$K^{\prime}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-submodule in
$K\langle\binom{\Psi^{\prime}}{s}\rangle$ generated by any element $\alpha$ is
contained in a submodule isomorphic to a finite direct sum
$\bigoplus_{\lambda}(x^{\lambda}K^{\prime})^{m_{\lambda}}\oplus\bigoplus_{i\geq
1}K^{\prime}\langle\binom{\Psi^{\prime}}{s_{i}}\rangle^{n_{i}}$.

As $F_{\Psi}$ is the fraction field of $F_{\Psi^{\prime}}\otimes_{k}F_{J}$, in
the case $K=F_{\Psi}$ it suffices to show that, for any field extension $L|k$
in which $k$ is the algebraically closed, any element $\alpha$ in the fraction
field of $F_{\Psi^{\prime}}\otimes_{k}L$ is contained in a
$K^{\prime}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi|J}\rangle$-submodule
isomorphic to a finite direct sum $\bigoplus_{i\geq
0}K^{\prime}\langle\binom{\Psi^{\prime}}{s_{i}}\rangle^{n_{i}}$.

For any group $G$ and a $G$-field $M$, let $M(t)|M$ be a simple (purely)
transcendental $G$-field extensions such that the line $tM$ is $G$-invariant.
Consider the partial fraction decomposition

(2)
$M(t)=\bigoplus_{n\in\mathbb{Z}}t^{n}M\oplus\bigoplus_{m=1}^{\infty}\bigoplus_{O}\bigoplus_{j=0}^{\deg
O-1}\left(t^{j}\bigoplus_{P\in O}P(t)^{-m}M\right),$

where $O$ runs over the $G$-orbits of (non-constant) irreducible polynomials
$P$ over $M$ such that $P(0)=1$. Here the $M\langle G\rangle$-module
$\bigoplus_{P\in O}P(t)^{-m}M$ is isomorphic to $M\langle
G/\mathrm{Stab}_{P}\rangle$.

We proceed by induction on minimal order $d$ of a subset $T\subset L$ such
that $\alpha$ is algebraic over the fraction field of
$F_{\Psi^{\prime}}\otimes_{k}k(T)$, the case $d=0$ being trivial. We may
assume that $L|k(T)$ is finite. Then, in
$\mathop{\mathrm{Sm}}\nolimits_{K^{\prime}}(\mathop{\mathfrak{S}}\nolimits_{\Psi|J})$,
the fraction field of $F_{\Psi^{\prime}}\otimes_{k}L$ is isomorphic to
$F_{\Psi^{\prime}}(T)^{[L:k(T)]}$. Fix some $t\in T$. Taking
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}}$ and
$M=F_{\Psi^{\prime}}(T\smallsetminus\\{t\\})$ in decomposition (2), we see
that $F_{\Psi^{\prime}}(T)=M(t)$ is a direct sum of objects isomorphic to
$M\langle G/U\rangle$ for open subgroups $U$ of $G$. By induction assumption,
$F_{\Psi^{\prime}}(T\smallsetminus\\{t\\})$ is already of required type, while
$\bigoplus_{P\in O}F_{\Psi^{\prime}}(T\smallsetminus\\{t\\})\cdot P(t)^{-m}$
is isomorphic to a direct sum of objects
$K^{\prime}\langle\binom{\Psi^{\prime}}{s_{i}}\rangle\otimes\mathbb{Z}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}}/\mathrm{Stab}_{P}\rangle$
for some open subgroups
$\mathrm{Stab}_{P}\subseteq\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}}$.
Applying Lemmas 3.7 and 3.15 and the above isomorphism
$K^{\prime}\langle\binom{\Psi^{\prime}}{n}\times\binom{\Psi^{\prime}}{s}\rangle\cong\bigoplus_{j=0}^{n+s}K^{\prime}\langle\binom{\Psi^{\prime}}{j}\rangle^{\oplus
N_{j}}$, we see that the latter object is of required type. This completes the
induction step.

In the case $K=K_{\mathrm{a}}$, $K$ is contained in $F_{\Psi}(\Psi\times S)$
which is isomorphic in
$\mathop{\mathrm{Sm}}\nolimits_{F_{\Psi^{\prime}}(\Psi^{\prime}\times
S)}(\mathop{\mathfrak{S}}\nolimits_{\Psi|J})$ to a direct sum of copies of
$F_{\Psi^{\prime}}(\Psi^{\prime}\times
S)\langle\binom{\Psi^{\prime}}{s}\rangle=F_{\Psi^{\prime}}(\Psi^{\prime}\times
S)\otimes_{K^{\prime}}K^{\prime}\langle\binom{\Psi^{\prime}}{s}\rangle$ for
$s\geq 0$. To show that $F_{\Psi^{\prime}}(\Psi^{\prime}\times S)$ is
isomorphic in
$\mathop{\mathrm{Sm}}\nolimits_{K^{\prime}}(\mathop{\mathfrak{S}}\nolimits_{\Psi|J})$
to a direct sum of copies of injective objects
$K^{\prime}\langle\binom{\Psi^{\prime}}{s}\rangle$ for $s\geq 1$ and
$x^{\lambda}K^{\prime}$, it suffices to check that any element of
$F_{\Psi^{\prime}}(\Psi^{\prime}\times S)$ belongs to a direct sum of
subobjects of such type. We may, thus, assume $S$ to be finite, and proceed by
induction on $\\#S$. As in the proof of Lemma 5.2, after a base change of
$\mathbb{Z}\langle S\rangle$, we may assume that $\Gamma$ is generated by the
elements $m(i)[i]$ for all $i\in S$. As $F_{\Psi^{\prime}}(\Psi^{\prime}\times
S)=K^{\prime}(x_{i}~{}|~{}i\in S)$ for any $x\in\Psi^{\prime}$, we again use
decomposition (2) with $M=K_{\mathrm{b},t}$ for some $t\in S$ and
$G=\mathop{\mathfrak{S}}\nolimits_{\Psi^{\prime}}$ to see that
$F_{\Psi^{\prime}}(\Psi^{\prime}\times S)$ is a direct sum of objects
isomorphic to $x_{t}^{d}K_{\mathrm{b},t}$ for all $d\in\mathbb{Z}$ or
$K_{\mathrm{b},t}\langle G/U\rangle$ for open subgroups $U$ of $G$. Applying
arguments similar to the above ones, we conclude that
$F_{\Psi^{\prime}}(\Psi^{\prime}\times
S)\langle\binom{\Psi^{\prime}}{s}\rangle$ admits a decomposition of desired
type. ∎

###  5.3. Proof of Theorem 1.3 in the cases $K=K_{\mathrm{b}}$ and
$K=K_{\mathrm{c}}$

For a field $k$ of characteristic $p$, consider the $k$-vector space
$\mathbb{D}$ of formal series $a_{0}+\sum_{i\geq 1}a_{i}D^{(i)}$, $a_{i}\in
k$, with multiplication $D^{(i)}D^{(j)}=\binom{i+j}{i}D^{(i+j)}$ for all
integer $i,j\geq 1$. This $k$-algebra is $k[[D^{(1)}]]$ if $p=0$. This
$k$-algebra is
$k[[D^{(1)},D^{(p)},D^{(p^{2})},\dots]]/((D^{(1)})^{p},(D^{(p)})^{p},(D^{(p^{2})})^{p},\dots)$
if $p>0$.

By E.Lucas’ theorem, if $n=\sum_{i\geq 0}n_{i}p^{i}$ and $m=\sum_{i\geq
0}m_{i}p^{i}$ for some $0\leq n_{i},m_{i}<p$ then
$\binom{n}{m}\equiv\prod_{i}\binom{n_{i}}{m_{i}}\pmod{p}$. The set
$\mathbb{N}$ is partially ordered by $n\succeq_{p}m$ if $p\nmid\binom{n}{m}$,
i.e. $m_{i}\leq n_{i}$ for all $i$.

######  Proposition 5.12.

The socle series $\Phi_{\bullet}$ of
$K_{\mathrm{c}}\left[\frac{x}{x-y}\right]$ is exhausting,
$\Phi_{0}=K_{\mathrm{c}}$ and $\Phi_{s}$ is the $K_{\mathrm{c}}$-linear
envelope of the products of the $s$-tuples of elements of $\Phi_{1}$. The
socle series of the object $K_{\mathrm{b}}[x]$ of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
is exhausting as well, and coincides with
$K_{\mathrm{b}}\otimes_{K_{\mathrm{c}}}\Phi_{\bullet}$. Let $p$ be the
characteristic of $k$.

If $p=0$ then $\Phi_{s}$ is the space of the polynomials of degree $\leq s$,
in particular, the module $K_{\mathrm{c}}[\frac{x}{x-y}]$ is uniserial (its
submodules are totally ordered by inclusion); $K_{\mathrm{b}}[x]^{<s}$ admits
the injective resolution $0\to K_{\mathrm{b}}[x]^{<s}\to
K_{\mathrm{b}}[x]\xrightarrow{\mathrm{d}^{s}/\mathrm{d}^{s}x}K_{\mathrm{b}}[x]\to
0$, so $\mathrm{Ext}^{\geq
2}_{K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(-,V)=0$
for any $V$ of finite length.

If $p>0$ then $\Phi_{s}$ is the $K_{\mathrm{c}}$-linear envelope of
$\frac{x^{n}}{(x-y)^{n}}$ for $n\geq 0$, where sum of the digits of $n$ in
base $p$ is $\leq s$. For any $s>0$, there exist indecomposable modules of
length $s+1$ with the socle $K_{\mathrm{b}}^{s}$.

The $k$-algebra
$\mathop{\mathrm{End}}\nolimits_{K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K_{\mathrm{b}}[x])$
is isomorphic to $\mathbb{D}$, so it is local;
$\mathop{\mathrm{Hom}}\nolimits_{K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K_{\mathrm{b}}[x],K_{\mathrm{b}})=0$.

Each smooth indecomposable
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $V$
of length 2 is isomorphic to a unique submodule of $K_{\mathrm{b}}[x]$, which
corresponds to a point of
$\mathbb{P}_{k}((K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}})$.

###### Proof.

The $K_{\mathrm{c}}$-linear map

$\Phi_{s}/\Phi_{s-1}\to\begin{cases}\bigoplus_{n:~{}\text{sum of digits of $n$
is $s$}}(x-y)^{-n}K_{\mathrm{c}},&\mbox{if $p>0$,}\\\
(x-y)^{-s}K_{\mathrm{c}},&\mbox{if
$p=0$,}\end{cases}\quad\left(\frac{x}{x-y}\right)^{n}\mapsto(x-y)^{-n},$

is an isomorphism, since

$\sigma\left(\frac{x}{x-y}\right)^{n}=\sigma\left(\left(\frac{x}{x-y}\right)^{p^{i_{1}}+\cdots+p^{i_{s}}}\right)=\prod_{j=1}^{s}\left(\left(\frac{x-y}{x^{\sigma}-y^{\sigma}}\right)^{p^{i_{j}}}\left(\frac{x}{x-y}\right)^{p^{i_{j}}}+\left(\frac{x^{\sigma}-x}{x^{\sigma}-y^{\sigma}}\right)^{p^{i_{j}}}\right)\\\
\equiv\left(\frac{x-y}{x^{\sigma}-y^{\sigma}}\right)^{n}\left(\frac{x}{x-y}\right)^{n}\bmod{\Phi_{s-1}\cap
K_{\mathrm{d}}\left[\frac{x}{x-y}\right]^{\leq n-p^{i_{1}}}}$

for any $n=p^{i_{1}}+\cdots+p^{i_{s}}$, $i_{1}\leq i_{2}\leq\cdots\leq i_{s}$,
and $\sigma\in\mathop{\mathfrak{S}}\nolimits_{\Psi}$. In particular,
$\Phi_{s}/\Phi_{s-1}$ and
$K_{\mathrm{b}}\otimes_{K_{\mathrm{c}}}(\Phi_{s}/\Phi_{s-1})$ are semisimple.
It remains to check that $\Phi_{s}/\Phi_{s-1}$ (resp.
$K_{\mathrm{b}}\otimes_{K_{\mathrm{c}}}(\Phi_{s}/\Phi_{s-1})$) is the socle of
$K_{\mathrm{c}}\left[\frac{x}{x-y}\right]/\Phi_{s-1}$ (resp. of
$K_{\mathrm{b}}[x]/K_{\mathrm{b}}\otimes_{K_{\mathrm{c}}}\Phi_{s-1}$). We
proceed by induction on $s\geq 0$ (with $\Phi_{-1}:=0$), the case $s=0$ being
evident.

The operators $D^{(i)}:K_{\mathrm{b}}[x]\to K_{\mathrm{b}}[x]$,
$x^{n}\mapsto\binom{n}{i}x^{n-i}$, commute with the
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action for all $i\geq 1$, so $D^{(i)}$
induce
$(K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}\to
K_{\mathrm{b}}[x]^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=k$. If $p>0$ and
$n=mp^{i}$ with $m>1$ prime to $p$ then
$D^{(p^{i})}x^{n}=\binom{n}{p^{i}}x^{n-p^{i}}=(-1)^{p-1}mx^{n-p^{i}}\neq 0$,
and therefore,
$(K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}\subseteq\\{\sum_{i=0}^{n}a_{i}x^{p^{i}}~{}|~{}a_{i}\in
K_{\mathrm{b}}\\}$. If
$Q=\sum_{i=0}^{n}a_{i}x^{p^{i}}\in(K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$
then
$Q^{\sigma}\equiv\sum_{i=0}^{n}a_{i}^{\sigma}x^{p^{i}}\pmod{K_{\mathrm{b}}}$
for any $\sigma\in\mathop{\mathfrak{S}}\nolimits_{\Psi}$, so $a_{i}\in
K_{\mathrm{b}}^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=k$. If $p=0$ then
$(K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=K_{\mathrm{b}}\cdot
x$. This identifies $\Phi_{1}$.

Let us check that the operators $D^{(i)}$ map $\Phi_{s}$ to $\Phi_{s-1}$ for
all $i\geq 1$. This is trivial if $p=0$, so assume that $p>0$. By E.Lucas’
theorem, $p\nmid\binom{n}{i}$ if and only if $n\succeq_{p}i$, so if
$x^{n}\in\Phi_{s}$ then either $D^{(i)}x^{n}=0$ or $n\succeq_{p}i$ and
$x^{n-i}\in\Phi_{s-1}$. Therefore, all $D^{(i)}$ induce
$K_{\mathrm{b}}[x]/\Phi_{s}\to K_{\mathrm{b}}[x]/\Phi_{s-1}$ and
$(K_{\mathrm{b}}[x]/\Phi_{s})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}\to(K_{\mathrm{b}}[x]/\Phi_{s-1})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=(\Phi_{s}/\Phi_{s-1})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$.
Moreover, if $x^{n}\notin\Phi_{s-1}$ and $n\succeq_{p}p^{j}$ then
$x^{n-p^{j}}\notin\Phi_{s-2}$. This shows that if an element does not belong
to $\Phi_{s+1}$ then some $D^{(i)}$ maps it outside of $\Phi_{s}$, so it is
not fixed by $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ modulo $\Phi_{s}$, thus
completing the induction step.

An endomorphism $\varphi$ of $K_{\mathrm{b}}[x]$ sends the element $x^{n}\in
K_{\mathrm{b}}[x]^{\mathop{\mathfrak{S}}\nolimits_{\Psi|x}}$ to some
$P_{n}^{x}(x)\in
K_{\mathrm{b}}[x]^{\mathop{\mathfrak{S}}\nolimits_{\Psi|x}}=F_{x}[x]$. Then
$y^{n}=\sum_{i=0}^{n}\binom{n}{i}x^{i}(y-x)^{n-i}$ is sent to
$\sum_{i=0}^{n}\binom{n}{i}(y-x)^{n-i}P_{i}^{x}(x)$, so
$P_{n}^{y}(y)=\sum_{i=0}^{n}\binom{n}{i}(y-x)^{n-i}P_{i}^{x}(x)$. Comparing
the coefficients of the polynomials, we see that $P_{n}^{y}(y)\in k[y]$. As
the left hand side is independent of $x$, we get
$P_{n}(y)=\sum_{i=0}^{n}\binom{n}{i}P_{i}(0)y^{n-i}$, and thus, the map
$\varphi$ is given by $\sum_{i\geq 0}a_{i}D^{(i)}$. If $p=0$ then
$D^{(i)}=D^{i}/i!$, so
$\mathop{\mathrm{End}}\nolimits_{K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(K_{\mathrm{b}}[x])=k[[D]]$.
If $p>0$ then $D^{(i)}:x^{n}\mapsto(-1)^{i}x^{n-i}$ for any
$n\equiv-1\pmod{p^{m}}$ with $p^{m}>i$, so $D^{(i)}$ are (topologically)
linearly independent.

