# Stable finiteness of endomorphism rings

Simone Virili  Universitat Autònoma de Barcelona, Departament de Matemàtiques,
Edifici C, Facultat de Ciències \- 08193 - Bellaterra (Barcelona), Spain.
<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
###### Abstract.

We combine a combinatorial idea of Benjamin Weiss and some localization theory
of Grothendieck categories to give a short and completely self-contained proof
of the following recent result of Hanfeng Li and Bingbing Liang: Given a left
Noetherian ring $R$ and a sofic group $G$, the group-ring $R[G]$ is stably
finite.

###### Key words and phrases:

Directly finite, stably finite, sofic, Kaplansky’s conjecture, Grothendieck
category.

###### 2020 Mathematics Subject Classification:

Primary 20C07, 18E15; secondary 16D10, 20C99, 18E35.

##  1\. Introduction

A function between two sets is said to be surjunctive if it is non-injective
or surjective. Let now $G$ be a group, $A$ a finite set and equip the product
$A^{G}$ with the product of the discrete topologies on each copy of $A$ (so
that $A^{G}$ is a compact space with a continuous $G$-action). A continuous
and $G$-equivariant map $\phi\colon A^{G}\to A^{G}$ is also called a cellular
automaton (on the finite alphabet $A$). A long standing open problem by
Gottschalk [7] is that of determining whether or not all cellular automata on
finite alphabets are surjunctive; we refer to this problem as the
Surjunctivity Conjecture. When $G$ is amenable this problem has been known for
a long time to have a positive solution, but it was just in 1999 when Gromov
[8] came out with a general theorem solving the problem in the positive for
the large class of sofic groups (see also [14]). The general case remains
open.

A related (or, actually, dual) problem is that of the stable finiteness of
group rings (see [3] for connections between surjunctivity, “linear
surjunctivity” and stable finiteness). Indeed, a ring $R$ is directly finite
if $xy=1$ implies $yx=1$ for all $x,y\in R$. Furthermore, $R$ is stably finite
if the ring of square $k\times k$ matrices $\mathrm{Mat}_{k}(R)$ is directly
finite for all $k\in\mathbb{N}_{\geq 1}$. In other words, a ring $R$ is stably
finite if, for each positive integer $k$, an endomorphism $\phi\colon R^{k}\to
R^{k}$ is (split) surjective if and only if it is bijective. A long standing
open problem due to Kaplansky [9] is to determine whether the group ring
$\mathbb{F}[G]$ is stably finite for any field $\mathbb{F}$, we refer to this
problem as the Stable Finiteness Conjecture. In case the field $\mathbb{F}$
has characteristic $0$, then the problem was solved in the positive by
Kaplansky. There was no progress in the positive characteristic case until the
year 2002, when Ara, O’Meara and Perera [1] proved that a group algebra $D[G]$
is stably finite whenever $G$ is residually amenable and $D$ is any division
ring. This last result was generalized by Elek and Szabó [4], that proved the
same result for $G$ a sofic group (see also [3] and [2] for alternative
proofs).

Recently, the author [13] proved that $R[G]$ is stably finite whenever $R$ is
left Noetherian and $G$ is amenable using a notion of “algebraic entropy” for
modules over $R[G]$. Soon after, Li and Liang [10] were able to extend this
entropy theory to modules over $R[G]$ with $G$ sofic and, with this new
invariant at hand, they could use similar ideas to [13] to verify that $R[G]$
is stably finite whenever $R$ is left Noetherian and $G$ sofic. In this short
note, we show that the main ideas of [14], combined with some localization
theory for Grothendieck categories, can be used to give a direct and
completely self-contained proof of this last result. In fact, we obtain an
even more general statement for representations of a sofic group $G$ on a
Grothendieck category $\mathcal{G}$ (see Corollary 3.4). The announced result
then follows just specializing to the case when $\mathcal{G}\cong
R\text{-}\mathrm{Mod}$ for a left Noetherian ring $R$ (see Corollary 3.5).

##  2\. The main ingredients of the proof

In this section we recall some basic definitions and facts that will be needed
for the proofs of the main results. In particular, we start fixing some
notations for labeled directed graphs and we recall a combinatorial result
proved by Weiss (see Lemma 2.2): this simple-looking result is key for the
proofs in Section 3. After this, we introduce Cayley graphs and we use these
to define the class of sofic groups. In the second part of the section, we
focus instead on Grothendieck categories, on their localizations, and on the
so-called Gabriel filtration of a Grothendieck category. Finally, in the third
and last part of the section, we concentrate on the category of actions of a
group on the objects of a given Grothendieck category.

###  2.1. Cayley Graphs and sofic groups

A labeled directed graph (digraph) $\Gamma$ is a pair $(V,E)$, where $V$ is a
set of vertices and $E\subseteq V\times V$ is a set of ordered pairs of
vertices, called edges, together with a map $\phi\colon E\to B$, from the set
of edges to a set of labelings $B$. If we need to make explicit the set of
labelings $B$ we say that $\Gamma$ is a $B$-labeled digraph.

Let $B$ be a set of labelings and consider two $B$-labeled digraphs
$\Gamma_{1}=(V_{1},E_{1})$ and $\Gamma_{2}=(V_{2},E_{2})$, with labelings
defined by $\phi_{1}\colon E_{1}\to B$ and $\phi_{2}\colon E_{2}\to B$,
respectively. A morphism of $B$-labeled digraphs
$\alpha\colon\Gamma_{1}\to\Gamma_{2}$ is a map $\alpha\colon V_{1}\to V_{2}$
that sends edges to edges (i.e., $(v,v^{\prime})\in E_{1}$ implies
$(\alpha(v),\alpha(v^{\prime}))\in E_{2}$, for all $v,\,v^{\prime}\in V_{1}$)
and it respects the labels (i.e.,
$\phi_{1}(v,v^{\prime})=\phi_{2}(\alpha(v),\alpha(v^{\prime}))$, for all
$(v,v^{\prime})\in E_{1}$). In particular, such an $\alpha$ is an isomorphism
if it is bijective and its inverse also sends edges to edges or, equivalently,
($(\alpha(v),\alpha(v^{\prime}))\in E_{2}$ implies $(v,v^{\prime})\in E_{1}$,
for all $v,\,v^{\prime}\in V_{1}$).

