Smooth Semilinear Representations

• Smooth semilinear representations are K-vector space representations of groups where the action is twisted by field automorphisms and each vector has an open stabilizer.
• They are organized in Grothendieck categories such as Sm_K(G), featuring distinct regimes with semisimple and locally noetherian behaviors depending on group compactness.
• Methodologies include permutation modules, Hilbert 90-type rigidity, Gabriel spectra, and invariant subfields, enabling detailed categorical classifications.

Smooth semilinear representations are representations of a topological group $G$ on a $K$-vector space in which the group acts by additive automorphisms twisted by a given action of $G$ on the field $K$, and every vector has open stabilizer. In the literature represented here, the central objects are the Grothendieck categories $\Sm_K(G)$ of smooth $K$-semilinear representations for permutation groups endowed with the topology of pointwise stabilizers, especially $G=\Sym(S)$ for an infinite set $S$ and $G=S_\infty$. The subject combines Hilbert 90–type rigidity, noetherian and injective structures in non-precompact settings, descriptions of Gabriel spectra, and explicit invariant-subfield constructions that produce both triviality phenomena and nontrivial finite-dimensional simple objects (Rovinsky, 2022).

1. Definitions and ambient categories

A permutation group $G$ on a set $K$0 is equipped with the topology whose open subgroups are the pointwise stabilizers

$K$1

as $K$2 runs over finite subsets of $K$3. Such $K$4 is totally disconnected (Rovinsky, 2022). If $K$5 is a field with a smooth action of $K$6 by field automorphisms, the skew-group algebra $K$7, also denoted $K$8, encodes the semilinear action through the rule

$K$9

A smooth $G$0-semilinear representation is then a left $G$1-module $G$2 such that $G$3 is a $G$4-vector space, $G$5 acts by additive automorphisms, and

$G$6

for all $G$7, $G$8, and $G$9, with each stabilizer $K$0 open (Rovinsky, 2014).

The resulting category $K$1 is abelian and, in the formulations emphasized by Rovinsky, a Grothendieck category (Rovinsky, 2022). A standard model is obtained by taking

$K$2

with $K$3 acting by permuting variables, $K$4. The associated category $K$5 of smooth $K$6-semilinear $K$7-representations serves as a basic test case for the general theory (Nagpal et al., 2019).

Two families of standard objects recur throughout the subject. For $K$8, one has

$K$9

where $\Sm_K(G)$0 has basis indexed by $\Sm_K(G)$1-element subsets of $\Sm_K(G)$2 (Nagpal et al., 2019). In the more general $\Sm_K(G)$3 setting, one uses the permutation modules

$\Sm_K(G)$4

which control both the injective theory and the level filtration (Rovinsky, 2022).

2. Hilbert 90, precompactness, and semisimplicity

A topological group $\Sm_K(G)$5 is precompact if every open subgroup has finite index. The decisive structural statement is a Hilbert 90–type criterion: if $\Sm_K(G)$6 acts smoothly on a field $\Sm_K(G)$7 of characteristic $\Sm_K(G)$8 and $\Sm_K(G)$9, then

$K$0

(Rovinsky, 2022). In the precompact case, the semilinear category is controlled by fixed vectors: one has

$K$1

for smooth semilinear representations, so the theory reduces to vector spaces over the fixed field (Rovinsky, 2014).

This places classical Hilbert 90 inside a broader categorical framework. If $K$2 is finite, then smooth semilinear $K$3-modules are direct sums of copies of the standard one-dimensional $K$4-module (Rovinsky, 2022). More generally, in the precompact regime every irreducible smooth semilinear representation is one-dimensional over $K$5, and the relevant $K$6-vanishing is formulated as

$K$7

for the corresponding cocycle description of semilinear actions (Rovinsky, 2014).

The non-precompact case is qualitatively different. The same sources explicitly note that there are non-semisimple smooth semilinear representations when $K$8 is not precompact, and the infinite symmetric group is the principal example (Rovinsky, 2015). The theory of smooth semilinear representations of $K$9 therefore separates into a rigid semisimple regime controlled by precompactness and a non-semisimple regime in which injectives, local splitting, and invariant subfields become essential organizing tools (Rovinsky, 2022).

3. The infinite symmetric group and local noetherian behavior

For $G=\Sym(S)$0 with $G=\Sym(S)$1 infinite and the usual topology of pointwise stabilizers, the category is not semisimple, but it remains highly structured. If $G=\Sym(S)$2 is any left noetherian ring with smooth $G=\Sym(S)$3-action, then every finitely generated object of $G=\Sym(S)$4 is noetherian over the open-stabilizer subrings; equivalently, $G=\Sym(S)$5 is locally noetherian (Rovinsky, 2022). In the formulation of Theorem 3.18, if $G=\Sym(S)$6 is any left-noetherian ring endowed with a smooth $G=\Sym(S)$7-action, then the category of smooth left $G=\Sym(S)$8-modules is locally noetherian, and in particular every smooth finitely generated $G=\Sym(S)$9-module is noetherian whenever $S$0 is a smooth $S$1-field (Rovinsky, 2015).

A finer feature is the “locally split” behavior of morphisms. For a map $S$2 between finitely generated objects, there exists a finite $S$3 such that, in $S$4, both

$S$5

split (Rovinsky, 2022). This does not make the ambient category semisimple, but it produces a controlled approximation to splitting after restriction to an open subgroup.