Given a smooth indecomposable
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module $V$
of length 2, an isomorphism of the socle of $V$ onto $K_{\mathrm{b}}\subset
K_{\mathrm{b}}[x]$ can be extended to a morphism to $K_{\mathrm{b}}[x]$, which
is injective, since the socle is essential. To check uniqueness, it suffices
to show that the submodules corresponding to distinct points of
$\mathbb{P}_{k}((K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}})$
are non-isomorphic. Indeed, as $K_{\mathrm{b}}[x]$ is injective, any
isomorphism between submodules of $K_{\mathrm{b}}[x]$ extends to an
endomorphism of $K_{\mathrm{b}}[x]$, which is scalar on
$(K_{\mathrm{b}}[x]/K_{\mathrm{b}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$.

For any $s\geq 0$, the dual of
$\\{a+b_{0}x+\cdots+b_{s}x^{p^{s}}~{}|~{}a,b_{0}\dots,b_{s}\in
K_{\mathrm{b}}\\}$ is
$\\{(a_{0}+bx,\dots,a_{s}+bx^{p^{s}})~{}|~{}a_{0},\dots,a_{s},b\in
K_{\mathrm{b}}\\}$ (where the pairing is
$\langle(a_{0}+bx,\dots,a_{s}+bx^{p^{s}}),a+b_{0}x+\cdots+b_{s}x^{p^{s}}\rangle=ab-
a_{0}b_{0}-\cdots-a_{s}b_{s}$; this pairing is equivariant). It is
indecomposable, since so is the former. ∎

######  Theorem 5.13.

In the notation of Theorem 1.3, the indecomposable injectives of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
are injective hulls $P_{s}^{(K_{\mathrm{b}})}$ of
$K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$ for all integer $s\geq 0$, where
$P_{0}^{(K_{\mathrm{b}})}=K_{\mathrm{b}}[x]$,
$P_{s}^{(K_{\mathrm{b}})}=K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$ if
$s\neq 0,2$,
$P_{2}^{(K_{\mathrm{b}})}=K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$
if $F=k$ and $p=2$ $($and
$\mathop{\mathrm{End}}\nolimits_{K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(P_{2}^{(K_{\mathrm{b}})})\cong
k[X]/(X^{2}))$,
$P_{2}^{(K_{\mathrm{b}})}=K_{\mathrm{b}}\langle\binom{\Psi}{2}\rangle$
otherwise.

###### Proof.

By Theorem 5.7, $F(X)_{\Psi}$ is an injective cogenerator of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
For any $x\in\Psi$, one has
$F(X)_{\Psi}=K_{\mathrm{b}}(x)=K_{\mathrm{b}}[x]\oplus\bigoplus_{R}\bigoplus_{m\geq
1}V^{(m)}_{R}$, where $R$ runs over the
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-orbits of non-constant irreducible
monic polynomials in $K_{\mathrm{b}}[x]$ and $V^{(m)}_{R}$ is the
$K_{\mathrm{b}}$-linear envelope of $P(x)/Q^{m}$ for all $Q\in R$ and $P\in
K_{\mathrm{b}}[x]$ with $\deg P<\deg Q$. As $F(X)_{\Psi}$ is injective, its
direct summand $K_{\mathrm{b}}[x]$ is also injective, as well as $V^{(m)}_{R}$
are for all $R$ and $m$.

Each $V^{(m)}_{R}$ is filtered by $V^{(j,m)}_{R}$, $0\leq j<\deg R$, where
$V^{(j,m)}_{R}$ is the $K_{\mathrm{b}}$-linear envelope of $P(x)/Q^{m}$ for
all $Q\in R$ and $P\in K_{\mathrm{b}}[x]$ with $\deg P\leq j$. Clearly, these
decomposition and filtration are independent of $x$. Moreover, for all $0\leq
j<\deg R$ and $m$ the map $V^{(j,1)}_{R}\to V^{(j,m)}_{R}$, $P(x)/Q\mapsto
P(x)/Q^{m}$, is an isomorphism of
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules.

We separate the case where $F=k$ and the characteristic $p$ of $k$ is 2. We
denote this case by $\star$.

Let us show that,

  * •

for any $R$ and $0\leq j<\deg R$, the
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module
$V^{(j,1)}_{R}$ is non-canonically isomorphic to
$(K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathrm{Stab}_{Q}\rangle)^{j+1}$,
where $Q\in R$ is arbitrary, but in the $\star$ case we exclude those $R$ with
$\mathrm{Stab}_{Q}=\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x,y\\}}$ for some
$x\neq y$ in $\Psi$;

  * •

in the $\star$ case, if
$\mathrm{Stab}_{Q}=\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x,y\\}}$ for some
$R$, $Q\in R$ and $x\neq y$ in $\Psi$, and $0\leq j<\frac{1}{2}\deg R$, the
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module
$V^{(2j+1,1)}_{R}$ is non-canonically isomorphic to
$(K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle)^{j+1}$.

(In particular, by Lemma 3.13, there are no simple
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodules
in $V^{(1)}_{R}$).

We proceed by induction on $0\leq j<\deg R$, the case $j=0$ being obvious,
since the morphism
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathrm{Stab}_{Q}\rangle\to
V^{(0,1)}_{R}$, $[g]\mapsto(gQ)^{-1}$, is an isomorphism.

Let $R$ contain $Q=x_{1}+\sum_{i=2}^{s}(x_{i}-x_{1})^{2}$ for some $s\geq 1$
and pairwise distinct $x_{1},\dots,x_{s}\in\Psi$. Then (i) $\mathrm{Stab}_{Q}$
is of index $s$ in
$\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x_{1},\dots,x_{s}\\}}$; (ii)
$V_{R}^{(1)}\cong K_{\mathrm{b}}\langle\Psi\rangle$ if $s=1$; (iii)
$V_{R}^{(1)}\cong
K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$ if $s=2$. By
Lemma 3.15, this implies that $V^{(1)}_{R}$ is isomorphic to
$K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle^{s}$, unless $F=k$, $p=2$ and
$s=2$. According to Example 3.16, this implies that
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/U\rangle$ is
injective for any proper open subgroup
$U\subset\mathop{\mathfrak{S}}\nolimits_{\Psi}$, with the exception of
$U=\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x,y\\}}$ for $x\neq y$ in $\Psi$ in
the $\star$ case.

Now $R$ is arbitrary. By the induction hypothesis, $V^{(j-1,m)}_{R}$ is
isomorphic to
$(K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathrm{Stab}_{Q}\rangle)^{j}$,
and thus, injective. Then the inclusion $V^{(j-1,1)}_{R}\hookrightarrow
V^{(j,1)}_{R}$ splits: $V^{(j,1)}_{R}\cong
V^{(j-1,1)}_{R}\oplus(V^{(j,1)}_{R}/V^{(j-1,1)}_{R})$. Then the canonical
isomorphisms
$x^{j}\cdot:V^{(0,m)}_{R}\xrightarrow{\sim}V^{(j,1)}_{R}/V^{(j-1,1)}_{R}$
complete the induction step.

Let $V\subseteq K_{\mathrm{b}}[x]$ be a non-zero
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodule.
Let $Q\in V$ be a monic polynomial in $x$. Then $V$ contains $Q-\sigma Q$ for
any $\sigma\in\mathop{\mathfrak{S}}\nolimits_{\Psi}$. If $\sigma Q\neq Q$ for
some $\sigma\in\mathop{\mathfrak{S}}\nolimits_{\Psi}$ then $Q-\sigma Q$ is a
nonzero polynomial of degree $<\deg Q$. In particular, if $Q$ is of minimal
degree then $Q-\sigma Q=0$ for any
$\sigma\in\mathop{\mathfrak{S}}\nolimits_{\Psi}$, which means that $Q\in k$,
i.e. $V$ contains $K_{\mathrm{b}}$. This implies that $K_{\mathrm{b}}[x]$ is
an injective hull of $K_{\mathrm{b}}$, in particular, all
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-submodules
of $K_{\mathrm{b}}[x]$ are indecomposable.

We know that any simple object is isomorphic to $K_{\mathrm{b}}$, so by Lemma
5.6, the injective hull $K_{\mathrm{b}}[x]$ of $K_{\mathrm{b}}$ is a
cogenerator.

The remaining indecomposable injectives are described in a way similar to the
case of $K_{\mathrm{a}}$. Namely, we have to decompose
$F_{\Psi}(\Psi)\langle\binom{\Psi}{s}\rangle=F_{\Psi}(\Psi)\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$
in
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
for all $s\geq 1$. The summands of type
$K_{\mathrm{b}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/U\rangle$ in
$F_{\Psi}(\Psi)$ give the same type of summands in
$F_{\Psi}(\Psi)\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$.
Now, if $K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$ is injective then, by
the above inductive argument,
$K_{\mathrm{b}}[x]^{<n}\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle\cong
K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle^{\oplus n}$, so
$K_{\mathrm{b}}[x]\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle\cong\bigoplus_{n\geq
0}K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$. Finally, if $p=2$ then
$K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle\to
K_{\mathrm{b}}[x]^{<2}\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{2}\rangle$,
$(x,y)\mapsto(x\\{x,y\\},y\\{x,y\\})$, is an isomorphism, so reasoning by
induction we see that
$K_{\mathrm{b}}[x]^{<2n}\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{2}\rangle$
is injective for any $n\geq 1$ and
$K_{\mathrm{b}}[x]^{<2n}\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{2}\rangle\cong
K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle^{\oplus n}$,
and thus,
$K_{\mathrm{b}}[x]\otimes_{K_{\mathrm{b}}}K_{\mathrm{b}}\langle\binom{\Psi}{2}\rangle\cong\bigoplus_{n\geq
0}K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$. ∎

######  Proposition 5.14.

Fix some $x\neq y$ in $\Psi$. In the notation of Theorem 1.3, any simple
object of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{c}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
is isomorphic to $(x-y)^{n}K_{\mathrm{c}}\subset F_{\Psi}(\Psi)$ for some
$n\in\mathbb{Z}$; the object
$M^{(n)}:=(x-y)^{n}K_{\mathrm{c}}\left[\frac{x}{x-y}\right]\subset
F_{\Psi}(\Psi)$ is an injective hull of $(x-y)^{n}K_{\mathrm{c}}$. The socle
series $\Phi^{(n)}_{\bullet}$ of $M^{(n)}$ coincides with
$(x-y)^{n}\Phi_{\bullet}$ of Proposition 5.12. In particular, it is
exhausting.

###### Proof.

Let $\mathop{\mathfrak{S}}\nolimits_{\Psi}$ act on $K_{\mathrm{c}}[X]$ as it
does on $K_{\mathrm{c}}[x-y]$, i.e. by
$\sigma:\sum_{i}a_{i}X^{i}\mapsto\sum_{i}a_{i}^{\sigma}\left(\frac{x^{\sigma}-y^{\sigma}}{x-y}\right)^{i}X^{i}$.
Then
$K_{\mathrm{b}}=K_{\mathrm{c}}(x-y)=\bigoplus_{n\in\mathbb{Z}}(x-y)^{n}K_{\mathrm{c}}\oplus\bigoplus_{R}\bigoplus_{m\geq
1}V_{R}^{(m)}$, where $R$ runs over the set of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-orbits of non-constant irreducible
polynomials $Q(X)\in K_{\mathrm{c}}[X]$ with $Q(0)=1$ and $V_{R}^{(m)}$ is the
$K_{\mathrm{c}}$-vector subspace of $K_{\mathrm{b}}$ spanned by
$(x-y)^{j}/Q(x-y)^{m}$ for all $0\leq j<\deg R$ and all $Q\in R$, so

$K_{\mathrm{b}}[x]=K_{\mathrm{b}}\otimes_{K_{\mathrm{c}}}K_{\mathrm{c}}\left[\frac{x}{x-y}\right]=\bigoplus_{n\in\mathbb{Z}}M^{(n)}\oplus\bigoplus_{R}\bigoplus_{m\geq
1}V_{R}^{(m)}\otimes_{K_{\mathrm{c}}}K_{\mathrm{c}}\left[\frac{x}{x-y}\right].$

In particular, $M^{(n)}$ and $V_{R}^{(1)}$ are injective for all
$n\in\mathbb{Z}$ and $R$. Note that for all $m$ the map $V^{(1)}_{R}\to
V^{(m)}_{R}$, $P(x-y)/Q(x-y)\mapsto P(x)/Q(x-y)^{m}$, is an isomorphism of
$K_{\mathrm{c}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules.

Any simple object $V$ of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{c}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
can be embedded into a simple object of
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{b}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
i.e. into $K_{\mathrm{b}}$. As $K_{\mathrm{b}}$ admits an
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant discrete valuation trivial
on $K_{\mathrm{c}}$ and positive on $x-y$, $K_{\mathrm{b}}$ embeds
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-equivariantly into
$K_{\mathrm{c}}((x-y))=\mathop{\underrightarrow{\lim}}\limits_{m}\prod\limits_{n\geq-m}(x-y)^{n}K_{\mathrm{c}}\subset\prod\limits_{n\in\mathbb{Z}}(x-y)^{n}K_{\mathrm{c}}$,
so $V$ is isomorphic to $(x-y)^{n}K_{\mathrm{c}}$ for some $n$.

The natural map
$\mathop{\mathrm{Hom}}\nolimits_{K_{\mathrm{c}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(M,M^{\prime})\to\mathop{\mathrm{Hom}}\nolimits_{K_{\mathrm{c}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}(M\otimes_{K_{\mathrm{c}}}\mathcal{L},M^{\prime}\otimes_{K_{\mathrm{c}}}\mathcal{L})$
is bijective for any invertible $\mathcal{L}$ in
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{c}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
so
$\mathop{\mathrm{Hom}}\nolimits_{K_{\mathrm{c}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle}((x-y)^{m}K_{\mathrm{c}},M^{(n)})\cong(M^{(n-m)})^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=\begin{cases}k,&\mbox{if
$n=m$,}\\\ 0,&\mbox{if $n\neq m$.}\end{cases}$ This means that the socle of
$M^{(n)}$ is $(x-y)^{n}K_{\mathrm{c}}$, so $M^{(n)}$ is an injective hull of
$(x-y)^{n}K_{\mathrm{c}}$. As $M^{(n)}$ is artinian, the socle series
$\Phi^{(n)}_{\bullet}$ of $M^{(n)}$ is exhausting.

Let $R$ contain
$Q(x_{1}-x_{2})=(x_{1}-x_{2})\prod_{i=3}^{s}\frac{x_{s}-x_{1}}{x_{s}-x_{2}}+1$
for some $s\geq 2$ and pairwise distinct $x_{1},\dots,x_{s}\in\Psi$. Then (i)
$V_{R}^{(1)}\cong
K_{\mathrm{c}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathrm{Stab}_{Q}\rangle$;
(ii) $\mathrm{Stab}_{Q}$ is of index $s(s-1)$ in
$\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x_{1},\dots,x_{s}\\}}$ for all $s\geq
2$, with the exception of $s=2,3$ when $p=2$, where $\mathrm{Stab}_{Q}$ is of
index $2s-3$ in
$\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x_{1},\dots,x_{s}\\}}$. By Lemma
3.15, if $U$ is an open subgroup in
$\mathop{\mathfrak{S}}\nolimits_{\Psi,\\{x_{1},\dots,x_{s}\\}}$ of a finite
index $\kappa$ for some $s\geq 3$ then
$K_{\mathrm{c}}\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/U\rangle$ is
isomorphic to $K_{\mathrm{c}}\langle\binom{\Psi}{s}\rangle^{\oplus\kappa}$ if
$s\geq 3$; $K_{\mathrm{c}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle\to
K_{\mathrm{c}}\langle\binom{\Psi}{2}\rangle\oplus(x-y)K_{\mathrm{c}}\langle\binom{\Psi}{2}\rangle$,
$(x,y)\mapsto(\\{x,y\\},(x-y)\\{x,y\\})$, is an isomorphism for $p\neq 2$. In
particular, $V^{(1)}_{R}\cong
K_{\mathrm{c}}\langle\binom{\Psi}{s}\rangle^{\oplus\kappa_{s}}$ if $s\geq 3$.