Our main example of a labeled digraph is the following: given a finitely
generated group $G$, fix a finite symmetric set of generators $B$ of $G$. The
Cayley graph $\Gamma(G,B)$ of $G$ with respect to $B$ is a $B$-labeled digraph
$\Gamma(G,B)=(V,E)$ such that the set of vertices $V$ coincides with $G$ and
the edges are the pairs of the form $(g,gb)$, for $g\in G$ and $b\in B$; such
an edge is labeled by $b$.

Given a (labeled) digraph $\Gamma=(V,E)$ we define the distance between
vertices

$d\colon V\times V\longrightarrow\mathbb{N}\cup\\{\infty\\},$

where $d(v,v^{\prime})$ is the number of edges in a shortest directed path
between $v$ and $v^{\prime}$, if some directed path exists; and
$d(v,v^{\prime})=\infty$ otherwise. For all $v\in V$ and $n\in\mathbb{N}$ we
let

$N_{n}(v):=\\{v^{\prime}\in V:d(v,v^{\prime})\leq n\\}$

be the $n$-th neighborhood of $v$ in $\Gamma$. In the case of a Cayley graph
$\Gamma(G,B)$, the distance function is denoted by $d_{B}(-,-)$. For all $g\in
G$ and $n\in\mathbb{N}$, we denote by $N_{n}(B,g)$ $n$-th neighborhood of $g$
in $\Gamma(G,B)$. If $g=1$ we usually denote $N_{n}(B,1)$ simply by
$N_{n}(B)$.

######  Example 2.1.

Let $G=\mathbb{Z}$ be the additive group of integers, with the set of
generators $B=\\{-1,0,1\\}$. Then $N_{n}(B)=\\{-n,\dots,-1,0,1,\dots,n\\}$.
More generally, if we take
$G=\mathbb{Z}^{k}=\mathbb{Z}e_{1}\times\dots\times\mathbb{Z}e_{k}$ for some
positive integer $k$, the canonical choice for the set of generators is
$B=\\{-e_{1},\dots,-e_{k},0,e_{1},\dots,e_{k}\\}$. Thus we have

(1)
$N_{n}(B)=\left\\{\sum_{i=1}^{k}\lambda_{i}e_{i}:\lambda_{i}\in\\{-n,\dots,0,\dots,n\\}\text{
and }\sum_{i=1}^{k}|\lambda_{i}|\leq n\right\\}\,.$

The following combinatorial lemma (which is Lemma 3.1 in [14]) will be
extremely important later on.

######  Lemma 2.2.

Let $G$ be a finitely generated group, $B$ a finite symmetric set of
generators of $G$ and $\Gamma(G,B)$ its Cayley graph. If $\Lambda=(V,E)$ is a
finite $B$-labeled digraph and, for at least half the vertices of $\Lambda$,
their $2r_{0}+1$-neighborhoods in $\Lambda$ are isomorphic to
$N_{2r_{0}+1}(B)$ (the $2r_{0}+1$-neighborhood of $1$ in $\Gamma(G,B)$), then
there is a subset $V_{1}$ of these vertices such that

  1. (1)

$|V_{1}|/|V|\geq 1/2|N_{2r_{0}+1}(B)|$;

  2. (2)

the minimal distance between any two vertices in $V_{1}$ is at least
$2r_{0}+1$.

Let us conclude this subsection with the definition of sofic group. Indeed, a
finitely generated group $G$ is sofic if, for each finite symmetric set of
generators $B$ of $G$, $\varepsilon\in(0,1)$ and $r\in\mathbb{N}$, there is a
finite $B$-labeled digraph $\Gamma=(V,E)$, which has a finite subset of its
vertices $V_{0}\subseteq V$ satisfying:

  1. (1)

$|V_{0}|\geq(1-\varepsilon)|V|$;

  2. (2)

for each $v\in V_{0}$, the $r$-neighborhood $N_{r}(v)$ of $v$ in $\Gamma$ is
isomorphic to $N_{r}(B)$.

An arbitrary group $G$ is sofic if each of its finitely generated subgroups is
sofic.

It is an easy exercise to prove that each finite and each Abelian group is
sofic. In fact, one can even prove that the class of sofic groups contains
both the class of amenable and the class of residually finite (or, more
generally, that of residually amenable) groups. It is still not known whether
any group is sofic (for more details see [14] and [5]).

###  2.2. Grothendieck categories, localization and Gabriel dimension

An Abelian category $\mathcal{G}$ is said to be a Grothendieck category if it
is cocomplete, it has a generator, and directed colimits are exact in
$\mathcal{G}$. One can prove that Grothendieck categories are also complete
and that they have injective envelopes.

A torsion theory in a Grothendieck category $\mathcal{G}$ is a pair
$\tau=(\mathcal{T},\mathcal{F})$ of full subcategories of $\mathcal{G}$ such
that

  * •

$\mathcal{T}\subseteq\mathcal{G}$ is closed under taking quotients, extensions
and coproducts;

  * •

$\mathrm{Hom}_{\mathcal{G}}(T,F)=0$, for each $T\in\mathcal{T}$ and
$F\in\mathcal{F}$;

  * •

for each $X\in\mathcal{G}$, there are $T_{X}\in\mathcal{T}$,
$F_{X}\in\mathcal{F}$ and a short exact sequence:

$0\to T_{X}\to X\to F_{X}\to 0.$

A torsion theory $\tau=(\mathcal{T},\mathcal{F})$ is said to be hereditary
provided $\mathcal{T}$ is closed under taking subobjects. Given a torsion
theory $\tau=(\mathcal{T},\mathcal{F})$ in a Grothendieck category
$\mathcal{G}$, the inclusion $\mathcal{T}\to\mathcal{G}$ has a right adjoint
$\mathbf{T}_{\tau}\colon\mathcal{G}\to\mathcal{T}$, which is a subfunctor of
the identity and it is called the torsion functor. For each $X\in\mathcal{G}$,
we have that $X/\mathbf{T}_{\tau}X\in\mathcal{F}$.