In the special case

$S$6

the category $S$7 of smooth semilinear $S$8-representations exhibits additional finiteness. Every finitely generated $S$9 has finite injective dimension, and one can build an injective resolution

$G=S_\infty$0

with $G=S_\infty$1 equal to the generation degree of $G=S_\infty$2 (Nagpal et al., 2019). The shift functor $G=S_\infty$3 and the decomposition

$G=S_\infty$4

drive the inductive arguments (Nagpal et al., 2019). This suggests that the infinite symmetric case, while not semisimple, still admits a robust homological calculus.

4. Injective objects, cogenerators, and Gabriel spectra

A principal construction starts with a field extension $G=S_\infty$5 such that $G=S_\infty$6 is algebraically closed in $G=S_\infty$7, and the fraction field

$G=S_\infty$8

on which $G=S_\infty$9 acts by permuting tensor factors (Rovinsky, 2022). For any $G$0-invariant subfield $G$1, the object $G$2 is an injective cogenerator of $G$3; equivalently, every smooth $G$4-module embeds into a product of copies of $G$5 (Rovinsky, 2022, Rovinsky, 2015).

The Gabriel spectrum $G$6 is described in terms of injective hulls of permutation modules. For each $G$7, the standard permutation module $G$8 has an injective hull $G$9, and these $K$00 are pairwise distinct points of the spectrum (Rovinsky, 2022). The closure relations are governed by the level filtration: $K$01 and infinite subsets $K$02 are dense (Rovinsky, 2022). In the corresponding description over $K$03, the indecomposable injectives are precisely the $K$04, and the closure of $K$05 is

$K$06

(Rovinsky, 2015). The spectrum is therefore noetherian in the Gabriel topology (Rovinsky, 2022).

In the $K$07-model $K$08, the classification is especially transparent: the indecomposable injectives are exactly the $K$09, $K$10, and a representation is injective if and only if it is a possibly infinite direct sum of the $K$11 (Nagpal et al., 2019). The Grothendieck group $K$12 has $K$13 as a $K$14-basis (Nagpal et al., 2019). These results align the spectral description of $K$15 with an explicit supply of permutation-theoretic injectives.

5. Finite-dimensional simples and invariant subfields

One of the sharpest distinctions in the subject concerns the dependence on the chosen $K$16-field. Over the full field $K$17, every finite-dimensional smooth representation is trivial: any simple $K$18-module embeds into $K$19 for some $K$20, but $K$21 has no nonzero $K$22-dimensional $K$23-submodules unless $K$24, so the only simple object is $K$25 itself with trivial $K$26-action (Rovinsky, 2022). In the earlier formulation for $K$27, any finite-length smooth semilinear $K$28-module is isomorphic to $K$29 for some $K$30, and every irreducible smooth semilinear representation is one-dimensional and trivial (Rovinsky, 2014).

That rigidity does not persist for all invariant subfields. For a cross-ratio field $K$31, defined as a subfield generated by cross-ratios, there exist irreducible finite-dimensional smooth representations of arbitrarily large dimension (Rovinsky, 2022). More precisely, finite-dimensional simple objects in $K$32 correspond naturally to finite-dimensional simple algebraic representations of $K$33, and if $K$34 is the unique simple of level $K$35, then

$K$36

viewed as a $K$37-representation via the usual action on $K$38 (Rovinsky, 2022). A statement such as “the infinite symmetric case has only trivial finite-dimensional smooth semilinear representations” is therefore correct over $K$39 or $K$40 in the cited finite-length settings, but not over invariant subfields such as $K$41.

Two further subfields furnish explicit model categories. For the degree-$K$42 subfield $K$43, the one-dimensional modules

$K$44

satisfy

$K$45

and the $K$46 form a system of injective cogenerators; every finite-length smooth $K$47-module is a direct sum $K$48 (Rovinsky, 2015). For the subfield generated by all differences $K$49, the injective envelope of the trivial module is $K$50, this object is indecomposable and a cogenerator, and for each finite length $K$51 there is a unique isomorphism class of indecomposable smooth $K$52-semilinear representations of length $K$53 (Rovinsky, 2015).

6. Alternative categorical models and methods

The semilinear category $K$54 for $K$55 is essentially equivalent to a simpler linear algebraic category $K$56 (Nagpal et al., 2019). Here $K$57 is the $K$58-linear category whose objects are finite sets and whose morphisms are generated by injections with coefficients in rational-function fields, and $K$59 is the category of $K$60-modules (Nagpal et al., 2019). Using the objects

$K$61

one obtains a fully faithful functor $K$62, and the functors

$K$63

induce mutually quasi-inverse contravariant equivalences

$K$64

(Nagpal et al., 2019). The category $K$65 is locally coherent abelian, though not noetherian, and its finitely presented objects are artinian (Nagpal et al., 2019).

The proof techniques emphasized across the literature are consistent. They include Hilbert 90 and Speiser’s theorem for semisimplicity criteria, permutation modules $K$66 and their direct-sum decompositions, a level filtration by subquotients of the standard modules $K$67, explicit invariant-subfield constructions such as $K$68, and Gabriel-spectrum analysis via injective hulls of permutation modules (Rovinsky, 2022). Another recurring tool is the construction, for any smooth $K$69-field $K$70, of a canonical extension

$K$71

for which $K$72 is a cogenerator of $K$73 (Rovinsky, 2022). In Rovinsky’s terminology, this is a “weak period” extension (Rovinsky, 2022).

Taken together, these results present smooth semilinear representation theory as a domain in which very large permutation groups exhibit both Hilbert 90–type collapse and unexpectedly rich behavior, depending on the topology of $K$74 and the choice of invariant field. The subject is therefore organized not by a single classification theorem, but by a precise trichotomy among precompact semisimple cases, non-precompact yet locally noetherian categories, and special invariant-subfield regimes supporting new finite-dimensional simple objects (Rovinsky, 2022).