As $K_{\mathrm{b}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$ and
$K_{\mathrm{b}}\langle\binom{\Psi}{s}\rangle$ are injective for $s\neq 0,2$
(it suffices to use this fact for $s=1$), the objects
$K_{\mathrm{c}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle\otimes_{K_{\mathrm{c}}}K_{\mathrm{c}}\langle\binom{\Psi}{s}\rangle$
and
$K_{\mathrm{c}}\langle\binom{\Psi}{t}\rangle\otimes_{K_{\mathrm{c}}}K_{\mathrm{c}}\langle\binom{\Psi}{s}\rangle$
are injective for $t\geq 3$ and $s\neq 0,2$. The
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-set $\binom{\Psi}{t}\times\Psi$
contains a $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-subset isomorphic to
$\mathop{\mathfrak{S}}\nolimits_{\Psi}/U$ for a subgroup $U$ of finite index
in $\mathop{\mathfrak{S}}\nolimits_{\Psi|\\{x_{1},\dots,x_{t}\\}}$, so
$P_{t}^{(K_{\mathrm{c}})}:=K_{\mathrm{c}}\langle\binom{\Psi}{t}\rangle$ are
injective for all $t\geq 3$. Similarly, the
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-set
$\Psi^{2}\smallsetminus\Delta_{\Psi}\times\Psi$ contains a
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-subset isomorphic to
$\Psi^{2}\smallsetminus\Delta_{\Psi}$, so
$K_{\mathrm{c}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$ (which is
$P_{2}^{(K_{\mathrm{c}})}$ if $p=2$,
$P_{2}^{(K_{\mathrm{c}})}\oplus{P_{2}^{\prime}}^{(K_{\mathrm{c}})}$ if $p\neq
2$) is injective. The remaining summands $(x-y)^{n}K_{\mathrm{c}}$ of
$K_{\mathrm{b}}$ give the injective objects
$P_{1,n}^{(K_{\mathrm{c}})}:=(x-y)^{n}K_{\mathrm{c}}\langle\Psi\rangle$ in
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{c}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
Clearly, the $P_{s}^{(K_{\mathrm{b}})}$ for $s\geq 1$ are direct sums of such
objects in
$\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{c}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

There remains to decompose
$V:=V_{R}^{(1)}\otimes_{K_{\mathrm{c}}}K_{\mathrm{c}}\left[\frac{x}{x-y}\right]$.
The object $K_{\mathrm{c}}\left[\frac{x}{x-y}\right]$ is filtered with the
successive quotients $(x-y)^{n}K_{\mathrm{c}}$ with $n\geq 0$. If
$V_{R}^{(1)}\cong K_{\mathrm{c}}\langle\binom{\Psi}{s}\rangle^{\oplus\kappa}$
with $s\geq 3$ then $V$ is isomorphic to a direct sum of copies of
$K_{\mathrm{c}}\langle\binom{\Psi}{s}\rangle$. Similarly, if $V_{R}^{(1)}$ is
isomorphic to (a direct summand of)
$K_{\mathrm{c}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$ then $V$ is
isomorphic to a direct sum of copies of (direct summands of)
$K_{\mathrm{c}}\langle\Psi^{2}\smallsetminus\Delta_{\Psi}\rangle$. ∎

###  5.4. Topology of the Gabriel spectra

Given a Grothendieck category $\mathcal{A}$, any proper closed subset of the
Gabriel spectrum $\mathrm{Spec}(\mathcal{A})$ of $\mathcal{A}$ is an
intersection of sets
$[X]^{c}:=\\{P~{}|~{}\mathop{\mathrm{Hom}}\nolimits(X,P)\neq 0\\}$ for a
collection of compact objects $X$.

######  Lemma 5.15.

  1. (1)

Let $X,Y$ be compact objects of $\mathcal{A}$ such that there is either an
epimorphism $X\to Y$ or a monomorphism $Y\to X$. Then
$[Y]^{c}\subseteq[X]^{c}$.

  2. (2)

If $\mathcal{A}$ is locally noetherian then the closure $\overline{\\{P\\}}$
of a point $P$ consists of the points $P^{\prime}$ such that $P$ admits a
monomorphism into a product of injectives presenting $P^{\prime}$.

  3. (3)

Let $T$ be a collection of compact generators of $\mathcal{A}$, and $S$ be a
subset in $\mathrm{Spec}(\mathcal{A})$ such that for any $W\in T$ there exists
$P\in S$ with $\mathop{\mathrm{Hom}}\nolimits(W,P)=0$. Then $S$ is dense.

  4. (4)

For any simple object of $\mathcal{A}$, the isomorphism class of its injective
hull is a closed point.

###### Proof.

Let $S$ be a non-dense subset. As the closure $\overline{S}$ of $S$ is the
intersection of all proper closed subsets containing $S$, one has

$\overline{S}=\bigcap_{X:\text{ compact,
$\mathop{\mathrm{Hom}}\nolimits(X,P)\neq 0$ for all $P\in
S$}}[X]^{c}=\bigcap_{X\subset\prod_{P\in S}P:\text{ compact,
$\mathrm{pr}_{P}(X)\neq 0$ for all $P\in S$}}[X]^{c}.$

If $S=\\{P\\}$ consists of a single point $P$ then we get
$\overline{\\{P\\}}=\bigcap_{0\neq X\subset P:\text{ compact}}[X]^{c}$. If
$P^{\prime}\in\overline{\\{P\\}}$ then for any non-zero subobject $X\subset P$
there is a non-zero morphism $X\to P^{\prime}$, so extending it to $P$ we see
that $P$ admits a monomorphism into a product of copies of $P^{\prime}$.
Conversely, if $P$ admits an embedding into a product of copies of
$P^{\prime}$ then any non-zero subobject $X\subset P$ admits a non-zero
morphism to $P^{\prime}$.

If $P$ is simple then $[P]^{c}$ is the class of $E(P)$, i.e. the class of
$E(P)$ is a closed point. ∎

######  Lemma 5.16.

Let $K$ be a smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field, and
$\mathrm{Spec}_{K}=\mathrm{Spec}(\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi}))$.

  1. (1)

Any proper closed subset in $\mathrm{Spec}_{K}$ consists of injective hulls of
objects of bounded level.

In particular, any infinite subset of $\\{P_{1}^{(K)},P_{2}^{(K)},\dots\\}$ is
dense in $\mathrm{Spec}_{K}$, where $P_{i}^{(K)}$ is the isomorphism class of
$E(K\langle\binom{\Psi}{i}\rangle)$.

  2. (2)

Let $Q=E(Y)$ be a point of $\mathrm{Spec}_{K}$ for an object $Y$ of level
$\leq n$. Then the closure of $Q$ does not contain points $P=E(X)$ such that
$N_{n}X=0$. In particular, $\overline{\\{P_{i}\\}}\not\ni P_{j}$ for $j>i$.

  3. (3)

One has
$\overline{\\{P_{n}^{(K_{?})}\\}}\supseteq\\{P_{0}^{(K_{?})},P_{1}^{(K_{?})},\dots,P_{n}^{(K_{?})}\\}$
for any $n\geq 0$ with the following exceptions, where $F=k:$
$?=\mathrm{a,b,c}$, $n=1$, and $\Gamma=0$ if $?=\mathrm{a};$ $?=\mathrm{d}$
and $n=2$.

  4. (4)

The points $P_{1}^{(K_{\mathrm{a}})}$ and $P_{1,m}^{(K_{\mathrm{c}})}$ for all
$m\in\mathbb{Z}$ are closed.

  5. (5)

For $?\in\\{\mathrm{a},\mathrm{c}\\}$, the closure of
$E(\mathrm{Pic}_{K_{?}})$ contains neither $P_{s}$ for $s>1$, nor
$P^{\prime}_{2}$ if $?=\mathrm{c}$ and $p\neq 2$, nor $P_{1,n}$ for
$n\in\mathbb{Z}$ if $?=\mathrm{c}$.

###### Proof.

For any compact object $X$ of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
there is an integer $n\geq 0$ such that $X=N_{n}X$. Then
$\mathop{\mathrm{Hom}}\nolimits(X,E(Y))=0$ for all $Y$ such that $N_{n}Y=0$,
in particular, $\mathop{\mathrm{Hom}}\nolimits(X,P_{j})=0$ for all $j>n$, i.e.
$[X]^{c}$ contains only finitely many $P_{s}$’s. This implies that any closed
set containing infinitely many $P_{s}$’s is not proper, i.e. coincides with
$\mathrm{Spec}_{K}$.

To describe closures of points, we apply Lemmas 5.15 (2) and 3.9 to
$U=\mathop{\mathfrak{S}}\nolimits_{\Psi,I}$,
$V=K\langle\binom{\Psi}{s}\rangle$ for $s<\\#I$, and
$V\xrightarrow{\lambda}K$,
$\sum_{J\in\binom{\Psi}{s}}a_{J}[J]\mapsto\sum_{J\in\binom{\Psi}{s}}a_{J}$. As
the subgroups $\mathop{\mathfrak{S}}\nolimits_{\Psi,I}$ are maximal, we have
only to check that $K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,I}}\neq
K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$ and that there is an element $A\in
K_{I}\langle\binom{I}{s}\rangle^{\mathop{\mathfrak{S}}\nolimits_{I}}$ with
$\lambda(A)=1$. The $\mathop{\mathfrak{S}}\nolimits_{I}$-action on
$K_{?}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|I}}$ is non-trivial, so
$K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,I}}\neq
K^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$, while in the exceptional cases
$K_{?}^{\mathop{\mathfrak{S}}\nolimits_{\Psi,I}}=k=K_{?}^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$.
Then the composition of
$K^{\mathop{\mathfrak{S}}\nolimits_{\Psi,I,J}}=K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I,J}}\xrightarrow{\sim}V^{U}$,
$a\mapsto
A_{a}:=\sum_{g\in\mathop{\mathfrak{S}}\nolimits_{I}/\mathop{\mathfrak{S}}\nolimits_{I,J}}a^{g}[gJ]\in
V^{U}$, with $\lambda$ is the trace
$\mathrm{tr}_{K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I,J}}|K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I}}}:K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I,J}}\to
K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I}}$. As the field extension
$K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I,J}}|K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I}}$
is separable, there is an element $a$ with
$\mathrm{tr}_{K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I,J}}|K_{I}^{\mathop{\mathfrak{S}}\nolimits_{I}}}(a)=1$,
and then $\lambda(A_{a})=1$.

As the $k$-vector space
$\mathop{\mathrm{Hom}}\nolimits((x-y)^{m}K_{?}\langle\Psi\rangle,(x-y)^{n}E(K_{?}))=\mathop{\mathrm{Hom}}\nolimits(K_{?}\langle\Psi\rangle,(x-y)^{n-m}E(K_{?}))$
is at most one-dimensional and $E(K_{?})$ is injective,
$\mathop{\mathrm{Hom}}\nolimits(\langle(x-y)^{2m-n}(y^{n-m}[x]-x^{n-m}[y])\rangle,E(K_{?}))=0$,

The $k$-vector space
$\mathop{\mathrm{Hom}}\nolimits((x-y)^{m}K_{\mathrm{c}}\langle\Psi\rangle,(x-y)^{n}E(K_{\mathrm{c}}))=\mathop{\mathrm{Hom}}\nolimits(K_{\mathrm{c}}\langle\Psi\rangle,(x-y)^{n-m}E(K_{\mathrm{c}}))$
is one-dimensional if $F=k$ and $n\geq m$; it vanishes if $F=k$ and $n<m$. As
$E(K_{\mathrm{c}})$ is injective,
$\mathop{\mathrm{Hom}}\nolimits(\langle(x-y)^{2m-n}(y^{n-m}[x]-x^{n-m}[y])\rangle,E(K_{\mathrm{c}}))=0$,
i.e. the $\mathrm{Pic}_{K_{\mathrm{c}}}$-orbit of $P_{0}$ does not meet the
closure of $P_{1,m}$. The remaining points are not in the closure of $P_{1,m}$
for trivial reasons.

For any $m\in\mathbb{Z}$, $\mathop{\mathrm{Hom}}\nolimits(P_{1,m},P_{0,n})\neq
0$ for all $n\in\mathbb{Z}$, i.e. $P_{0,n}\in[P_{1,m}]^{c}$, while
$\mathop{\mathrm{Hom}}\nolimits(P_{1,m},P_{s})=0$ for all $s>1$, i.e.
$P_{s}\notin[P_{1,m}]^{c}$, meaning that the closure of
$\\{P_{0,n}~{}|~{}n\in\mathbb{Z}\\}\subset\mathrm{Cl}_{K_{?}}$ contains
neither $P_{s}$ for $s>1$.

As $\mathop{\mathrm{Hom}}\nolimits(P_{1},P^{\prime}_{2})=0$, the closure of
$\\{P_{0,n}~{}|~{}n\in\mathbb{Z}\\}\subset\mathrm{Cl}_{K_{\mathrm{c}}}$ does
not contain $P^{\prime}_{2}$ if $p\neq 2$.

As $\mathop{\mathrm{Hom}}\nolimits(P_{1,m},P_{1,n})=0$ for all $n\neq m$, the
closure of
$\\{P_{0,n}~{}|~{}n\in\mathbb{Z}\\}\subset\mathrm{Cl}_{K_{\mathrm{c}}}$
contains neither $P_{1,n}$ for $n\in\mathbb{Z}$. ∎

###  5.5. (Finite) $G$-extensions and finite-dimensional representations

######  Lemma 5.17.

Let $G$ be a group, $\widetilde{K}$ be a $G$-field, and
$K\subseteq\widetilde{K}$ be a $G$-invariant subfield. Set
$k:=\widetilde{K}^{G}$. Let $V\neq 0$ be a $K\langle G\rangle$-module of a
finite dimension $d$. Then (i)
$\dim_{k}\mathop{\mathrm{Hom}}\nolimits_{K\langle
G\rangle}(V,\widetilde{K})\leq d$; (ii) $V$ is simple and
$\mathop{\mathrm{End}}\nolimits_{K\langle G\rangle}(V)=k$ if $k\subseteq K$
and there is a $K\langle G\rangle$-module embedding $V^{\oplus
d}\hookrightarrow\widetilde{K}$. In particular, if $\mathcal{L}$ and
$\mathcal{L}^{\prime}$ are isomorphic invertible $K\langle
G\rangle$-submodules of $\widetilde{K}$ then
$\mathcal{L}=\mathcal{L}^{\prime}$ if $k\subseteq K$.

###### Proof.

The multiplication map
$\widetilde{K}\otimes_{k}(\widetilde{K}\otimes_{K}V^{\vee})^{G}\to\widetilde{K}\otimes_{K}V^{\vee}$
is injective, so $\dim_{k}\mathop{\mathrm{Hom}}\nolimits_{K\langle
G\rangle}(V,\widetilde{K})=\dim_{k}(\widetilde{K}\otimes_{K}V^{\vee})^{G}=\dim_{\widetilde{K}}(\widetilde{K}\otimes_{k}(\widetilde{K}\otimes_{K}V^{\vee})^{G})\leq\dim_{\widetilde{K}}(\widetilde{K}\otimes_{K}V^{\vee})=\dim_{K}V=:d$.

If $V^{\prime}\subset V$ is a non-zero submodule then any embedding $V^{\oplus
d}\hookrightarrow\widetilde{K}$ restricts to an embedding
${V^{\prime}}^{\oplus d}\hookrightarrow\widetilde{K}$, so $k^{\oplus
d}\subseteq(K^{G})^{\oplus d}\subseteq\mathop{\mathrm{End}}\nolimits_{K\langle
G\rangle}(V^{\prime})^{\oplus d}=\mathop{\mathrm{Hom}}\nolimits_{K\langle
G\rangle}(V^{\prime},{V^{\prime}}^{\oplus
d})\hookrightarrow\mathop{\mathrm{Hom}}\nolimits_{K\langle
G\rangle}(V^{\prime},\widetilde{K})$, and therefore, $V^{\prime}=V$ and
$\mathop{\mathrm{End}}\nolimits_{K\langle G\rangle}(V)=k$ by the above
argument. ∎

######  Example 5.18.

Fix some $x\in\Psi$. Let $k$ be a field, $n\geq 3$ be an integer,
$L:=k(x^{n},u/x~{}|~{}u\in\Psi\smallsetminus\\{x\\})$, $a\in k^{\times}$, and
$K:=L^{\langle\iota\rangle}$ be the
($\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant) fixed subfield of the
$k$-field involution $\iota:k(\Psi)\to k(\Psi)$, $u\mapsto a/u$ for all
$u\in\Psi$. As before, $\mathrm{Spec}_{K}$ is the Gabriel spectrum of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.
Then

  * •

the isomorphism classes of 2-dimensional simple objects are
$xL,\dots,x^{[(n-1)/2]}L$, they are pairwise distinct and injective;

  * •

the remaining simple objects are invertible, they are all injective if
$p:=\mathrm{char}(k)\neq 2$;

  * •

injective hulls $E(\mathcal{L})$ of invertible objects $\mathcal{L}$ are
2-dimensional and $E(\mathcal{L})/\mathcal{L}\cong\mathcal{L}$ if $p=2$;

  * •

$\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\cong\\{\pm
1\\}^{\oplus(\frac{3+(-1)^{n}}{2}-\delta_{2,p})}$, it fixes the non-closed
points of $\mathrm{Spec}_{K}$ and $[x^{n/4}L]$ if $4|n$;

  * •

the isomorphism classes of $x^{i}L$ for $1\leq i\leq[(n-1)/2]$, $i\neq n/4$,
is fixed by an order $2-\delta_{2,p}$ subgroup of
$\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$;

  * •

the closed points of $\mathrm{Spec}_{K}$ are presented by injective hulls of
the simple objects;

  * •

the remaining points of $\mathrm{Spec}_{K}$ are non-closed, presented by
$K\langle\binom{\Psi}{s}\rangle$ for all integer $s>0$.