Recall now that a Giraud subcategory of a Grothendieck category $\mathcal{G}$,
is a reflective subcategory $\mathcal{C}\subseteq\mathcal{G}$ such that the
inclusion functor $\mathbf{S}\colon\mathcal{C}\to\mathcal{G}$ has an exact
left adjoint $\mathbf{Q}\colon\mathcal{G}\to\mathcal{C}$. To such a Giraud
subcategory, one can associate a unique hereditary torsion theory, whose
torsion class is:

$\ker(\mathbf{Q}):=\\{X\in\mathcal{G}:\mathbf{Q}(X)=0\\}.$

On the other hand, given a hereditary torsion theory
$\tau=(\mathcal{T},\mathcal{F})$ one can construct a Giraud subcategory of
$\mathcal{G}$ as follows. We say that an object $X\in\mathcal{G}$ is
$\tau$-local if both $X\in\mathcal{F}$ and $E(X)/X\in\mathcal{F}$. The full
subcategory of $\mathcal{G}$ of all the $\tau$-local objects is denoted by
$\mathcal{G}/\mathcal{T}$ and the inclusion functor by
$\mathbf{S}_{\tau}\colon\mathcal{G}/\mathcal{T}\to\mathcal{G}$; one can give
an explicit construction of the left adjoint
$\mathbf{Q}_{\tau}\colon\mathcal{G}\to\mathcal{G}/\mathcal{T}$ (called the
$\tau$-quotient functor) to $\mathbf{S}_{\tau}$, and prove that this functor
is exact. Let us remark that we adopt the notation “$\mathcal{G}/\mathcal{T}$”
because this category is equivalent to the Gabriel quotient of $\mathcal{G}$
over $\mathcal{T}$ (for more details on these constructions see [12], [6] or
[11]).

Let us now recall that the Gabriel filtration of a Grothendieck category
$\mathcal{G}$ is a transfinite chain
$0=\mathcal{G}_{-1}\subseteq\mathcal{G}_{0}\subseteq\dots\subseteq\mathcal{G}_{\alpha}\subseteq\dots$
of hereditary torsion classes, where:

  * •

$\mathcal{G}_{-1}:=0$;

  * •

suppose that $\alpha$ is an ordinal for which $\mathcal{G}_{\alpha}$ has
already been defined. An object $X\in\mathcal{G}$ is said to be
$\alpha$-cocritical if $X$ is $\tau_{\alpha}$-torsion free and every proper
quotient of $X$ is $\tau_{\alpha}$-torsion, where $\tau_{\alpha}$ is the
unique torsion theory whose torsion class is $\mathcal{G}_{{\alpha}}$. We let
$\mathcal{G}_{\alpha+1}$ be the smallest hereditary torsion class containing
$\mathcal{G}_{{\alpha}}$ and all the ${\alpha}$-cocritical objects;

  * •

if $\lambda$ is a limit ordinal, $\mathcal{G}_{\lambda}$ is the smallest
hereditary torsion class containing
$\bigcup_{\alpha<\lambda}\mathcal{G}_{\alpha}$.

For any ordinal $\alpha$, we let
$\mathbf{T}_{\alpha}\colon\mathcal{G}\to\mathcal{G}_{\alpha}$ and
$\mathbf{Q}_{\alpha}\colon\mathcal{G}\to\mathcal{G}/\mathcal{G}_{\alpha}$ be
the $\tau_{\alpha}$-torsion and the $\tau_{\alpha}$-quotient functor,
respectively. Note that the ascending chain of subcategories
$\mathcal{G}_{\alpha}\subseteq\mathcal{G}$ eventually stabilizes (one way to
see this is to note that $\mathcal{G}_{\alpha}$ is generated by the
$\tau_{\alpha}$-torsion quotients of a fixed generator $G$ of $\mathcal{G}$).
Hence, it makes sense to consider
$\widetilde{\mathcal{G}}:=\bigcup_{\alpha}\mathcal{G}_{\alpha}$. Let us remark
that, even for $\mathcal{G}=R\text{-}{\mathrm{Mod}}$ a category of modules, it
may happen that $\widetilde{\mathcal{G}}\neq\mathcal{G}$ (e.g., take $R$ to be
a non-discrete valuation domain).

######  Lemma 2.3.

Let $\mathcal{G}$ be a Grothendieck category, and
$X\in\widetilde{\mathcal{G}}$. Then, $X=0$ if, and only if,
$\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(X))=0$, for all $\alpha$.

###### Proof.

By definition of $\widetilde{\mathcal{G}}$, there is an ordinal $\delta\geq-1$
such that $X\in\mathcal{G}_{\delta}$. We proceed by transfinite induction on
such a $\delta$. Indeed, if $\delta=-1$, then $X=0$ and so there is nothing to
prove. If $\delta=\alpha+1$ (so that $X=\mathbf{T}_{\alpha+1}(X)$), then there
is a short exact sequence

$0\to\mathbf{T}_{\alpha}(X)\to X\to X/\mathbf{T}_{\alpha}(X)\to 0.$

Now, $\mathbf{T}_{\alpha}(X)=0$ by inductive hypothesis, while
$X/\mathbf{T}_{\alpha}(X)\leq\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(X))=0$,
by hypothesis. Hence, $X=0$ as desired. Finally, suppose that $\delta$ is a
limit ordinal. In this case,
$X\cong\bigcup_{\alpha<\delta}\mathbf{T}_{\alpha}(X)$ and, by inductive
hypothesis, $\mathbf{T}_{\alpha}(X)=0$ for each $\alpha<\delta$, showing that
$X=0$. ∎

Recall that a Grothendieck category $\mathcal{G}$ is said to be semi-Artinian
if, and only if, every non-trivial object $X\in\mathcal{G}$ has a simple
(i.e., $(-1)$-cocritical) subobject. In particular, given an object $X$ in a
semi-Artinian Grothendieck category $\mathcal{G}$, there exists an ordinal
$\alpha$ and a continuous chain $\\{X_{\beta}:\beta<\alpha\\}$ of subobjects
of $X$ such that $X=\bigcup_{\beta<\alpha}X_{\beta}$ and where
$X_{\beta+1}/X_{\beta}$ is a simple object for all $\beta<\alpha$. Note that a
Grothendieck category is semi-Artinian if, and only if,
$\mathcal{G}=\mathcal{G}_{0}$.

######  Lemma 2.4.

Given a Grothendieck category $\mathcal{G}$, the category
$\mathcal{G}_{\alpha+1}/\mathcal{G}_{\alpha}$ is semi-Artinian for each
ordinal $\alpha$.

###### Proof.

By construction, we have that
$\mathcal{G}_{\alpha+1}/\mathcal{G}_{\alpha}\cong(\mathcal{G}/\mathcal{G}_{\alpha})_{0}$.
∎

Let $\mathcal{G}$ be a Grothendieck category, $X\in\mathcal{G}$, and denote by
$\mathcal{L}(X)$ the lattice of subobjects of $X$. We say that $X$ is
Noetherian, Artinian or of finite length if the lattice $\mathcal{L}(X)$ has
the ascending chain condition, the descending chain condition or both,
respectively. In case $X$ is of finite length, we denote by
$\ell(X)\in\mathbb{N}$ the composition length of the lattice $\mathcal{L}(X)$.

######  Lemma 2.5.

Let $\mathcal{G}$ be a Grothendieck category, and $0\neq N\in\mathcal{G}$ a
Noetherian object. Then, the following statements hold true:

  1. (1)

$N\in\widetilde{\mathcal{G}}$;

  2. (2)

$\mathbf{T}_{\alpha}(N)$ is a Noetherian object in $\mathcal{G}_{\alpha}$, for
each ordinal $\alpha$;

  3. (3)

$\mathbf{Q}_{\alpha}(N)$ is a Noetherian object in
$\mathcal{G}/\mathcal{G}_{\alpha}$, for each ordinal $\alpha$;

  4. (4)

$\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(N))$ is an object of finite length
in $\mathcal{G}_{\alpha+1}/\mathcal{G}_{\alpha}$, for each ordinal $\alpha$.

###### Proof.

(1). Suppose, looking for a contradiction, that
$N\notin\widetilde{\mathcal{G}}$. By Noetherianity, we can select a maximal
subobject $K\leq N$ such that $N/K\notin\widetilde{\mathcal{G}}$. Then, for
each $K\lneq H\leq N$ there exists an ordinal $\alpha_{H}$ such that
$N/H\in\mathcal{G}_{\alpha_{H}}$, let $\alpha:=\sup_{H}\alpha_{H}$. By
construction, $N/K$ is $\alpha$-cocritical, so
$N/K\in\mathcal{G}_{\alpha+1}\subseteq\widetilde{\mathcal{G}}$, which is a
contradiction.

(2). The lattice $\mathcal{L}_{\mathcal{G}_{\alpha}}(\mathbf{T}_{\alpha}(N))$
of subobjects of $\mathbf{T}_{\alpha}(N)$ in $\mathcal{G}_{\alpha}$ can be
identified with the sublattice of $\mathcal{L}(N)$ of those
$K\in\mathcal{L}(N)$ such that $K\leq\mathbf{T}_{\alpha}(N)$. In particular,
any ascending chain in
$\mathcal{L}_{\mathcal{G}_{\alpha}}(\mathbf{T}_{\alpha}(N))$ induces an
ascending chain in $\mathcal{L}(N)$ and, therefore, it stabilizes after a
finite number of steps.

(3). The lattice
$\mathcal{L}_{\mathcal{G}/\mathcal{G}_{\alpha}}(\mathbf{Q}_{\alpha}(N))$ of
subobjects of $\mathbf{Q}_{\alpha}(N)$ in $\mathcal{G}/\mathcal{G}_{\alpha}$
can be identified with the sublattice of $\mathcal{L}(N)$ of those
$K\in\mathcal{L}(N)$ such that $N/K$ is $\tau_{\alpha}$-torsion free (this is
proved in [12, Corollary IX.4.4] for categories of modules, and the argument
carries over to general Grothendieck categories). In particular, any ascending
chain in
$\mathcal{L}_{\mathcal{G}/\mathcal{G}_{\alpha}}(\mathbf{Q}_{\alpha}(N))$
induces an ascending chain in $\mathcal{L}(N)$ and, therefore, it stabilizes
after a finite number of steps.

(4). By (2) and (3) we deduce that
$\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(N))$ is a Noetherian object in
$\mathcal{G}_{\alpha+1}/\mathcal{G}_{\alpha}$. As we have previously observed,
$\mathcal{G}_{\alpha+1}/\mathcal{G}_{\alpha}$ is semi-Artinian and, therefore,
$\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(N))$ can be written as the union of
a continuous chain whose subfactors are simple objects. By Noetherianity, any
such ascending chain has to be finite. ∎

###  2.3. Group actions on objects of a Grothendieck category

Let $\mathcal{G}$ be a Grothendieck category and consider a group $G$ as a
category with one object. The category of representations of $G$ on
$\mathcal{G}$ is the functor category
$\mathcal{G}^{G}:=\mathrm{Fun}(G,\mathcal{G})$. An alternative, but
equivalent, description of this category can be given as follows:

  * •

each object of $\mathcal{G}^{G}$ can be thought of as a pair
${}_{G}X:=(X,\lambda\colon G\to\mathrm{Aut}_{\mathcal{G}}(X))$, with $X$ in
$\mathcal{G}$ and $\lambda$ a homomorphism to the automorphism group of $X$;

  * •

given two objects ${}_{G}X_{i}:=(X_{i},\lambda_{i}\colon
G\to\mathrm{Aut}_{\mathcal{G}}(X_{i}))$ (with $i=1,2$), a morphism
$\phi\colon{}_{G}X_{1}\to{}_{G}X_{2}$ in $\mathcal{G}^{G}$ is a
$G$-equivariant morphism $\phi\colon X_{1}\to X_{2}$ in $\mathcal{G}$, that
is, the following square commutes for all $g\in G$:

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{1}(g)}$$\scriptstyle{\phi}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{2}(g)}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{X_{2}.}$

As an example, one can take the trivial group $G=\mathbf{1}$ and, in that
case, it is easy to verify that $\mathcal{G}^{\mathbf{1}}\cong\mathcal{G}$.
Consider a group $G$ and take the obvious inclusion $\iota\colon\mathbf{1}\to
G$. This induces a restriction functor (also called a forgetful functor)