###### Proof.

As $\iota$ commutes with the $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action,
$K$ is $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant. Obviously,
$k(\Psi)=\bigoplus_{i=0}^{n-1}x^{i}L$ is a decomposition into a direct sum of
subobjects. If $p\neq 2$ then $L$ splits into the sum of eigenspaces of
$\iota$ as $L=K\oplus(x^{n}-a^{n}x^{-n})K$. If $p=2$ then the sequence $0\to
K\to L\xrightarrow{\mathrm{tr}_{L|K}}K\to 0$ does not split by Lemma 5.17. The
involution $\iota$ induces isomorphisms
$\iota_{i}:x^{i}L\xrightarrow{\sim}x^{n-i}L$. By Lemma 5.17, this implies that
$x^{i}L$ is irreducible if $i\neq n-i$, and $x^{i}L$’s are pairwise non-
isomorphic for $1\leq i<n/2$. If $i=n-i$ then the $K$-eigenspaces
$\mathcal{L}_{\pm}:=(x^{n/2}\pm a^{n/2}x^{-n/2})K$ of $\iota_{i}$ are
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant. If $p=2$ then the sequence
$0\to\mathcal{L}_{+}\to
x^{n/2}L\xrightarrow{id_{\mathcal{L}_{+}}\otimes_{K}\mathrm{tr}_{L|K}}\mathcal{L}_{+}\to
0$ is not split, again by Lemma 5.17.

The multiplication induces an isomorphism
$(x^{n}-a^{n}x^{-n})K\otimes_{K}x^{i}L\xrightarrow{\sim}x^{i}L$ and an
isomorphism
$\mathcal{L}_{+}\otimes_{K}x^{n/4}L\xrightarrow[\sim]{\iota_{3n/4}}x^{n/4}L$
if $4|n$.

The indecomposable injectives in
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ are
direct summands of the indecomposable injectives in
$\mathop{\mathrm{Sm}}\nolimits_{k(\Psi)}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
so their list comes from the isomorphism
$K\langle\binom{\Psi}{s}\rangle^{\oplus
2n}\xrightarrow{\sim}k(\Psi)\otimes_{K}K\langle\binom{\Psi}{s}\rangle$ for
$s>0$. ∎

####  5.5.1. Torsion of the Picard group, and its trivialization

######  Lemma 5.19.

Let $G$ be a permutation group, $\widetilde{K}$ be a smooth $G$-field, and $H$
be a profinite automorphism group of the $G$-field $\widetilde{K}$. Set
$k:=\widetilde{K}^{G}$. Then there is a natural exact sequence
$0\to\mathrm{Pic}_{k}(H)\to\mathrm{Pic}_{\widetilde{K}^{H}}(G)\to\mathrm{Pic}_{\widetilde{K}}(G)^{H}$.

###### Proof.

By Proposition 2.2, $\mathrm{Pic}_{\widetilde{K}}(G\times
H)=\mathrm{Pic}_{K}(G)$, so the above exact sequence is the inflation-
restriction sequence $0\to H^{1}_{\text{{\rm cont}}}(H,k^{\times})\to
H^{1}_{\text{{\rm cont}}}(G\times H,\widetilde{K}^{\times})\to
H^{1}_{\text{{\rm cont}}}(G,\widetilde{K}^{\times})^{H}$. ∎

For an integer $n>0$, denote by ${}_{n}\mathrm{Pic}_{K}(G)$ the $n$-torsion
subgroup in $\mathrm{Pic}_{K}(G)$.

######  Proposition 5.20.

Let $G$ be a permutation group, $K$ be a smooth $G$-field and $n>0$ be an
integer. Set $k:=K^{G}$ and $\mu_{n}:=\\{z\in K^{\times}~{}|~{}z^{n}=1\\}$.
Then there is a natural exact sequence

$H^{1}_{\text{{\rm
cont}}}(G,\mu_{n})\to{}_{n}\mathrm{Pic}_{K}(G)\xrightarrow{\beta}(K^{\times}/K^{\times
n})^{G}/k^{\times}\xrightarrow{\xi}H^{2}_{\text{{\rm cont}}}(G,\mu_{n}).$

Assume that $G$ admits no open subgroups of finite index. Then any invertible
object $\mathcal{L}$ of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ of order $n$ is
contained in a $G$-field $K(a^{1/n})$ for some $a\in(K^{\times}/K^{\times
n})^{G}$.

###### Proof.

If $\mathcal{L}$ is an object of $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ with
$\mathcal{L}^{\otimes_{K}^{n}}\cong K$, choose non-zero $\pi\in\mathcal{L}$
and $\lambda\in(\mathcal{L}^{\otimes_{K}^{n}})^{G}$, and set
$\beta([\mathcal{L}]):=[\pi^{\otimes n}/\lambda]$. In terms of 1-cocycles,
$\beta((f_{\sigma}))=[g]$, where $g^{\sigma}/g=f_{\sigma}^{n}$ for all
$\sigma\in G$.

Define $\xi$ by $a\mapsto[(b_{\sigma}b_{\tau}^{\sigma}b_{\sigma\tau}^{-1})]$,
where $\sigma a/a=b_{\sigma}^{n}$ for some 1-cochain $(b_{\sigma})$ with
values in $K^{\times}$. Obviously, $\xi\beta=0$. If $\xi a=0$ then
$(b_{\sigma}\zeta_{\sigma})$ is a 1-cocycle for some 1-cochain
$(\zeta_{\sigma})$ with values in $\mu_{n}$, so $(b_{\sigma}\zeta_{\sigma})$
defines an element of $\mathrm{Pic}_{K}(G)$. Obviously, it is of order $n$.

If $\beta((f_{\sigma}))=0$ then there exists $g\in K^{\times}$ such that
$(g^{n})^{\sigma}/g^{n}=f_{\sigma}^{n}$ for all $\sigma\in G$, and therefore,
$(gf_{\sigma}/g^{\sigma})$ is a 1-cocycle with values in $\mu_{n}$.

Let $R=R_{\mathcal{L},\lambda}$ be the quotient of the symmetric $K$-algebra
of $\mathcal{L}$ by the ideal generated by $\lambda-1$. The natural map
$\bigoplus_{i=0}^{n-1}\mathcal{L}^{\otimes_{K}^{i}}\to R$ is a $K\langle
G\rangle$-module isomorphism, so the $K$-algebra $R$ is finite-dimensional
($\dim_{K}R=n$), and thus, it has only finitely many maximal ideals.

If $G$ admits no open subgroups of finite index then the $G$-action on the set
of maximal ideals is trivial, i.e. any maximal ideal is $G$-invariant. As
$\mathcal{L}^{\otimes_{K}^{i}}$ are pairwise non-isomorphic for $0\leq i<n$,
any $G$-invariant $K$-vector subspace in $R$ is a direct sum of several
$\mathcal{L}^{\otimes_{K}^{i}}$. As any non-empty direct sum of
$\mathcal{L}^{\otimes_{K}^{i}}$’s contains invertible elements, we conclude
that 0 is the only maximal ideal in $R$, i.e. $R$ is a field. Fix a non-zero
$\pi\in\mathcal{L}$ and set $a:=\pi^{n}/\lambda\in K^{\times}$, so $R\cong
K[\pi]/(\pi^{n}-a)$. As $\pi^{\sigma}/\pi\in K^{\times}$ for any $\sigma\in
G$, one has $(\pi^{\sigma}/\pi)^{n}=a^{\sigma}/a\in K^{\times n}$, so $a$ is
fixed by $G$ modulo $K^{\times n}$. ∎

Proposition 5.20 means that torsion in the Picard group is a particular case
of finite smooth $G$-field extensions $L|K$ with $L^{G}=K^{G}$.

######  Example 5.21.

The following $G$-fields $K$ admit no non-trivial finite smooth $G$-field
extension $L|K$ with $L^{G}=K^{G}$, so the Picard group is torsion free: (i)
any $K$ such that it is a cogenerator of
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, e.g. $K_{\mathrm{a}}$ with $\Xi=0$,
(ii) any $K$ of characteristic 0 such that all simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ are isomorphic, (iii) $K_{\mathrm{a}}$
with $\Gamma=0$, (iv) $F_{\Psi}(x-y~{}|~{}x,y\in\Psi)$.

###### Proof.

(i) If $L$ is a finite smooth $G$-extension of $K$ then $L$ is a trivial
$K$-semilinear representation of $G$, so $1=\dim_{K^{G}}L^{G}=[L:K]$.

(ii) If $L$ is a non-trivial finite smooth $G$-extension of $K$ and $[L:K]$ is
invertible in $K$ then $[L:K]^{-1}\mathrm{tr}_{L|K}$ splits the inclusion
$K\hookrightarrow L$, so the socle of $L$ contains a subobject isomorphic to
$K\oplus K$, contradicting $\dim_{K^{G}}L^{G}=1$.

(iii) If $L$ is a finite smooth
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-extension of $K_{\mathrm{a}}$ then
$L\cong\bigoplus x^{\lambda}K_{\mathrm{a}}$ is an integral ring with a
$K_{\mathrm{a}}$-semilinear action of $\mathop{\mathfrak{S}}\nolimits_{\Psi}$.
As $L\otimes_{K_{\mathrm{a}}}L\xrightarrow{\times}L$ is injective on each
$K_{\mathrm{a}}\cdot x^{\lambda}\otimes_{K_{\mathrm{a}}}K_{\mathrm{a}}\cdot
x^{\lambda}\cong K_{\mathrm{a}}\cdot x^{2\lambda}$, taking $\lambda$ with
maximal absolute value of $i$-th coordinate, we see that $L\cong
K_{\mathrm{a}}^{[L:K_{\mathrm{a}}]}$. By the argument of (i),
$L=K_{\mathrm{a}}$.

(iv) Let $K=F_{\Psi}(x-y~{}|~{}x,y\in\Psi)$. By Theorem 1.3 (• ‣ 4), $K[x]$ is
injective in $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$, so if $L$ is a finite
smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-extension of $K$ then the
inclusion $K\hookrightarrow K[x]$ extends to a morphism $\varphi:L\to K[x]$ in
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. As
$L^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=k$, $K$ is the only simple object
and $\ker\varphi$ is of finite length, the map $\varphi$ is injective. We may
thus assume that $L\subseteq K[x]^{<d}$ for a minimal $d\geq 1$. As $K[x]$ is
injective, any commutative $K$-algebra structure $L\otimes_{K}L\to L\subset
K[x]$ extends to a morphism $\xi:K[x]\otimes_{K}K[x]^{<d}\to K[x]$ in
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. One has
$K[x]\otimes_{K}K[x]^{<d}=\bigoplus_{i=0}^{d-1}(K[x]\otimes_{K}K)(1\otimes
x-x\otimes 1)^{i}\xrightarrow{\sim}K[x]^{\oplus d}$, $f\otimes
g=\sum_{i=0}^{d-1}(fD^{(i)}g\otimes 1)(1\otimes x-x\otimes
1)^{i}\mapsto(fg,fD^{(1)}g,fD^{(2)}g,\dots,fD^{(d-1)}g)$, in
$\mathop{\mathrm{Sm}}\nolimits_{K}(G)$. By Proposition 5.12, $\xi(f\otimes
g)=\sum_{i=0}^{\infty}\sum_{j=0}^{d-1}a_{ij}D^{(i)}f\cdot D^{(j)}g$ for some
$a_{ij}\in k$. As $\xi(1\otimes 1)\neq 0$, the element $a_{00}$ is non-zero,
and therefore, $\deg\xi(f\otimes g)=\deg f+\deg g$. If $\deg f$ is maximal on
$L$ then $\deg\xi(f\otimes f)=2\deg f\leq\deg f$, i.e. $\deg f=0$, and thus,
$L=K$. ∎

Consider a finite abelian group $A$ as a trivial $\mathfrak{S}_{\Psi}$-module.
By [4, Theorem 3.5 and Corollary 3.7], $H^{>0}_{\text{{\rm
cont}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi},A)=0$. For each integer $n>0$,
the sequences $0\to\mathbb{Z}\xrightarrow{\times
n}\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}\to 0$ and
$0\to\mathbb{Z}/n\to\mathbb{Q}/\mathbb{Z}\xrightarrow{\times
n}\mathbb{Q}/\mathbb{Z}\to 0$ show that
$H^{>0}_{\text{cont}}(\mathfrak{S}_{\Psi},\mathbb{Z})$ and
$H^{>0}_{\text{cont}}(\mathfrak{S}_{\Psi},\mathbb{Q}/\mathbb{Z})$ are uniquely
$n$-divisible, i.e., they are $\mathbb{Q}$-vector spaces. Obviously,
$H^{1}_{\text{cont}}(\mathfrak{S}_{\Psi},\mathbb{Z})=H^{1}_{\text{cont}}(\mathfrak{S}_{\Psi},\mathbb{Q}/\mathbb{Z})=0$.

####  5.5.2. Representations over fixed subfields of $k(\Psi)$

Let $G$ be a permutation group. The most straightforward source of finite-
dimensional semilinear representations of $G$ comes from the following
setting.

Let $\widetilde{K}$ be a smooth $G$-field, $H$ be a finite group of $G$-field
automorphisms of $\widetilde{K}$, and $K:=\widetilde{K}^{H}$.

By the primitive element theorem, there is an isomorphism of $K[H]$-modules
$K[H]\xrightarrow[\sim]{\alpha}\widetilde{K}$. In particular, any cyclic
$K[H]$-module is a quotient of $\widetilde{K}$. As the $K$-linear pairing
$\widetilde{K}\times\widetilde{K}\to K$,
$(x,y)\mapsto\mathrm{tr}_{\widetilde{K}|K}(xy)$, is $G\times H$-equivariant
and non-degenerate, $\widetilde{K}\in\mathop{\mathrm{Sm}}\nolimits_{K}(G\times
H)$ is self-dual. By the self-duality of $\widetilde{K}$, any simple
$K[H]$-module is contained in $\widetilde{K}$.

For each subfield $k\subseteq K$, let $\Pi_{k}=\Pi_{k}(H)$ be the set of
isomorphism classes of $k$-linear irreducible representations of $H$. Fix some
‘sufficiently big’ $k$, so that the coefficient extension to $K$ induces a
bijection $K\otimes_{k}(-):\Pi=\Pi_{k}\xrightarrow{\sim}\Pi_{K}$. Set
$V_{\rho}:=\mathop{\mathrm{Hom}}\nolimits_{k[H]}(\rho,\widetilde{K})$ for each
$k$-linear representation $\rho$ of $H$. This is a smooth $K$-semilinear
representation of $G$. Then $V_{\rho}\neq 0$ for any irreducible $\rho$.

Assume that the characteristic $p$ of $K$ does not divide $\\#H$. Then
$\widetilde{K}=\bigoplus_{\bar{\rho}\in\Pi}W_{\rho}$ is a decomposition into a
direct sum of objects of $\mathop{\mathrm{Sm}}\nolimits_{K}(G\times H)$, where
$D_{\rho}:=\mathop{\mathrm{End}}\nolimits_{k[H]}(\rho)$ and
$W_{\rho}:=V_{\rho}\otimes_{D_{\rho}}\rho$. Using the $H$-bimodule
decomposition
$k[H]=\bigoplus_{\bar{\rho}\in\Pi}\rho^{\vee}\otimes_{D_{\rho}}\rho$, we see
that $\dim_{K}V_{\rho}=\dim_{k}\rho$. As
$\mathop{\mathrm{Hom}}\nolimits_{K[H]}(W_{\rho}\otimes_{K}W_{\rho^{\prime}},K)\neq
0$ if and only if $\rho^{\prime}\cong\rho^{\vee}$ (i.e. $\rho$ and
$\rho^{\prime}$ are dual), $W_{\rho}$ and $W_{\rho^{\prime}}$ are dual if and
only if $\rho^{\prime}\cong\rho^{\vee}$. In particular, $V_{\rho}$ and
$V_{\rho^{\prime}}$ are dual in $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$ if and
only if $\rho^{\prime}\cong\rho^{\vee}$.

In the following examples, $\widetilde{K}=k(\Psi)$ and $H$ is a subgroup of
$\mathrm{PGL}_{2}(k)$ acting ‘diagonally’ on $k(\Psi)$. Such $H$ can be
cyclic, dihedral, $\mathfrak{A}_{4}$, $\mathfrak{S}_{4}$ or
$\mathfrak{A}_{5}$. Except for the dihedral groups, there is only one
conjugacy class for each of these groups (with no exceptions if
$k^{\times}=k^{\times 2}$), see [1, Theorem 4.2].