$\downarrow_{\mathbf{1}}^{G}\colon\mathcal{G}^{G}\to\mathcal{G},$

mapping a $G$-representation ${}_{G}X\mapsto X$. The left adjoint to
$\downarrow_{\mathbf{1}}^{G}$ is the extension functor

$\uparrow_{\mathbf{1}}^{G}\colon\mathcal{G}\to\mathcal{G}^{G}.$

Given an object $X\in\mathcal{G}$, we have that
$\uparrow_{\mathbf{1}}^{G}(X)\cong(X^{(G)},\beta\colon
G\to\mathrm{Aut}_{\mathcal{G}}(X^{(G)}))$, where $X^{(G)}$ is the coproduct of
$|G|$-many copies of $X$ in $\mathcal{G}$, and $\beta$ is the usual Bernoulli
shift on $X^{(G)}$, that is, for each $g\in G$, $\beta^{g}\colon X^{(G)}\to
X^{(G)}$ can be represented by the column-finite square matrix
$\beta^{g}=(\beta^{g}_{g_{2},g_{1}})_{g_{1},g_{2}\in
G}\in\mathrm{Mat}_{|G|}(\mathrm{End}_{\mathcal{G}}(X))$:

$\beta^{g}_{g_{2},g_{1}}:=\begin{cases}\mathrm{id}_{X}&\text{if
$g_{2}=gg_{1}$;}\\\ 0&\text{otherwise.}\end{cases}$

######  Example 2.6.

Let $R$ be a ring and $G$ a group. The group-ring $R[G]$ is defined as the
Abelian group $\bigoplus_{g\in G}\underline{g}R$, with the sum defined
componentwise, and product defined as follows: given $x=\sum_{g\in
G}\underline{g}x_{g}$ and $y=\sum_{g\in G}\underline{g}y_{g}\in R[G]$,

$x\cdot y:=\sum_{g\in
G}\underline{g}\left(\sum_{h_{1}h_{2}=g}x_{h_{1}}y_{h_{2}}\right).$

With these operations, $R[G]$ becomes an associative and unitary ring.
Consider now the category of left $R[G]$-modules $R[G]\text{-}{\mathrm{Mod}}$.
Essentially by definition, $R[G]\text{-}{\mathrm{Mod}}$ is equivalent to the
category of additive functors $\mathrm{Add}(R[G],\mathrm{Ab})$ from $R[G]$,
viewed as a one-object preadditive category, to the category of Abelian groups
$\mathrm{Ab}$. Similarly, these categories are both equivalent to
$\mathrm{Add}(\mathbb{Z}[G],R\text{-}{\mathrm{Mod}})$ and also to the category
of (non-necessarily additive) functors
$\mathrm{Fun}(G,R\text{-}{\mathrm{Mod}})$, that is,
$R[G]\text{-}{\mathrm{Mod}}\cong(R\text{-}{\mathrm{Mod}})^{G}$. In this
special example, the restriction and induction functors are called,
respectively, the forgetful functor and the extension of scalars:

$\downarrow_{\mathbf{1}}^{G}\cong\mathrm{Hom}_{R[G]}({}_{R[G]}R[G]_{R},-)\qquad\text{and}\qquad\uparrow_{\mathbf{1}}^{G}\cong{}_{R[G]}R[G]_{R}\otimes_{R}-.$

Take now a Grothendieck category $\mathcal{G}$ and a group $G$. Limits and
colimits in $\mathcal{G}^{G}$ can be computed “as in $\mathcal{G}$”. More
explicitly, given a small category $I$ and a functor $F\colon
I\to\mathcal{G}^{G}$, its (co)limit is the (co)limit of
$\downarrow^{G}_{\mathbf{1}}\circ F\colon I\to\mathcal{G}$, endowed with the
unique compatible $G$-action. In particular, the fact that $\mathcal{G}$ is a
cocomplete Abelian category with exact directed colimits implies that
$\mathcal{G}^{G}$ has the same properties. Furthermore, given a generator
$X\in\mathcal{G}$, the induced representation $\uparrow^{G}_{\mathbf{1}}X$ is
a generator of $\mathcal{G}^{G}$, which is therefore a Grothendieck category.

An object $X\in\mathcal{G}$ is said to be compact if the functor
$\mathrm{Hom}_{\mathcal{G}}(X,-)\colon\mathcal{G}\to\mathrm{Ab}$ commutes with
coproducts. In particular, if $X$ is Noetherian (or, more generally, finitely
generated, that is, $\mathrm{Hom}_{\mathcal{G}}(X,-)$ commutes with
monomorphic directed colimits) then it is compact. Observe that, given a
compact object $X\in\mathcal{G}$, an object $Y\in\mathcal{G}$ and a set $I$,
we have the following natural isomorphisms:

$\textstyle{\mathrm{Hom}_{\mathcal{G}}(X^{(I)},Y^{(I)})\cong\prod_{I}\mathrm{Hom}_{\mathcal{G}}(X,Y^{(I)})\cong\prod_{I}\bigoplus_{I}\mathrm{Hom}_{\mathcal{G}}(X,Y),}$

that is, a morphism $\phi\colon X^{(I)}\to Y^{(I)}$ can be naturally
considered as a column-finite matrix with coefficients in
$\mathrm{Hom}_{\mathcal{G}}(X,Y)$.