######  Remark 5.22.

The case of dihedral groups is covered by Example 5.18. Namely, if $k$
contains a primitive $n$-th root of unity $\zeta$ then each such group is
conjugated to $H_{a}=\langle\zeta,\iota_{a}\rangle\cong\mu_{n}\rtimes\mu_{2}$,
where $\zeta$ acts as $z\mapsto\zeta z$, and $\iota_{a}:z\mapsto a/z$ for some
$a\in k^{\times}$, so $k(\Psi)^{\langle\zeta\rangle}=L$ and
$K=k(\Psi)^{H_{a}}$.

######  Example 5.23.

Let $k$ be a field of characteristic $p\neq 2,3$, and
$L:=k(x^{2}+\frac{1}{x^{2}},\frac{u}{x}+\frac{x}{u},\frac{x^{2}-x^{-2}}{u/x-x/u}~{}|~{}u\in\Psi\smallsetminus\\{x\\})$
for some $x\in\Psi$. Then (i) the automorphism
$\theta:u\mapsto\frac{u+\sqrt{-1}}{u-\sqrt{-1}}$ of the field
$k(\sqrt{-1})(\Psi)$ over $k(\sqrt{-1})$ preserves $L$, (ii) the subfield
$K=K_{A4}$ of $L$ fixed by $\theta$ is
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant, (iii)

  * •

$(x-\frac{1}{x})L$ presents the unique isomorphism class of simple
3-dimensional objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,

  * •

if $\sqrt{-3}\in k$ then the remaining simple objects are invertible and
$\\#\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})=3$,

  * •

if $\sqrt{-3}\notin k$ then
$\frac{x^{12}+15x^{8}+15x^{4}+1}{x^{2}(x^{4}-1)^{2}}K\oplus\frac{x^{8}+14x^{4}+1}{(x^{4}-1)^{2}}K\subset
L$ presents the unique remaining isomorphism class of non-invertible simple
objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ and
$\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})=0$.

###### Proof.

One has $L(x^{2})=k(x^{2},u/x~{}|~{}u\in\Psi)$ and $[L(x^{2}):L]=2$ (since $L$
is fixed by $\iota:k(\Psi)\to k(\Psi)$, $u\mapsto 1/u$ for all $u\in\Psi$,
while $x^{2}$ is not), $[L(x):L(x^{2})]=2$ (since $L(x^{2})$ is fixed by
$u\mapsto-u$, while $x$ is not), $[L(x):L]=4$. This means that
$L=k(\Psi)^{\langle-z,\iota\rangle}$, and therefore, $L$ contains
$u^{2}+u^{-2}$, $u/v+v/u$, $(u^{2}-u^{-2})(u/v-v/u)$, $\frac{u\pm u^{-1}}{x\pm
x^{-1}}$ and $\frac{u^{2}-u^{-2}}{x^{2}-x^{-2}}$ for all $u,v\in\Psi$. In
particular, (i) the subfield $L\subset k(\Psi)$ is
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant, (ii)
$k(\Psi)=L\oplus(x-x^{-1})L\oplus(x+x^{-1})L\oplus(x^{2}-x^{-2})L$ is a
decomposition into a sum of $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant
$L$-vector subspaces, (iii) $\theta$ preserves $L$ even if $\sqrt{-1}\notin k$
(since the embeddings $\sigma\theta$ and $\theta\iota$ of $k(\Psi)$ into
$k(\sqrt{-1})(\Psi)$ coincide, where $\sigma$ is the generator of
$\mathrm{Gal}(L(\sqrt{-1})|L)$, while $\theta\iota$ and $\theta$ coincide on
$L$).

Assuming that $p\neq 3$, if $k$ does not contain a third root of unity
$\zeta\neq\zeta^{-1}$ then we adjoin it, so $K(\zeta)$,
$\left(\frac{(x^{2}-1)^{2}}{4x^{2}}-\frac{4\zeta
x^{2}}{(x^{2}+1)^{2}}-\frac{(x^{2}+1)^{2}}{\zeta(x^{2}-1)^{2}}\right)K(\zeta)$,
$\left(\frac{(x^{2}-1)^{2}}{4x^{2}}-\frac{4x^{2}}{\zeta(x^{2}+1)^{2}}-\frac{\zeta(x^{2}+1)^{2}}{(x^{2}-1)^{2}}\right)K(\zeta)$
are the ($\mathop{\mathfrak{S}}\nolimits_{\Psi}$-invariant) eigenspaces of
$\theta$ in the 3-dimensional $K(\zeta)$-vector space $L(\zeta)$. By Lemma
5.17, the isomorphisms
$(x-x^{-1})L\xrightarrow[\widetilde{\phantom{--}}]{\sqrt{-1}\theta}(x+x^{-1})L\xrightarrow[\widetilde{\phantom{--}}]{\theta}(x^{2}-x^{-2})L$,
$x-x^{-1}\mapsto-\frac{4}{x+x^{-1}}\mapsto-2\frac{x+x^{-1}}{x-x^{-1}}$, imply
that $(x-x^{-1})L$, $(x+x^{-1})L$, $(x^{2}-x^{-2})L$ are simple.

[Alternatively. If $(x-x^{-1})L(\zeta)$ (resp. $(x+x^{-1})L(\zeta)$, resp.
$(x^{2}-x^{-2})L(\zeta)$) is reducible then either itself or its dual contains
an invertible subobject. But this is impossible, since by Lemma 5.19,
$\\#\mathrm{Pic}_{K(\zeta)}(\mathop{\mathfrak{S}}\nolimits_{\Psi})=3$, and the
invertible subobjects in $k(\zeta)(\Psi)$ are multiplicity-free.] ∎

######  Example 5.24.

Let $k$ be a field, $H$ be a finite subgroup of
$\mathop{\mathrm{Aut}}\nolimits_{k[\mathop{\mathfrak{S}}\nolimits_{\Psi}]}(k(\Psi))\cong\mathrm{PGL}_{2}k$
and $K:=k(\Psi)^{H}$. Assume that the characteristic $p$ of $k$ does not
divide $\\#H$, and the irreducible representations of $H$ over $k$ are
absolutely irreducible. Then
$\rho\mapsto\mathop{\mathrm{Hom}}\nolimits_{k[H]}(\rho,k(\Psi))$ is a
dimension-preserving bijection between $\Pi_{k}(H)$ and the isomorphism
classes of simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

In particular,

  * •

if $p\neq 2,3$, $k$ contains $\sqrt{-1}$ and $H=H_{S4}\supset H_{A4}$ is
isomorphic to $\mathfrak{S}_{4}$ then there are 5 isomorphism classes of
simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$:
one of dimension 2 (fixed by
$\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\cong\\{\pm 1\\}$),
and two free $\mathrm{Pic}_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$-orbits
of classes of dimensions 1 and 3;

  * •

if $p\neq 2,3,5$, $k$ contains $\sqrt{-1}$ and $\sqrt{5}$, and
$H=H_{A5}\supset H_{A4}$ is isomorphic to $\mathfrak{A}_{5}$ then there are 5
isomorphism classes of simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$:
$K$, unique ones of dimensions 4 and 5, and two classes of dimension 3.

This follows from Lemma 5.17 and the classification of irreducible
representations of $\mathfrak{A}_{5}$.

###  5.6. Isotrivial finitely generated
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-extensions of $F_{\Psi}$

######  Proposition 5.25.

Let $F|k$ be a field extension with $k$ algebraically closed in $F$, and $Y$
be an absolutely irreducible $k$-variety, and
$\mathop{\mathfrak{S}}\nolimits_{\Psi}\xrightarrow{\tau\mapsto\tau^{\iota}}\mathop{\mathrm{Aut}}\nolimits_{\mathrm{field}}(K|k)$
be a smooth $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action on the field
$K:=F_{\Psi}(Y)$ extending the $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-action
on $F_{\Psi}$. Then $K^{\mathfrak{S}_{\Psi}}\cong k(Y^{\prime})$ for an
absolutely irreducible $k$-variety $Y^{\prime}$ such that $Y_{k^{\prime}}$ and
$Y^{\prime}_{k^{\prime}}$ are birational for a finite extension
$k^{\prime}|k$.

###### Proof.

As the pointwise stabilizer of $k(Y)$ is open, it contains
$\mathfrak{S}_{\Psi|J}$ for a finite subset $J\subset\Psi$. For each
$\tau\in\mathfrak{S}_{\Psi}$, $\tau^{\iota}(k(Y))$ is fixed by
$\mathfrak{S}_{\Psi|\tau(J)}$, so $\tau^{\iota}(k(Y))\subseteq
K^{\mathfrak{S}_{\Psi|\tau(J)}}\subseteq
K^{\mathfrak{S}_{\Psi|J\cup\tau(J)}}=F_{J\cup\tau(J)}(Y)$. If
$\eta\in\mathfrak{S}_{\Psi}$ is an involution such that
$J\cap\eta(J)=\varnothing$ then $\eta$ and $\mathfrak{S}_{\Psi|J}$ generate
$\mathfrak{S}_{\Psi}$. Let $F^{\prime}\subseteq F$ be a finitely generated
field extension of $k$ such that $\eta^{\iota}(k(Y))\subseteq
F^{\prime}_{J\cup\eta(J)}(Y)$. Then $\tau^{\iota}(k(Y))\subseteq
F_{J\cup\tau(J)}(Y)\cap F^{\prime}_{\Psi}(Y)=F^{\prime}_{J\cup\tau(J)}(Y)$ for
any $\tau\in\mathfrak{S}_{\Psi}$, so we may further assume that
$F=F^{\prime}$. As $F_{J\cup\tau(J)}\tau^{\iota}(k(Y))\subseteq
F_{J\cup\tau(J)}(Y)$ and
$F_{J\cup\tau^{-1}(J)}(\tau^{-1})^{\iota}(k(Y))\subseteq
F_{J\cup\tau^{-1}(J)}(Y)$ (or equivalently, $F_{J\cup\tau(J)}(Y)\subseteq
F_{J\cup\tau(J)}\tau^{\iota}(k(Y))$), we get
$F_{J\cup\tau(J)}\tau^{\iota}(k(Y))=F_{J\cup\tau(J)}(Y)$.

This gives rise to a field automorphism
$\alpha_{\tau}\in\mathop{\mathrm{Aut}}\nolimits(F_{J\cup\tau(J)}(Y)|F_{J\cup\tau(J)})$
extending the $F_{J\cup\tau(J)}$-algebra homomorphism
$F_{J\cup\tau(J)}\otimes_{k}k(Y)\xrightarrow{id\cdot\tau^{\iota}}F_{J\cup\tau(J)}(Y)$.

For each subset $I\subset\Psi$, the group
$\mathop{\mathrm{Aut}}\nolimits(F_{I}(Y)|F_{I})$ is embedded into
$\mathop{\mathrm{Aut}}\nolimits(F_{\Psi}(Y)|F_{\Psi})$, and
$\mathfrak{S}_{\Psi}$ is embedded into
$\mathop{\mathrm{Aut}}\nolimits(F_{\Psi}(Y)|k(Y))$ by the trivial action on
$k(Y)$. Then $\mathfrak{S}_{\Psi}$ acts on
$\mathop{\mathrm{Aut}}\nolimits(F_{\Psi}(Y)|F_{\Psi})$ by conjugation:
$\sigma:\alpha\mapsto\alpha^{\sigma}:=\sigma\alpha\sigma^{-1}$ for all
$\sigma\in\mathfrak{S}_{\Psi}$ and
$\alpha\in\mathop{\mathrm{Aut}}\nolimits(F_{\Psi}(Y)|F_{\Psi})$.

The composition
$F_{J\cup\tau(J)}(Y)\xrightarrow{\alpha_{\tau}}F_{J\cup\tau(J)}(Y)\xrightarrow{\sigma}F_{\sigma(J)\cup\sigma\tau(J)}(Y)\xrightarrow{\alpha_{\sigma}}F_{J\cup\sigma(J)\cup\sigma\tau(J)}(Y)$
coincides with $\alpha_{\sigma\tau}\sigma$, i.e.
$\alpha_{\sigma}\sigma\alpha_{\tau}=\alpha_{\sigma\tau}\sigma$, so
$\alpha_{\sigma}\alpha_{\tau}^{\sigma}=\alpha_{\sigma\tau}$.

Fix some $\sigma\in\mathfrak{S}_{\Psi}$ such that
$|J\cup\sigma(J)\cup\sigma\eta(J)|=3|J|$. As the field $F_{J\cup\eta(J)}(Y)$
is finitely generated over $k$, there is a finite Galois extension
$k^{\prime}|k$ and a place $F_{J}\dashrightarrow k^{\prime}$ such that the
compositions of $\alpha_{\sigma}$ and of $\alpha_{\sigma\eta}$ with the
induced place $s:F_{J\cup\sigma(J)\cup\sigma\eta(J)}(Y)\dashrightarrow
F_{\sigma(J)\cup\sigma\eta(J)}(Y)\otimes_{k}k^{\prime}$ are well-defined. Then
the cocycle condition implies
$s(\alpha_{\sigma})\alpha_{\eta}^{\sigma}=s(\alpha_{\sigma\eta})$, where
$s(\alpha_{\sigma})\in\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{\sigma(J)}(Y)|k^{\prime}F_{\sigma(J)})$
and
$s(\alpha_{\sigma\eta})\in\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{\sigma\eta(J)}(Y)|k^{\prime}F_{\sigma\eta(J)})$.
Equivalently, $\alpha_{\eta}=s_{\sigma}^{-1}s_{\sigma\eta}^{\eta}$, where
$s_{\xi}:=s(\alpha_{\xi})^{\xi^{-1}}\in\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{J}(Y)|k^{\prime}F_{J})$
for both $\xi\in\\{\sigma,\sigma\eta\\}$. For any $\varphi\in k(Y)$ and
$\beta\in\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{J}(Y)|k^{\prime}F_{J})$,
one has
$\eta:\varphi\circ\beta\mapsto\varphi^{\eta}\circ\beta^{\eta}=(\varphi\circ\beta)\circ(\beta^{-1}\alpha_{\eta}\beta^{\eta})$.
Set $\beta:=(s(\alpha_{\sigma\eta})^{\eta\sigma^{-1}})^{-1}$.

Replacing the subfield $k^{\prime}(Y)$ of the field $K\otimes_{k}k^{\prime}$
by $\beta(k^{\prime}(Y))$, we may assume further that
$\alpha_{\eta}\in\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{J}(Y)|k^{\prime}F_{J})$
and still $\alpha_{\sigma}=1$ for all $\sigma\in\mathfrak{S}_{\Psi|J}$. The
1-cocycle relation $\alpha_{\eta}\alpha_{\eta}^{\eta}=1$ implies that
$\alpha_{\eta}=(\alpha_{\eta}^{\eta})^{-1}\in\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{J}(Y)|k^{\prime}F_{J})\cap\mathop{\mathrm{Aut}}\nolimits(k^{\prime}F_{\eta(J)}(Y)|k^{\prime}F_{\eta(J)})=\mathop{\mathrm{Aut}}\nolimits(k^{\prime}(Y)|k^{\prime})$.
Therefore, the map
$\mathfrak{S}_{\Psi}\to\mathop{\mathrm{Aut}}\nolimits(k^{\prime}(Y)|k^{\prime})$,
$\xi\mapsto\alpha_{\xi}$, is a homomorphism with open kernel, which is trivial
by the simplicity of $\mathfrak{S}_{\Psi}$. This means that
$(K\otimes_{k}k^{\prime})^{\iota(\mathfrak{S}_{\Psi})}=k^{\prime}(Y)$. As the
$\mathrm{Gal}(k^{\prime}|k)$-action on $K\otimes_{k}k^{\prime}$ commutes with
the $\mathfrak{S}_{\Psi}$-action,
$K^{\mathfrak{S}_{\Psi}}=((K\otimes_{k}k^{\prime})^{\mathrm{Gal}(k^{\prime}|k)})^{\mathfrak{S}_{\Psi}}=((K\otimes_{k}k^{\prime})^{\mathfrak{S}_{\Psi}})^{\mathrm{Gal}(k^{\prime}|k)}=k^{\prime}(Y)^{\mathrm{Gal}(k^{\prime}|k)}=:k(Y^{\prime})$.
Then $K$ is a finite $\mathfrak{S}_{\Psi}$-extension of
$F_{\Psi}(Y^{\prime})$, but $K\cong
F_{\Psi}(Y^{\prime})^{\oplus[K:F_{\Psi}(Y^{\prime})]}$ in
$\mathop{\mathrm{Sm}}\nolimits_{F_{\Psi}(Y^{\prime})}(\mathfrak{S}_{\Psi})$,
so $k(Y^{\prime})=K^{\mathfrak{S}_{\Psi}}\cong
k(Y^{\prime})^{\oplus[K:F_{\Psi}(Y^{\prime})]}$, and thus,
$K=F_{\Psi}(Y^{\prime})$. ∎

######  Remark 5.26.

The $k$-varieties $Y$ and $Y^{\prime}$ in Proposition 5.25 need not to be
birational. E.g. a plane quadric with no rational points $Y$ and
$Y^{\prime}:=\mathbb{P}^{1}_{k}$ are not birational, while $Y_{F}$ has a
rational point if $F\cong k(Y)$, so $F_{\Psi}(Y)=F_{\Psi}(Y^{\prime})$.