######  Proposition 2.7.

Let $\mathcal{G}$ be a Grothendieck category, $X,\,Y\in\mathcal{G}$ compact
objects, and let $G$ be a group. Consider a morphism
$\phi=(\phi_{g_{2},g_{1}})_{g_{1},g_{2}\in G}\colon X^{(G)}\to Y^{(G)}$ in
$\mathcal{G}$. Then, $\phi$ represents a morphism
$\uparrow^{G}_{\mathbf{1}}X\to\uparrow^{G}_{\mathbf{1}}Y$ in $\mathcal{G}^{G}$
(that is, it is $G$-equivariant with respect to the Bernoulli actions) if and
only if:

$\phi_{g^{-1}g_{2},g_{1}}=\phi_{g_{2},gg_{1}},\qquad\text{for all
$g,\,g_{1},\,g_{2}\in G$.}$

Furthermore, given a morphism
$\psi=(\psi_{g_{2},g_{1}})_{g_{1},g_{2}}\colon\uparrow_{\mathbf{1}}^{G}Y\to\uparrow^{G}_{\mathbf{1}}X$
in $\mathcal{G}^{G}$, so that $\psi_{g^{-1}g_{2},g_{1}}=\psi_{g_{2},gg_{1}}$
for all $g,\,g_{1},\,g_{2}\in G$, we have that
$\phi\circ\psi=\mathrm{id}_{\uparrow_{\mathbf{1}}^{G}Y}$ if, and only if, the
following equality is verified, for all $g_{1},\,g_{2}\in G$:

$\sum_{h\in
G}\phi_{g_{2},h}\psi_{h,g_{1}}=\begin{cases}\mathrm{id}_{Y}&\text{if
$g_{1}=g_{2}$;}\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

For the first claim about $\phi$, we have to verify that
$\beta^{g}\phi=\phi\beta^{g}$ for all $g\in G$, and it is easily seen that,
for each $g,\,g_{1},\,g_{2}\in G$:

$\displaystyle(\beta^{g}\phi)_{g_{2},g_{1}}=\sum_{h\in
G}\beta^{g}_{g_{2},h}\phi_{h,g_{1}}=\phi_{g^{-1}g_{2},g_{1}}\qquad\text{and}$
$\displaystyle(\phi\beta^{g})_{g_{2},g_{1}}=\sum_{h\in
G}\phi_{g_{2},h}\beta^{g}_{h,g_{1}}=\phi_{g_{2},gg_{1}}.$

It is also clear that $\phi\circ\psi=\mathrm{id}_{\uparrow_{\mathbf{1}}^{G}Y}$
if, and only if, $(\phi\circ\psi)_{g_{2},g_{1}}=\mathrm{id}_{Y}$ if
$g_{1}=g_{2}$ and $(\phi\circ\psi)_{g_{2},g_{1}}=0$ if $g_{1}\neq g_{2}\in G$.
Furthermore, $(\phi\circ\psi)_{g_{2},g_{1}}=\sum_{h\in
G}\phi_{g_{2},h}\psi_{h,g_{1}}$∎

Let us conclude with a brief discussion about the behavior of representations
with respect to torsion and localization:

######  Proposition 2.8.

Let $\mathcal{G}$ be a Grothendieck category, $\tau=(\mathcal{T},\mathcal{F})$
a hereditary torsion theory in $\mathcal{G}$ and $G$ a group. Define:

  * •

$\mathcal{T}^{G}:=\\{(X,\lambda\colon
G\to\mathrm{End}_{\mathcal{G}}(X))\in\mathcal{G}^{G}:X\in\mathcal{T}\\}$;

  * •

$\mathcal{F}^{G}:=\\{(X,\lambda\colon
G\to\mathrm{End}_{\mathcal{G}}(X))\in\mathcal{G}^{G}:X\in\mathcal{F}\\}$;

  * •

$\tau^{G}:=(\mathcal{T}^{G},\mathcal{F}^{G})$.

Then, $\tau^{G}$ is a hereditary torsion theory in $\mathcal{G}^{G}$ and,
given $(X,\lambda\colon G\to\mathrm{End}_{\mathcal{G}}(X))\in\mathcal{G}^{G}$,

  1. (1)

$\mathbf{T}_{\tau^{G}}(X,\lambda)=(\mathbf{T}_{\tau}(G),\mathbf{T}_{\tau}(\lambda))$,
where $(\mathbf{T}_{\tau}\lambda)_{g}:=\mathbf{T}_{\tau}(\lambda_{g})$, for
all $g\in G$;

  2. (2)

there is an equivalence
$\mathcal{G}^{G}/\mathcal{T}^{G}\cong(\mathcal{G}/\mathcal{T})^{G}$ and,
identifying these categories, we have that
$\mathbf{Q}_{\tau^{G}}(X,\lambda)=(\mathbf{Q}_{\tau}(G),\mathbf{Q}_{\tau}(\lambda))$,
where $(\mathbf{Q}_{\tau}\lambda)_{g}:=\mathbf{Q}_{\tau}(\lambda_{g})$, for
all $g\in G$.

###### Proof.

Let $(X,\lambda)\in\mathcal{G}^{G}$ and consider
$(\mathbf{T}_{\tau}X,\mathbf{T}_{\tau}\lambda)\in\mathcal{T}^{G}$ and
$(X/\mathbf{T}_{\tau}X,\bar{\lambda})\in\mathcal{F}^{G}$, such that
$(\mathbf{T}_{\tau}\lambda)_{g}:=\mathbf{T}_{\tau}(\lambda_{g})$ and
$\bar{\lambda}_{g}\colon X/\mathbf{T}_{\tau}X\to X/\mathbf{T}_{\tau}X$ is the
map induced on the quotient (for each $g\in G$). It is easily seen that
$\iota\colon\mathbf{T}_{\tau}X\to X$ is $G$-equivariant with respect to
$\mathbf{T}_{\tau}(\lambda)$ and $\lambda$, and that $\pi\colon X\to
X/\mathbf{T}_{\tau}X$ is $G$-equivariant with respect to $\lambda$ and
$\bar{\lambda}$. We then obtain the following short exact sequence in
$\mathcal{G}^{G}$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbf{T}_{\tau}X,\mathbf{T}_{\tau}\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{(X,\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{(X/\mathbf{T}_{\tau}X,\bar{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Furthermore, one can use the closure properties of $\mathcal{T}$ to show that
$\mathcal{T}^{G}$ is a hereditary torsion class. Finally, given
${}_{G}T\in\mathcal{T}^{G}$ and ${}_{G}F\in\mathcal{F}^{G}$,

$\mathrm{Hom}_{\mathcal{G}^{G}}({}_{G}T,{}_{G}F)\subseteq\mathrm{Hom}_{\mathcal{G}}(T,F)=0,$

proving that $\tau^{G}$ is a hereditary torsion theory in $\mathcal{G}^{G}$.
Take now the adjunction