###  5.7. $K_{0}$ of categories of compact objects in
$\mathop{\mathrm{Sm}}\nolimits_{K}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$

For a Roelcke precompact $G$, denote by
$\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(G)$ the full subcategory of
compact objects in $\mathop{\mathrm{Sm}}\nolimits_{K}(G)$.

For each extension $L|K$ of smooth $G$-fields, the faithful functor
$L\otimes_{K}(-):\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(G)\to\mathop{\mathrm{Sm}}\nolimits_{L}^{\mathrm{fp}}(G)$
induces a $\lambda$-ring homomorphism
$K_{0}(\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(G))\to
K_{0}(\mathop{\mathrm{Sm}}\nolimits_{L}^{\mathrm{fp}}(G))$. This gives rise to
a functor $\beta_{G}:K\mapsto
K_{0}(\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(G))$ from the category
$\mathrm{SmFields}(G)$ of smooth $G$-fields to the category of
$\lambda$-rings.

Denote by $\Lambda_{{\mathbb{Z}}}$ the category of $\lambda$-rings endowed
with a surjective $\lambda$-ring homomorphism onto the binomial ring
$\mathrm{Int}(\mathbb{Z})\subset\mathbb{Q}[X]$ of integer-valued polynomials
in one variable. Its final object is the identity automorphism of
$\mathrm{Int}(\mathbb{Z})$. Let $\mathrm{SmFields}^{+}(G)$ be the subcategory
of non-trivial $G$-fields.

The following result generalizes [8, Theorem 4.18].

######  Theorem 5.27.

The functor $\beta_{\mathop{\mathfrak{S}}\nolimits_{\Psi}}$ factors through a
functor
$\mathrm{SmFields}^{+}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\xrightarrow{\xi}\Lambda_{{\mathbb{Z}}}$,
sending to the final object the fields $F_{k,\Psi}$ for all non-trivial field
extensions $F|k$ with $k$ algebraically closed in $F$.

###### Proof.

By Corollary 4.9, for any smooth non-trivial
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field $K$ and each object $V$ of
$\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
there is a unique polynomial $P_{V}(X)\in\mathbb{Q}[X]$ 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 $T\subset\Psi$.

By Corollary 4.19, $P_{V_{2}}=P_{V_{1}}+P_{V_{3}}$ for any short exact
sequence $0\to V_{3}\to V_{2}\to V_{1}\to 0$ in
$\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$,
and thus, $V\mapsto P_{V}$ induces an additive homomorphism $\varphi_{K}$ from
$K_{0}(\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(\mathop{\mathfrak{S}}\nolimits_{\Psi}))$.

As the polynomials $P_{K\langle\binom{\Psi}{s}\rangle}(X)=\binom{X}{s}$ are
linearly independent for all $s\geq 0$, the classes of
$K\langle\binom{\Psi}{s}\rangle$ are linearly independent as well. As the
polynomials $\binom{X}{s}$ additively generate the target, $\varphi_{K}$ is
surjective.

To check the multiplicativity and the compatibity with $\lambda$-structures,
it suffices to notice that
$(V_{1}\otimes_{K}V_{2})^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}=V_{1}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}\otimes_{K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}}V_{2}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}$
and
$(\bigwedge^{r}_{K}V)^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}=\bigwedge^{r}_{K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}}V^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}$
for all sufficiently big $J$. Namely, by Proposition 4.6, this is reduced to
the case of $V_{i}=K\langle\binom{\Psi}{s_{i}}\rangle$, which is clear.

If $K=F_{k,\Psi}$ then, as in [8], $V$ admits a finite resolution by finite
direct sums of $K\langle\binom{\Psi}{s}\rangle$’s, so the classes of
$K\langle\binom{\Psi}{s}\rangle$’s form an additive basis of
$K_{0}(\mathop{\mathrm{Sm}}\nolimits_{K}^{\mathrm{fp}}(\mathop{\mathfrak{S}}\nolimits_{\Psi}))$,
and thus, $\varphi_{K}$ is bijective. ∎

######  Example 5.28.

According to Example 3.16, the classes of
$K_{\mathrm{a}}\langle\binom{\Psi}{s}\rangle$ for all $s\geq 0$ generate a
subring of
$K_{0}(\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}^{\mathrm{fp}}(\mathop{\mathfrak{S}}\nolimits_{\Psi}))$
isomorphic to $\mathrm{Int}(\mathbb{Z})$; the classes of
$x^{\lambda}K_{\mathrm{a}}$ for all $\lambda\in\Xi$ generate a subring
isomorphic to the group ring $\mathbb{Z}[\mathrm{Pic}_{K_{\mathrm{a}}}]$. Let
$\varepsilon:\mathbb{Z}[\mathrm{Pic}_{K_{\mathrm{a}}}]\to\mathbb{Z}$ be given
by $\sum_{i}m_{i}[\mathcal{L}_{i}]\mapsto\sum_{i}m_{i}$. By Lemma 3.21 (ii),
$([K_{\mathrm{a}}]-[x^{\lambda}K_{\mathrm{a}}])[K_{\mathrm{a}}\langle\binom{\Psi}{s}\rangle]=0$
for all $s>0$, so
$K_{0}(\mathop{\mathrm{Sm}}\nolimits_{K_{\mathrm{a}}}^{\mathrm{fp}}(\mathop{\mathfrak{S}}\nolimits_{\Psi}))$
is isomorphic to the ring $\mathrm{Int}(\mathbb{Z})\oplus\ker\varepsilon$ with
the multiplication given by
$(P,\alpha)(Q,\beta)=(PQ,Q(0)\alpha+P(0)\beta+\alpha\beta)$.

###  5.8. Examples of non-injective tensor units of type $F_{S}$ in positive
characteristic

######  Proposition 5.29.

Let $k$ be a field of characteristic $p>0$, $F^{\prime}$ and $F$ be its field
extensions in which $k$ is algebraically closed, $F^{\prime}\neq k$, and
$s\geq p>t\geq 1$ be integers. Then (i) any morphism of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-modules
$\varphi:F^{\prime}_{\binom{\Psi}{t}}\to F_{\binom{\Psi}{s}}$ vanishes at 1;
(ii) the object $F_{\binom{\Psi}{s}}$ is not injective in
$\mathop{\mathrm{Sm}}\nolimits_{F_{\binom{\Psi}{s}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$.

###### Proof.

Fix some $x\in F^{\prime}\smallsetminus k$ and $J\in\binom{\Psi}{u}$ for some
$t<u\leq s$. For each $I\in\binom{\Psi}{t}$, denote by $x_{I}$ the
decomposable tensor in $\bigotimes_{k,\binom{\Psi}{t}}F^{\prime}\subset
F^{\prime}_{\binom{\Psi}{t}}$ whose only non-identical tensor multiple is $x$
on the $I$-th place. For such $I$ with $I\subset J$, let
$a_{I}:=\varphi\left(\frac{x_{I}}{\sum_{I^{\prime}\subset
J}x_{I^{\prime}}}\right)\in F_{\binom{\Psi}{s}}$. Then $a_{I}\in
F_{\binom{\Psi}{s}}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}=F_{\binom{J}{s}}$,
so either $a_{I}\in k$ or $u=s$ and $a_{I}$ is a decomposable tensor in
$\bigotimes_{k,\binom{\Psi}{s}}F\subset F_{\binom{\Psi}{s}}$ whose only non-
identical tensor multiple is on the $J$-th place.

As
$F_{\binom{\Psi}{s}}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}=F_{\binom{\Psi}{s}}^{\mathop{\mathfrak{S}}\nolimits_{\Psi,J}}$,
$a_{I}=a$ is independent of $I$, and thus $\sum_{I\subset
J}\frac{x_{I}}{\sum_{I^{\prime}\subset J}x_{I^{\prime}}}\in
F^{\prime}_{\binom{\Psi}{t}}$ is sent to $\binom{u}{t}a$. Taking $u=p$, we get
that 1 is sent to 0.

As any morphism $F^{\prime}_{\Psi}\to F_{\binom{\Psi}{s}}$ in
$\mathop{\mathrm{Sm}}\nolimits_{k}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
vanishes at 1, the restriction of any morphism
$\xi\in\mathop{\mathrm{Hom}}\nolimits_{F_{\binom{\Psi}{s}}}(F_{\binom{\Psi}{s}}\otimes_{k}F^{\prime}_{\Psi},F_{\binom{\Psi}{s}})$
to $k\otimes_{k}F^{\prime}_{\Psi}$ vanishes at 1, so the restriction of $\xi$
to $F_{\binom{\Psi}{s}}\otimes_{k}k=F_{\binom{\Psi}{s}}$ cannot be identical,
and thus, the natural embedding $F_{\binom{\Psi}{s}}\xrightarrow{(-)\otimes
1}F_{\binom{\Psi}{s}}\otimes_{k}F^{\prime}_{\Psi}$ in
$\mathop{\mathrm{Sm}}\nolimits_{F_{\binom{\Psi}{s}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
admits no splitting, which means that the object $F_{\binom{\Psi}{s}}$ of
$\mathop{\mathrm{Sm}}\nolimits_{F_{\binom{\Psi}{s}}}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
is not injective. ∎

######  Remarks 5.30.

1\. Now Lemma A.9 implies that the object $k$ of the category
$\mathop{\mathrm{Sm}}\nolimits_{k}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$ is
injective if and only if the characteristic of $k$ is $0$.

2\. For any tower of fields $k\subset F\subseteq\widetilde{F}$ with $k$
algebraically closed in $\widetilde{F}$ and any pair of integers $s>t\geq 1$,
there are no $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-field embeddings of
$F_{\binom{\Psi}{s}}$ into $\widetilde{F}_{\binom{\Psi}{t}}$ over $k$. [If
there is one, we may restrict it to $F^{\prime}_{\binom{\Psi}{s}}$ for a
subfield $k\neq F^{\prime}=k(t)$. Then, we may choose a subfield
$\widetilde{F}^{\prime}\subseteq\widetilde{F}$ finitely generated over $k$
such that, for some $I\in\binom{\Psi}{s}$,
${F^{\prime}_{\binom{\Psi}{s}}}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|I}}=F^{\prime}$
is embedded into
$\widetilde{F}^{\prime}_{\binom{\Psi}{t}}{}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|I}}=\mathrm{Frac}(\widetilde{F}^{\prime}{}^{\otimes_{k}^{\binom{I}{t}}})$.
But
$\mathrm{tr.deg}(F^{\prime}_{\binom{\Psi}{s}}{}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}|k)=\binom{\\#J}{s}$
for any finite $J\subset\Psi$, while
$\mathrm{tr.deg}(\widetilde{F}^{\prime}_{\binom{\Psi}{t}}{}^{\mathop{\mathfrak{S}}\nolimits_{\Psi|J}}|k)=\binom{\\#J}{t}\cdot\mathrm{tr.deg}(\widetilde{F}^{\prime}|k)$
is less than $\binom{\\#J}{s}$ for sufficiently large $J$.]

##  6\. Relation between representations of automorphism groups of universal
domains and of symmetric groups: some examples

We keep notations of §3.1. Let $C$ be an algebraically closed extension of
infinite transcendence degree of an algebraically closed field $k$ of
characteristic 0, $G$ be the group of all automorphisms of the field $C$
leaving $k$ fixed.

###  6.1. 0-cycles and representations

Denote by $\mathcal{I}_{G}(k)$ the full subcategory of
$\mathop{\mathrm{Sm}}\nolimits_{k}(G)$ consisting of all objects $V$ such that
$V^{G_{L}}=V^{G_{L^{\prime}}}$ for any purely transcendental field
subextension $L^{\prime}|L$ in $C|k$.

There are some reasons to expect that the following holds ([11, Conjecture on
p.513, Prop.7.6]).

######  Conjecture 6.1.

Any simple object of $\mathcal{I}_{G}(k)$ can be embedded into the exterior
algebra $\Omega^{\bullet}_{C|k}=\bigwedge_{C}^{\bullet}\Omega^{1}_{C|k}$ of
the space $\Omega^{1}_{C|k}$ of Kähler differentials on $C$ over $k$.

This conjecture has consequences for the Chow groups $CH_{0}(-)^{0}$ of
0-cycles of degree 0.

######  Corollary 6.2 ([11], Corollary 7.9; [15], Corollary 3.2).

Assume that Conjecture 6.1 holds and a rational map $f:Y\dasharrow X$ of
smooth proper $k$-varieties induces injections
$\Gamma(X,\Omega^{q}_{X|k})\to\Gamma(Y,\Omega^{q}_{Y|k})$ for all $q\geq 0$.
Then $f$ induces a surjection $CH_{0}(Y)\to CH_{0}(X)$.

If $\Gamma(X,\Omega^{q}_{X|k})=0$ for all $q\geq 2$ then the Albanese map
induces an isomorphism between $CH_{0}(X)^{0}$ and the group of $k$-points of
the Albanese variety of $X$. (The converse for ‘big’ fields $k$, due to
Mumford, is well-known.)

######  Example 6.3.

Let $r\geq 1$ be an integer and $X$ be a smooth proper $k$-variety with
$\Gamma(X,\Omega^{j}_{X|k})=0$ for all $r<j\leq\dim X$. Let $Y$ be a
sufficiently general $r$-dimensional plane section of $X$. Then the inclusion
$Y\to X$ induces an injection
$\Gamma(X,\Omega^{\bullet}_{X|k})\to\Gamma(Y,\Omega^{\bullet}_{Y|k})$.

######  Remark 6.4.

Though the direct summands of $\bigotimes_{C}^{\bullet}\Omega^{1}_{C|k}$ are
the only known explicit simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{C}(G)$, there is a continuum of others, at
least if $C$ is countable, cf. [14, Prop.3.5.2]. However, Conjecture 6.1
relates the ‘interesting’ simple objects of
$\mathop{\mathrm{Sm}}\nolimits_{C}(G)$ to Kähler differentials.

###  6.2. A remark on structure of objects of $\mathcal{I}_{G}(k)$

Let $\Psi$ be a transcendence base of $C$ over $k$, so $G_{\Psi}$ is the
subgroup of $G$ consisting of the elements identical on $\Psi$ (or
equivalently, $G_{\Psi}$ is the absolute Galois group of the subfield
$k(\Psi)$ of $C$ generated over $k$ by $\Psi$).

Let $\Pi$ be the set of isomorphism classes of smooth irreducible
$k$-representations of $G_{\Psi}$. For any $\overline{\rho}\in\Pi$ the
subgroup $\ker\overline{\rho}\subseteq G_{\Psi}$ is open and normal, so the
subfield $C^{\ker\overline{\rho}}$ is a finite Galois extension of $k(\Psi)$,
and thus, it coincides with $L_{\overline{\rho}}(\Psi\smallsetminus
T_{\overline{\rho}})$ for a finite subset $T_{\overline{\rho}}\subset\Psi$ and
a finite Galois extension $L_{\overline{\rho}}$ of $k(T_{\overline{\rho}})$.
Then $\overline{\rho}$ factors through the restriction
$G_{\Psi}\to\mathrm{Gal}(L_{\overline{\rho}}|k(T_{\overline{\rho}}))$.

The set $\Pi$ is endowed with a natural action of the group
$\mathrm{Cr}_{\Psi}:=N_{G}(G_{\Psi})/G_{\Psi}$. This
$\mathrm{Cr}_{\Psi}$-action is smooth, since, for each point
$\overline{\rho}\in\Pi$, its stabilizer
$\overline{\mathrm{Stab}}_{\overline{\rho}}$ contains the subgroup
$\mathrm{Cr}_{\Psi}\cap\mathop{\mathfrak{S}}\nolimits_{k(\Psi)|T_{\overline{\rho}}}$.

We restrict the objects of $\mathcal{I}_{G}(k)$ to the subgroup
$N_{G}(G_{\Psi})=\\{g\in G~{}|~{}g(k(\Psi))=k(\Psi)\\}$.

For any $\overline{\rho}\in\Pi$ and any smooth $k$-representation $W$ of $G$,
denote by
$W_{\overline{\rho}}=\rho\otimes_{k}\mathop{\mathrm{Hom}}\nolimits_{k\langle
G_{\Psi}\rangle}(\rho,W)$ the $\overline{\rho}$-isotypical part of $W$, where
$\rho$ is a representation in $\overline{\rho}$, so
$W=\bigoplus_{\overline{\rho}\in\Pi}W_{\overline{\rho}}=\bigoplus_{O}W_{O}$,
where $O$ runs over the $\mathrm{Cr}_{\Psi}$-orbits on the set $\Pi$ and
$W_{O}=\bigoplus_{\overline{\rho}\in O}W_{\overline{\rho}}$.