$\mathbf{Q}_{\tau}:\mathcal{G}\leftrightarrows\mathcal{G}/\mathcal{T}:\mathbf{S}_{\tau},$

identify $\mathcal{G}/\mathcal{T}$ with the essential image of the fully
faithful functor $\mathbf{S}_{\tau}$ (which is then identified with the
inclusion in $\mathcal{G}$). For each $X\in\mathcal{G}$, we denote by
$\eta_{X}\colon X\to\mathbf{Q}_{\tau}X$ the $X$-component of the unit of the
above adjunction. Consider now two functors:

$\mathbf{Q}^{G}_{\tau}:\mathcal{G}^{G}\leftrightarrows(\mathcal{G}/\mathcal{T})^{G}:\mathbf{S}^{G}_{\tau},$

where
$\mathbf{Q}^{G}_{\tau}(X,\lambda):=(\mathbf{Q}_{\tau}X,\mathbf{Q}_{\tau}\lambda)$
for each $(X,\lambda)\in\mathcal{G}^{G}$, with
$(\mathbf{Q}_{\tau}\lambda)_{g}:=\mathbf{Q}_{\tau}(\lambda_{g})$ for each
$g\in G$, and with $\mathbf{S}^{G}_{\tau}$ the inclusion of the full
subcategory $(\mathcal{G}/\mathcal{T})^{G}$ of $\mathcal{G}^{G}$ (with the
same identifications above). Let $(X,\lambda)\in\mathcal{G}^{G}$ and note that
$\eta_{X}\colon X\to\mathbf{Q}_{\tau}X$ is $G$-equivariant (when taking the
$G$-action $\mathbf{Q}_{\tau}\lambda$ on $\mathbf{Q}_{\tau}X$) by naturality.
Let now $(Y,\lambda^{\prime})\in(\mathcal{G}/\mathcal{T})^{G}$ and take a
morphism $\phi\colon{}_{G}X\to{}_{G}Y$ in $\mathcal{G}^{G}$. By the adjunction
$\mathbf{Q}_{\tau}\dashv\mathbf{S}_{\tau}$, there exists a unique morphism
$\bar{\phi}\colon\mathbf{Q}_{\tau}X\to Y$ in $\mathcal{G}$ such that
$\bar{\phi}\circ\eta_{X}=\phi$. Let us show that $\bar{\phi}$ is
$G$-equivariant. Indeed, let $g\in G$, then:

$\displaystyle\lambda_{g}^{\prime}\circ\bar{\phi}\circ\eta_{X}$
$\displaystyle=\lambda_{g}^{\prime}\circ\phi$ since
$\bar{\phi}\circ\eta_{X}=\phi$; $\displaystyle=\phi\circ\lambda_{g}$ since
$\phi$ is $G$-equivariant;
$\displaystyle=\bar{\phi}\circ\eta_{X}\circ\lambda_{g}$ since
$\bar{\phi}\circ\eta_{X}=\phi$;
$\displaystyle=\bar{\phi}\circ\mathbf{Q}_{\tau}\lambda_{g}\circ\eta_{X}$ since
$\eta_{X}$ is $G$-equivariant.

Since $\eta_{X}$ is the unit and $Y\in\mathcal{G}/\mathcal{T}$, the equality
$\lambda_{g}^{\prime}\circ\bar{\phi}\circ\eta_{X}=\bar{\phi}\circ\mathbf{Q}_{\tau}\lambda_{g}\circ\eta_{X}\colon
X\to Y$ implies that
$\lambda_{g}^{\prime}\circ\bar{\phi}=\bar{\phi}\circ\mathbf{Q}_{\tau}\lambda_{g}$,
as desired. It is also easy to show that $\mathbf{Q}_{\tau}^{G}$ is exact
(using the exactness of $\mathbf{Q}_{\tau}$) and, therefore,
$(\mathcal{G}/\mathcal{T})^{G}$ becomes a Giraud subcategory of
$\mathcal{G}^{G}$, associated with the hereditary torsion class

$\ker(\mathbf{Q}_{\tau}^{G})=\\{{}_{G}X\in\mathcal{G}^{G}:\mathbf{Q}_{\tau}(X)=0\\}=\mathcal{T}^{G}.$

In particular,
$(\mathcal{G}/\mathcal{T})^{G}\cong\mathcal{G}^{G}/\mathcal{T}^{G}$. ∎

######  Corollary 2.9.

Let $\mathcal{G}$ be a Grothendieck category, $G$ a group, $N\in\mathcal{G}$ a
Noetherian object and $M:=\uparrow^{G}_{\mathbf{1}}N\in\mathcal{G}^{G}$. Let
${}_{G}K\leq M$ be a subobject of $M$ in $\mathcal{G}^{G}$, then, $K=0$ if,
and only if, $\mathbf{Q}^{G}_{\alpha}(\mathbf{T}^{G}_{\alpha+1}(K))=0$, for
all $\alpha$.

###### Proof.

By the above proposition it is clear that
$\mathbf{Q}^{G}_{\alpha}(\mathbf{T}^{G}_{\alpha+1}(K))=0$, for all $\alpha$,
if and only if
$\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(\downarrow^{G}_{\mathbf{1}}K))=0$,
for all $\alpha$. By Lemma 2.3, this is equivalent to say that
$\downarrow^{G}_{\mathbf{1}}K=0$ in $\mathcal{G}$. Finally, note that $K=0$ in
$\mathcal{G}^{G}$ if, and only if, $\downarrow^{G}_{\mathbf{1}}K=0$ in
$\mathcal{G}$. ∎

##  3\. Stable finiteness of endomorphism rings

###  3.1. The semi-Artinian case

This subsection contains the main application of Weiss’ idea to
representations of a sofic group $G$ on a Grothendieck category $\mathcal{G}$.
The other results in this section will be proved by reducing to this case with
a combination of suitable torsion and quotient functors.

######  Proposition 3.1.

Let $\mathcal{G}$ be a Grothendieck category, $G$ a finitely generated sofic
group, $X\in\mathcal{G}$ an object of finite length, and let
${}_{G}M:=\uparrow^{G}_{\mathbf{1}}X\in\mathcal{G}^{G}$. Then,
$\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ is directly finite.