Let $\mathrm{Stab}_{\overline{\rho}}$ be the pullback of
$\overline{\mathrm{Stab}}_{\overline{\rho}}$ in $N_{G}(G_{\Psi})$, so
$\mathrm{Stab}_{\overline{\rho}}=\\{g\in
N_{G}(G_{\Psi})~{}|~{}g(W_{\overline{\rho}})=W_{\overline{\rho}}\\}$ if
$W_{\overline{\rho}}\neq 0$. Then
$\mathrm{Stab}_{\overline{\rho}}^{1}:=\\{g\in
N_{G}(G_{\Psi})~{}|~{}g|_{W_{\overline{\rho}}}=id\\}$ is a normal subgroup of
$\mathrm{Stab}_{\overline{\rho}}$ and $W_{O}=k\langle
N_{G}(G_{\Psi})\rangle\otimes_{k\langle\mathrm{Stab}_{\overline{\rho}}\rangle}W_{\overline{\rho}}=k\langle
N_{G}(G_{\Psi})/\mathrm{Stab}_{\overline{\rho}}^{1}\rangle\otimes_{k\langle\mathrm{Stab}_{\overline{\rho}}/\mathrm{Stab}_{\overline{\rho}}^{1}\rangle}W_{\overline{\rho}}$.

If $W\in\mathcal{I}_{G}(k)$ then $W_{\overline{\rho}}\subseteq
W^{G_{L_{\overline{\rho}}}}$, so $\mathrm{Stab}_{\overline{\rho}}^{1}$ is
open, since contains
$N_{G}(G_{\Psi})\cap\mathop{\mathfrak{S}}\nolimits_{C|L_{\overline{\rho}}}$.
Then the $k\langle N_{G}(G_{\Psi})\rangle$-module $W_{O}$ is isomorphic to
$k\langle N_{G}(G_{\Psi})/U\rangle\otimes_{k\langle H\rangle}W^{\prime}$ for
an open subgroup $U$, a subgroup $H$ of $N_{N_{G}(G_{\Psi})}(U)/U$ and a
(faithful) $k$-representation $W^{\prime}$ of $H$.

###  6.3. The functor $H^{0}(G_{\Psi},-)$

As it follows from Proposition 2.2, the functor $H^{0}(G_{\Psi},-)$ from the
category $\mathop{\mathrm{Sm}}\nolimits_{C}(G)$ to the category
$\mathop{\mathrm{Sm}}\nolimits_{k(\Psi)}(\mathop{\mathfrak{S}}\nolimits_{\Psi})$
is faithful and exact. However, it is not full: $\Omega^{1}_{C|k}$ and
$\mathrm{Sym}^{2}_{C}\Omega^{1}_{C|k}$ are distinct simple smooth $C\langle
G\rangle$-modules, while
$H^{0}(G_{\Psi},\Omega^{1}_{C|k})=\Omega^{1}_{k(\Psi)|k}$ and
$H^{0}(G_{\Psi},\mathrm{Sym}^{2}_{C}\Omega^{1}_{C|k})\cong\Omega^{1}_{k(\Psi)|k}\oplus\Omega^{2}_{k(\Psi)|k}$.

For any smooth $k(\Psi)$-semilinear representation $V$ of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$, $V\otimes_{k(\Psi)}C$ is naturally a
smooth $C$-semilinear representation of $\Upsilon:=\\{g\in
G~{}|~{}g(\Psi)=\Psi\\}$. Assume that $W$ is $V\otimes_{k(\Psi)}C$ endowed
with a smooth $C$-semilinear $G$-action extending the $\Upsilon$-action. It
follows from [13, Proposition 2.5] that any open subgroup of $G$ containing
$\Upsilon_{T}$ contains the subgroup $G_{T}$.

######  Example 6.5.

1\. Let $W_{0}:=V^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}\otimes_{k}C$
correspond to the sum of all copies of $K$ in $V$. Then
$W_{0}=W^{\Upsilon}\otimes_{k}C$. On the other hand, any vector of
$W^{\Upsilon}$ is fixed by an open subgroup of $G$ containing $\Upsilon$,
i.e., $W^{\Upsilon}=W^{G}$, and thus, $W_{0}=W^{G}\otimes_{k}C$ is a direct
sum of copies of $C$.

2\. If $H^{0}(G_{\Psi},W)\cong\Omega^{i}_{K|k}$ then $W\cong\Omega^{i}_{C|k}$,
at least if $k$ is the field of algebraic numbers.

###### Proof.

Indeed, $W$ is admissible and irreducible, so we can apply [12]. ∎

######  Proposition 6.6.

For any admissible $C\langle G\rangle$-module $V$ (i.e.,
$\dim_{C^{U}}V^{U}<\infty$ for any open subgroup $U\subset G$) one has
$H^{0}(G_{\Psi},V)\cong\bigoplus_{i=0}^{\infty}(\Omega^{i}_{k(\Psi)|k})^{\oplus
m_{i}}$ as
$k(\Psi)\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules for some
integer $m_{i}\geq 0$.

###### Proof.

By Lemma 4.13, the objects $k(\Psi)\langle\binom{\Psi}{i}\rangle$ are
injective, so using Corollary 4.15 and identifications
$k(\Psi)\langle\binom{\Psi}{i}\rangle\xrightarrow{\sim}\Omega^{i}_{k(\Psi)|k}$
of Example 3.20, we may assume that $V$ is cyclic. For any finite
$T\subset\Psi$ consider the $k(T)$-semilinear representation $V^{G_{T}}$ of
the group of $k$-linear automorphisms of the $k$-linear span of $T$. As it
follows from [12], $V^{G_{T}}$ admits a filtration whose quotients are direct
summands of $k(T)$-tensor powers (Schur functors) of $\Omega^{1}_{k(T)|k}$.
Moreover, for $T\subset T^{\prime}$ one has
$V^{G_{T}}\otimes_{k(T)}k(T^{\prime})\subseteq V^{G_{T^{\prime}}}$ and these
filtrations are compatible, thus giving rise to an ascending filtration on
$V^{G_{\Psi}}$ whose quotients are direct summands of $k(\Psi)$-tensor powers
of $\Omega^{1}_{k(\Psi)|k}$, so it remains to show that
$k(\Psi)\langle\Psi\rangle^{\otimes_{k(\Psi)}^{N}}$ is isomorphic to
$\bigoplus_{i=0}^{N}k(\Psi)\langle\binom{\Psi}{i}\rangle^{\oplus a_{i,N}}$ for
any integer $N\geq 0$, where
$t^{N}=:\sum_{i=0}^{N}a_{i,N}\binom{t}{i}\in\mathbb{Z}[t]$.

We proceed by induction on $N$, the cases $N\leq 1$ being trivial. For the
induction step it suffices to construct a bijective morphism

$\alpha:k(\Psi)\langle\binom{\Psi}{N}\rangle^{\oplus N}\oplus
k(\Psi)\langle\binom{\Psi}{N+1}\rangle^{\oplus(N+1)}\longrightarrow
k(\Psi)\langle\binom{\Psi}{N}\rangle\otimes_{k(\Psi)}k(\Psi)\langle\Psi\rangle=k(\Psi)\langle\binom{\Psi}{N}\times\Psi\rangle.$

Denote by $\sigma_{s}$ the elementary symmetric polynomials and set
$\alpha([S]_{s}):=\sum_{t\in S}\sigma_{s}(S\smallsetminus\\{t\\})[S,t]$,
$0\leq s<N$, and $\alpha([T]_{s}):=\sum_{t\in
T}\sigma_{s}(T\smallsetminus\\{t\\})[T\smallsetminus\\{t\\},t]$, $0\leq s\leq
N$. As the elementary symmetric polynomials are algebraically independent,
$\alpha$ is injective. The surjectivity follows from the coincidence of
$k(T)$-dimensions of $k(T)\langle\binom{T}{N}\rangle^{\oplus N}\oplus
k(T)\langle\binom{T}{N+1}\rangle^{\oplus(N+1)}$ and of
$k(T)\langle\binom{T}{N}\rangle\otimes_{k(T)}k(T)\langle T\rangle$ for all
finite subsets $T\subset\Psi$. ∎

######  Proposition 6.7.

Let $W\in\mathcal{I}_{G}(k)$. Then $H^{0}(G_{\Psi},W\otimes_{k}C)$ is
injective as a smooth
$k(\Psi)\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-module.

###### Proof.

We keep notation of §6.2. Then
$H^{0}(G_{\Psi},W\otimes_{k}C)=\bigoplus_{\overline{\rho}\in\Pi}H^{0}(G_{\Psi},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})=\bigoplus_{O}V_{O}$,
where $\overline{\rho}^{\vee}$ is the dual of $\overline{\rho}$, $O$ runs over
the $\mathop{\mathfrak{S}}\nolimits_{\Psi}$-orbits in $\Pi$ and
$V_{O}=\bigoplus_{\overline{\rho}\in
O}H^{0}(G_{\Psi},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})$.
For any $\overline{\rho}\in\Pi$ one has $W_{\overline{\rho}}\subseteq
W^{\ker\overline{\rho}}\subseteq W^{G_{L_{\overline{\rho}}}}$, so the
pointwise stabilizer $\mathrm{Stab}^{1}_{\overline{\rho}}$ of
$W_{\overline{\rho}}$ is open.

The image $\overline{\mathrm{Stab}^{1}_{\overline{\rho}}}$ of
$\mathrm{Stab}^{1}_{\overline{\rho}}$ in $\mathrm{Cr}_{\Psi}$ is an open
subgroup, i.e.,
$\overline{\mathrm{Stab}^{1}_{\overline{\rho}}}\supseteq\mathrm{Cr}_{\Psi}\cap\mathop{\mathfrak{S}}\nolimits_{k(\Psi)|T_{\overline{\rho}}}$
for a finite $T_{\overline{\rho}}\subset\Psi$ such that
$\overline{k(T_{\overline{\rho}})}\supseteq L_{\overline{\rho}}$.

Denote by $\mathrm{St}^{1}_{\overline{\rho}}(W)$ the image of
$\mathrm{Stab}^{1}_{\overline{\rho}}(W)\cap\Upsilon$ in
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$. Then
$\mathrm{St}^{1}_{\overline{\rho}}(W)$ is an open subgroup of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$, i.e.,
$\mathrm{St}_{\overline{\rho}}\supseteq\mathop{\mathfrak{S}}\nolimits_{\Psi|T_{\overline{\rho}}}$
for a finite $T_{\overline{\rho}}\subset\Psi$ such that
$\overline{k(T_{\overline{\rho}})}\supseteq L_{\overline{\rho}}$, so
$H^{0}(G_{\Psi},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})$ is
a smooth $K$-semilinear representation of $\mathrm{St}_{\overline{\rho}}$ with
“trivial” restriction to
$\mathop{\mathfrak{S}}\nolimits_{\Psi|T_{\overline{\rho}}}$, i.e.,
$H^{0}(G_{\Psi},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})=H^{0}(G_{\Psi},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T_{\overline{\rho}}}}\otimes_{k(T_{\overline{\rho}})}K$.
By Lemma 4.26 the $K\langle\mathrm{St}_{\overline{\rho}}\rangle$-module
$H^{0}(G_{F|K},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})$ is
“trivial”, i.e.,
$H^{0}(G_{F|K},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})=H^{0}(G_{F|K},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T_{\overline{\rho}}}}\otimes_{K^{\mathop{\mathfrak{S}}\nolimits_{\Psi|T_{\overline{\rho}}}}}K$,
and therefore,
$V_{O}=H^{0}(G_{\Psi},W_{\overline{\rho}}\otimes_{k}C_{\overline{\rho}^{\vee}})^{\mathrm{St}_{\overline{\rho}}}\otimes_{K^{\mathrm{St}_{\overline{\rho}}}}K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathrm{St}_{\overline{\rho}}\rangle$
is a direct sum of several copies of
$K\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}/\mathrm{St}_{\overline{\rho}}\rangle$.
∎

######  Remark 6.8.

It is not true in general that $H^{0}(G_{\Psi},W)$ is injective, even if
$W=V\otimes C$ for a $\mathbb{Q}$-linear smooth representation $V$ of $G$.
E.g., if $V$ is the kernel of the degree morphism $\mathbb{Q}\langle
C\smallsetminus k\rangle\to\mathbb{Q}$ then one has an exact sequence $0\to
H^{0}(G_{\Psi},W)\to H^{0}(G_{\Psi},C\langle C\smallsetminus k\rangle)\to K\to
0$, which is not split, since $H^{0}(G_{\Psi},C\langle C\smallsetminus
k\rangle)^{\mathop{\mathfrak{S}}\nolimits_{\Psi}}=H^{0}(G,C\langle
C\smallsetminus k\rangle)=0$.

###  6.4. Smooth semilinear representations of symmetric groups with quasi-
trivial connections

For field extensions $K|k$ and $L|K$, a $K$-vector space $V$ and a connection
$\nabla:V\to V\otimes_{K}\Omega_{K|k}$, denote by $\nabla_{L}:V\otimes_{K}L\to
V\otimes_{K}\Omega_{L|k}$ the unique extension of $\nabla$. If $V$ is endowed
with an action of a group $H$ then a connection on $V$ is called an
$H$-connection if it commutes with the $H$-action.

A connection $\nabla:V\to V\otimes_{K}\Omega_{K|k}$ is called trivial (resp.,
quasi-trivial) if the natural map $\ker\nabla\otimes_{k}K\to V$ is surjective
(resp., if $\nabla_{\overline{K}}$ is trivial).

If $k$ is algebraically closed and $H$ is a group of automorphisms of $K$ then
the functor of horizontal sections
$\ker\nabla_{\overline{K}}:(V,\nabla)\mapsto\ker\nabla_{\overline{K}}$ is an
equivalence of categories

$\begin{array}[]{ccc}\left\\{\begin{array}[]{c}\mbox{smooth}\\\
\mbox{$k\langle\widetilde{H}\rangle$-modules}\end{array}\right\\}&\xleftarrow[\sim]{\ker\nabla_{\overline{K}}}&\left\\{\begin{array}[]{c}\mbox{smooth
$K\langle H\rangle$-modules with}\\\ \mbox{quasi-trivial $H$-connection over
$k$}\end{array}\right\\}\end{array},$

where $\widetilde{H}$ is the group of all field automorphisms of
$\overline{K}$ inducing elements of $H$ on $K$, so $\widetilde{H}$ is the
extension of $H$ by $\mathrm{Gal}(\overline{K}|K)$. The inverse functor is
given by
$W\mapsto((W\otimes_{k}\overline{K})^{\mathrm{Gal}(\overline{K}|K)},\nabla_{W})$,
where $\nabla_{W}$ is restriction of the connection on
$W\otimes_{k}\overline{K}$ vanishing on $W$.

Consider the following diagram of functors.

$\begin{array}[]{ccccc}\mathop{\mathrm{Sm}}\nolimits_{k}(G)&\xrightarrow[\sim]{\otimes_{k}C}&\left\\{\begin{array}[]{c}\mbox{smooth}\\\
\mbox{$C\langle G\rangle$-modules with}\\\ \mbox{trivial
$G$-connection}\end{array}\right\\}&\xrightarrow{\mbox{for}}&\mathop{\mathrm{Sm}}\nolimits_{C}(G)\\\
\downarrow\hbox
to0.0pt{$\displaystyle\mbox{restriction}$\hss}&&\downarrow\hbox
to0.0pt{$\displaystyle H^{0}(G_{\Psi},-)$\hss}&&\downarrow\hbox
to0.0pt{$\displaystyle H^{0}(G_{\Psi},-)$\hss}\\\
\mathop{\mathrm{Sm}}\nolimits_{k}(\Upsilon)&\xleftarrow[\sim]{\ker\nabla_{C}}&\left\\{\begin{array}[]{c}\mbox{smooth
$k(\Psi)\langle\mathop{\mathfrak{S}}\nolimits_{\Psi}\rangle$-modules}\\\
\mbox{with quasi-trivial}\\\
\mbox{$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-connection}\end{array}\right\\}&\xrightarrow{\mbox{for}}&\mathop{\mathrm{Sm}}\nolimits_{k(\Psi)}(\mathop{\mathfrak{S}}\nolimits_{\Psi})\\\
\downarrow\hbox to0.0pt{$\displaystyle\mbox{restriction}$\hss}&&\\\
\mathop{\mathrm{Sm}}\nolimits_{k}(G_{\Psi})&&\end{array}$

By [13, Lemma 4.14], restriction to $\mathcal{I}_{G}(k)$ of the composition of
the upper row is fully faithful.

It is explained in [14, §4.5], that some conjectures (a conjectural relation
to Chow groups of 0-cycles of projective generators of the category
$\mathcal{I}_{G}(k)$ and the motivic conjectures) imply that there are only
finitely many (or no) isomorphism classes of simple objects of
$\mathcal{I}_{G}(k)$ containing a given irreducible representation of
$G_{\Psi}$. In fact, (in the spirit of Howe–Bushnell–Kutzko–et al.) one can
expect that any simple object of $\mathcal{I}_{G}(k)$ is determined uniquely
by its restriction to $G_{\Psi}$.

From this point of view, the smooth $k(\Psi)$-semilinear representations of
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$ with quasi-trivial
$\mathop{\mathfrak{S}}\nolimits_{\Psi}$-connection should carry interesting
information on the corresponding simple objects of $\mathcal{I}_{G}(k)$.

##  Appendix A Smooth linear representations of infinite symmetric groups

###  A.1. Induction and coinduction

Let $A\to R$ be a continuous homomorphism of associative unital topological
rings, admitting a base of neighbourhoods of 0 given by a collection of left
ideals.

######  Example A.1.

Let $G$ be a group, $U\subseteq G$ be a subgroup and $K$ be a ring endowed
with a smooth $G$-action. Our only examples of such continuous homomorphisms
$A\to R$ will be inclusions of skew group rings $K\langle
U\rangle\hookrightarrow K\langle G\rangle$ for open subgroups $U\subset G$;
the base of open left ideals $\\{I_{H}\\}$ is indexed by a base $B$ of open
subgroups $H\subset G$, where $I_{H}$ is generated by elements $u-1$ for all
$u\in H$.

Let $R\mbox{-mod}^{\mathrm{sm}}$ be the category of left $R$-modules such that
any element is annihilated by an open left ideal. For any $R$-module $W$
denote by $W^{\mathrm{sm}}:=\mathop{\underrightarrow{\lim}}\limits_{I}\\{w\in
W~{}|~{}Iw=0\\}$ the smooth part of $W$, where $I$ runs over the open left
ideals in $R$.

The restriction functor $\mathrm{Res}^{R}_{A}:R\mbox{-mod}^{\mathrm{sm}}\to
A\mbox{-mod}^{\mathrm{sm}}$ admits a right adjoint (coinduction), sending a
smooth $A$-module $V$ to the smooth part of the $R$-module
$\mathop{\mathrm{Hom}}\nolimits_{A}(R,V)$:

(3)
$\mathop{\mathrm{Hom}}\nolimits_{A}(\mathrm{Res}^{R}_{A}W,V)=\mathop{\mathrm{Hom}}\nolimits_{R}(W,\mathop{\mathrm{Hom}}\nolimits_{A}(R,V)^{\mathrm{sm}})$

for any smooth $A$-module $V$ and any smooth $R$-module $W$, where
$\mathop{\mathrm{Hom}}\nolimits_{A}(R,V)$ is endowed with the standard left
$R$-module structure:
$R\times\mathop{\mathrm{Hom}}\nolimits_{A}(R,V)\to\mathop{\mathrm{Hom}}\nolimits_{A}(R,V)$,
$(r,\varphi)\mapsto[r^{\prime}\mapsto\varphi(r^{\prime}r)]$.

If $A\to R$ is an open embedding then the functor $\mathrm{Res}^{R}_{A}$
admits a left adjoint (induction):
$\mathop{\mathrm{Hom}}\nolimits_{A}(V,\mathrm{Res}^{R}_{A}W)=\mathop{\mathrm{Hom}}\nolimits_{R}(R\otimes_{A}V,W)$
for any $V$ and $W$ as above.

It follows from the adjunction (3) that if a smooth $A$-module $V$ is
injective then $\mathop{\mathrm{Hom}}\nolimits_{A}(R,V)^{\mathrm{sm}}$ is an
injective smooth $R$-module. In particular, if the restriction to $A$ of a
smooth $R$-module $W$ is injective the adjunction morphism
$W\to\mathop{\mathrm{Hom}}\nolimits_{A}(R,W)^{\mathrm{sm}}$ gives an embedding
into an injective smooth $R$-module.

Notation. For a group $H$ and a pair of $H$-sets $M,N$ denote by
$\mathop{\mathrm{Maps}}\nolimits_{H}(M,N)$ the set of $H$-equivariant maps
from $M$ to $N$.

######  Example A.2.

One has $\mathop{\mathrm{Hom}}\nolimits_{K\langle U\rangle}(K\langle
G\rangle,V)^{\mathrm{sm}}=\mathop{\mathrm{Maps}}\nolimits_{U}(G,V)^{\mathrm{sm}}$
for any open subgroup $U\subset G$ and any $K\langle U\rangle$-module $V$. The
standard $K\langle G\rangle$-action on
$\mathop{\mathrm{Maps}}\nolimits_{U}(G,V)^{\mathrm{sm}}$ is given by $r\cdot
g:\varphi\mapsto[\xi\mapsto r^{\xi}\varphi(\xi g)]$.

######  Lemma A.3 ([9], Lemma 4.2).

For any group $G$, any finite collection $H_{1},\dots,H_{N}$ of subgroups of
$G$ of infinite index and any finite collection of $\xi_{ij}\in G$ one has
$G\neq\bigcup_{i=1}^{N}(\coprod_{j=1}^{M}\xi_{ij}H_{i})$.

###### Proof.

We proceed by induction on $N\geq 1$, the case $N=1$ being trivial. Suppose
that $G=\bigcup_{i=1}^{N}(\coprod_{j=1}^{M}\xi_{ij}H_{i})$ for some
$\xi_{ij}\in G$. If $H_{s}$ and $H_{t}$ are commensurable for some $s\neq t$
then $G=\bigcup_{i\neq t}(\coprod_{j=1}^{M[H_{t}:H_{s}\cap
H_{t}]}\xi_{ij}^{\prime}H_{i})$ for some $\xi_{ij}^{\prime}\in G$,
contradicting the induction assumption, so we further assume that $H_{i}$ are
pairwise non-commensurable.

On a set of pairwise non-commensurable subgroups of $G$ define the following
relation $S_{1}\preceq S_{2}$ if $[S_{1}:S_{1}\cap S_{2}]<\infty$. This is a
partial order: if $S_{1}\preceq S_{2}$ and $S_{2}\preceq S_{3}$ then
$[S_{1}\cap S_{2}:S_{1}\cap S_{2}\cap S_{3}]\leq[S_{2}:S_{2}\cap
S_{3}]<\infty$ (since the natural map $S_{1}\cap S_{2}/S_{1}\cap S_{2}\cap
S_{3}\to S_{2}/S_{2}\cap S_{3}$ is injective).

Let $H_{s}$ be maximal among $H_{i}$ with respect to $\preceq$. Fix $\xi\in G$
outside the set $\xi_{s1}H_{s}\coprod\cdots\coprod\xi_{sM}H_{s}$. Then $\xi
H_{s}\subseteq\bigcup_{i\neq s}(\coprod_{j=1}^{M}\xi_{ij}H_{i})$. Omitting
those $\xi_{ij}$ for which $\xi H_{s}\cap\xi_{ij}H_{i}=\varnothing$ and
replacing appropriately the remaining $\xi_{ij}$’s, we may assume that
$\xi^{-1}\xi_{ij}\in H_{s}$.

Finally, $H_{s}=\bigcup_{i\neq s}(\coprod_{j=1}^{M}\xi^{-1}\xi_{ij}H_{i}\cap
H_{s})$, contradicting induction assumption. ∎

######  Lemma A.4.

Let $G$ be a group, $R$ be a ring, $A$ be a subring of the centre of $R$ and
$E$ be an $R$-module. We endow $R$ and $E$ with the trivial $G$-action.

For any subgroup $H\subseteq G$ the map
$\mu:\mathop{\mathrm{Maps}}\nolimits_{H}(G,A)\otimes_{A}E\to\mathop{\mathrm{Maps}}\nolimits_{H}(G,E)$,
$\varphi\otimes e\mapsto[g\mapsto\varphi(g)e]$, is an injective morphism of
$R\langle G\rangle$-modules; there is a morphism of $R\langle
G\rangle$-modules

$\kappa_{H}:E\langle G/H\rangle\to\mathop{\mathrm{Maps}}\nolimits_{H}(G,E),\
\text{{\rm given by}}\
e[g]\mapsto\left(\xi\mapsto\left\\{\begin{array}[]{ll}e&\text{{\rm if}
$\xi^{-1}\in[g]$}\\\ 0&\text{{\rm otherwise}}\end{array}\right.\right)\
\text{{\rm for any $e\in E$, $g\in G$}}.$

Let $\\{H_{i}\\}$ be a collection of pairwise non-commensurable subgroups of
$G$. Then the $G$-equivariant map
$\sum_{i}\kappa_{H_{i}}:\bigoplus_{i}E\langle
G/H_{i}\rangle\to\mathop{\mathrm{Maps}}\nolimits_{\bigcap_{i}H_{i}}(G,E)$ is
injective.

###### Proof.

For any $f\in\mathop{\mathrm{Maps}}\nolimits_{H}(G,E)$ and any $[g]\in
H\backslash G$ the element $f(\tilde{g})\in E$ is independent of a choice of
lifting $\tilde{g}$ of $[g]$ to $G$, so there is a canonical isomorphism of
$A$-modules
$\mathop{\mathrm{Maps}}\nolimits_{H}(G,E)\xrightarrow{\sim}\mathop{\mathrm{Maps}}\nolimits_{\\{1\\}}(H\backslash
G,E)=\prod_{H\backslash G}E$, given by $f\mapsto[[g]\mapsto
f(\tilde{g})]=(f(\tilde{g}))_{[g]\in H\backslash G}$. Then the injectivity of
$\mu$ follows from the injectivity of the natural map $(\prod_{H\backslash
G}A)\otimes_{A}E\to\prod_{H\backslash G}E$.

The $G$-action on $\mathop{\mathrm{Maps}}\nolimits_{\\{1\\}}(H\backslash G,E)$
is given by $f^{g}:[\xi]\mapsto f([\xi g])$.

Suppose that $f\in\mathop{\mathrm{Maps}}\nolimits_{\\{1\\}}(H\backslash
G,E)^{U}$ for a subgroup $U\subseteq G$ such that $H\backslash G/U$ is finite,
i.e., $H\backslash G/U=\coprod_{i=1}^{N}Hg_{i}U$ for some $g_{i}\in G$. Then
$f=\sum_{i=1}^{N}f_{i}$, where $f_{i}$ is supported on $Hg_{i}U$. Then
$f_{i}=\mu(\delta_{Hg_{i}U}\otimes f_{i}(g_{i})):hg_{i}u\mapsto f_{i}(g_{i})$

Suppose that $f\in\mathop{\mathrm{Maps}}\nolimits_{H}(G,E)^{U}$ for a subgroup
$U\subseteq G$ such that $H\backslash G/U$ is finite, i.e., $H\backslash
G/U=\coprod_{i=1}^{N}Hg_{i}U$ for some $g_{i}\in G$. Then
$f=\sum_{i=1}^{N}f_{i}$, where
$f_{i}\in\mathop{\mathrm{Maps}}\nolimits_{H}(G,E)$ is supported on $Hg_{i}U$.
As $f_{i}(g_{i})=f_{i}(g_{i}U)=f_{i}((g_{i}Ug_{i}^{-1})g_{i})$, the element
$f_{i}(g_{i})$ is fixed by $H\cap(g_{i}Ug_{i}^{-1})$ and
$f_{i}(hg_{i}u)=f_{i}(g_{i})$ for all $h\in H$ and $u\in U$.

To check the $G$-equivariantness
($\kappa_{H}(e[g^{\prime}g])(\xi)=\kappa_{H}(e[g])(\xi g^{\prime})$), it
suffices to note that $\xi^{-1}\in[g^{\prime}g]$ if and only if $(\xi
g^{\prime})^{-1}\in[g]$. As $\kappa_{H}$ transforms $e[g]$ to the function
supported on the left $H$-coset $[g^{-1}]\in H\backslash G$ and taking there
the value $e$, the map $\kappa_{H}(e[g])$ is constant on $H\xi$, i.e.,
$\kappa_{H}$ factors through $\mathop{\mathrm{Maps}}\nolimits_{H}(G,E)$.

Suppose that $\sum_{i}\kappa_{H_{i}}$ is not injective, so its kernel contains
an element $\sum_{t=1}^{N}\alpha_{t}$ for some non-zero $\alpha_{t}\in
E\langle G/H_{i_{t}}\rangle$. Let $H_{i_{s}}$ be a maximal subgroup with
respect to the partial order $\preceq$ defined in the proof of Lemma A.3. Then
looking at the support of both sides of the equality
$-\kappa_{H_{i_{s}}}(\alpha_{s})=\sum_{t\neq
s}\kappa_{H_{i_{t}}}(\alpha_{t})$, we get an inclusion
$H_{i_{s}}\subseteq\bigcup_{t\neq s}(\coprod_{j}H_{i_{t}}\xi_{tj})$ into a
finite union of cosets. But this contradicts Lemma A.3, so
$\sum_{i}\kappa_{H_{i}}$ is injective. ∎

######  Lemma A.5.

Let $G$ be a permutation group and $H\subseteq G$ be an open subgroup such
that $H\backslash G/U$ is finite for any open subgroup $U\subset G$. Let $R$
be an associative unitary ring endowed with a smooth $G$-action and $E$ be a
smooth left $R\langle H\rangle$-module. Suppose that there is a collection
$\\{G_{\Lambda}\\}$ of pairwise non-commensurable subgroups of $G$ containing
$H$ such that $\sum_{\Lambda}\\#\\{\text{{\rm finite $U$-orbits in
$G/G_{\Lambda}$}}\\}=\\#(H\backslash G/U)$ for any $U$ from a base of open
subgroups of $G$. Then the left $R\langle G\rangle$-module $W$ coinduced by
$E$ $($i.e., the smooth part of $\mathop{\mathrm{Hom}}\nolimits_{R\langle
H\rangle}(R\langle G\rangle,E)=\mathop{\mathrm{Maps}}\nolimits_{H}(G,E))$ is
isomorphic to $\bigoplus_{\Lambda}E\langle G/G_{\Lambda}\rangle$.

###### Proof.

Lemma A.4 provides a natural injective morphism
$\sum_{\Lambda}\kappa_{\Lambda}:\bigoplus_{\Lambda}E\langle
G/G_{\Lambda}\rangle\to W$. To check its surjectivity, we verify the
surjectivity of the induced maps $\bigoplus_{\Lambda}E\langle
G/G_{\Lambda}\rangle^{U}\to W^{U}=\mathop{\mathrm{Maps}}\nolimits_{H}(G/U,E)$,
where $U$ runs over a base of open subgroups of $G$. One has (i)
$\mathop{\mathrm{Maps}}\nolimits_{H}(G/U,E)=\prod_{O}\mathop{\mathrm{Maps}}\nolimits_{H}(O,E)$,
where $O$ runs over the $H$-orbits in $G/U$ and (ii) $E^{sUs^{-1}\cap
H}\xrightarrow{\sim}\mathop{\mathrm{Maps}}\nolimits_{H}(O,E)$ for a fixed
$s\in O$, where $e\mapsto[hs\mapsto he]$. One has (i)
$\bigoplus_{\Lambda}E\langle
G/G_{\Lambda}\rangle^{U}=\bigoplus_{\Lambda}\bigoplus_{\omega\subset
G/G_{\Lambda}}E\langle\omega\rangle^{U}$, where $\omega$ runs over the finite
$U$-orbits in $G/G_{\Lambda}$; (ii) $E^{\sigma G_{\Lambda}\sigma^{-1}\cap
U}\xrightarrow{\sim}E\langle\omega\rangle^{U}$ for a fixed $\sigma\in\omega$,
where $e\mapsto\sum_{u\in U/\sigma G_{\Lambda}\sigma^{-1}\cap U}e^{u}\cdot
u\sigma$.

As $W^{U}$ is spanned by the delta-functions on $G_{J}\backslash G/U$, it
suffices to check the surjectivity in the case $E=\mathbb{Z}$, which in turn
is equivalent to the cases where $E$ runs over all prime fields. If $E$ is a
field then the surjectivity is equivalent to the coincidence of dimensions of
the source and the target, i.e., to the equality
$\sum_{\Lambda}\\#\\{\text{finite $U$-orbits in
$G/G_{\Lambda}$}\\}=\\#(H\backslash G/U)$. ∎

######  Proposition A.6.

Let $G$ be the group of automorphisms of a countable $\Psi$, which is either a
set or a vector space over a finite field, fixing a finite subset of $\Psi$.
Let $R$ be an associative unitary ring endowed with the trivial $G$-action,
$J\subset\Psi$ be a $G$-closed finite subset and $E$ be a left $R$-module
considered as a trivial $G_{J}$-module. Then the left $R\langle
G\rangle$-module coinduced by $E$ is isomorphic to