###### Proof.

Let $\phi=(\phi_{g_{2},g_{1}})_{g_{1},g_{2}\in G}$ and
$\psi=(\psi_{g_{2},g_{1}})_{g_{1},g_{2}\in
G}\in\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ be two endomorphisms such that
$\phi\circ\psi=\mathrm{id}_{M}$, and let us verify that $\phi$ is a
monomorphism. Indeed, we suppose, looking for a contradiction, that
$\ker(\phi))\neq 0$, and we choose a sufficiently big $r_{0}\in\mathbb{N}$ and
a sufficiently small $\varepsilon\in(0,1)$ in order to apply the definition of
sofic group (Step 1 below) to obtain a statement about objects of finite
length (Step 2), from which the contradiction can be seen (Step 3).

Step 1. The choice of $r_{0}$ and $\varepsilon$. Consider the following finite
subset of $G$:

$S:=\\{1\\}\cup\\{g\in G:\phi_{g,1}\neq 0\\}\cup\\{g\in G:\psi_{g,1}\neq
0\\}.$

We denote by $r_{1}$ the minimal positive integer such that $S\cdot S\subseteq
N_{r_{1}}(B)$ (where $B$ is a fixed finite symmetric set of generators for
$G$). For each subset $N\subseteq G$, identify $X^{(N)}$ with a sub-coproduct
of $X^{(G)}$ and note that $X^{(G)}=\bigcup_{n\in\mathbb{N}}X^{N_{n}(B)}$ is
an increasing direct union. Therefore,
$\ker(\phi)=\bigcup_{n\in\mathbb{N}}(X^{N_{n}(B)}\cap\ker(\phi))$ and so there
exists a minimal positive integer $r_{2}$ such that
$X^{N_{n}(B)}\cap\ker(\phi)\neq 0$ for all $n\geq r_{2}$. We let

  * •

$r_{0}:=\max\\{r_{1},r_{2}\\}$;

  * •

and we choose $\varepsilon\in(0,1)$ such that $\varepsilon\lneq
1/(2\,\ell(X)|N_{2r_{0}+1}(B)|)$.

The reason for this specific choice of $\varepsilon$ will be clear at the end
of the proof. Note that, by the choice of $r_{0}$ and Proposition 2.7, we have
the following relation:

(2) $\sum_{g\in N_{r_{0}}(B)}\phi_{gh,1}\psi_{1,g}=\sum_{g\in
N_{r_{0}}(B)}\phi_{h,g^{-1}}\psi_{g^{-1},1}=\sum_{g\in
G}\phi_{h,g}\psi_{g,1}=\begin{cases}\mathrm{id}_{X}&\text{if $h=1$;}\\\
0&\text{otherwise.}\end{cases}$

Step 2. Reduction to objects of finite length. By definition of sofic group,
there exists a finite $B$-labeled directed graph $\Gamma=(V,E)$, and a subset
$V_{0}\subseteq V$ such that

  * •

$|V_{0}|\geq(1-\varepsilon)|V|$;

  * •

there is an isomorphisms of labeled digraphs $N_{2r_{0}+1}(v)\cong
N_{2r_{0}+1}(B)$, for all $v\in V_{0}$.

Consider the following two subsets of $V$:

$V^{\prime}:=\\{v\in V:\text{ $N_{r_{0}}(v)\cong
N_{r_{0}}(B)$}\\}\qquad\text{and}\qquad V^{\prime\prime}:=\\{v\in
V^{\prime}:\text{ $N_{r_{0}}(v)\subseteq V^{\prime}$}\\}.$

Note that $V_{0}\subseteq V^{\prime\prime}\subseteq V^{\prime}\subseteq V$.
For each $v^{\prime}\in V^{\prime}$, let $\varepsilon_{v^{\prime}}\colon
N_{r_{0}}(v^{\prime})\to N_{r_{0}}(B)$ be a fixed isomorphism of $B$-labeled
digraphs and define two morphism

$\bar{\phi}:=(\bar{\phi}_{v,v^{\prime}})_{V\times V^{\prime}}\colon
X^{(V^{\prime})}\to
X^{(V)}\qquad\text{and}\qquad\bar{\psi}:=(\bar{\psi}_{v^{\prime},v^{\prime\prime}})_{V^{\prime}\times
V^{\prime\prime}}\colon X^{(V^{\prime\prime})}\to
X^{(V^{\prime})},\quad\text{by}$
$\bar{\phi}_{v,v^{\prime}}:=\begin{cases}\phi_{\varepsilon_{v^{\prime}}(v),1}&\text{if
$v\in N_{r_{0}}(v^{\prime})$;}\\\
0&\text{otherwise;}\end{cases}\quad\text{and}\quad\bar{\psi}_{v^{\prime},v^{\prime\prime}}:=\begin{cases}\psi_{1,\varepsilon_{v^{\prime}}(v^{\prime\prime})}&\text{if
$v^{\prime\prime}\in N_{r_{0}}(v^{\prime})$;}\\\
0&\text{otherwise.}\end{cases}$

By (2) and the definition of $\bar{\phi}$ and $\bar{\psi}$, we have that,
given $v_{1},\,v_{2}\in V^{\prime\prime}$,

$\displaystyle\sum_{v^{\prime}\in
V^{\prime}}\bar{\phi}_{v_{2},v^{\prime}}\bar{\psi}_{v^{\prime},v_{1}}$
$\displaystyle=\sum_{v^{\prime}\in N_{r_{0}}(v_{1})\cap
N_{r_{0}}(v_{2})}\phi_{\varepsilon_{v^{\prime}}(v_{2}),1}\psi_{1,\varepsilon_{v^{\prime}}(v_{1})}=\begin{cases}\mathrm{id}_{X}&\text{if
$v_{1}=v_{2}$;}\\\ 0&\text{otherwise.}\end{cases}$

In particular, $X^{(V_{0})}\leq
X^{(V^{\prime\prime})}\leq\mathrm{Im}(\bar{\phi})$.

Step 3. Where the contradiction comes. Taking the composition lengths, we
obtain the following lower bound for $\ell(\mathrm{Im}(\bar{\phi}))$

$(1-\varepsilon)|V|\ell(X)\leq|V_{0}|\ell(X)=\ell(X^{(V_{0})})\leq\ell(\mathrm{Im}(\bar{\phi}))\,.$

The contradiction will follow by showing that
$\ell(\mathrm{Im}(\bar{\phi}))\lneq(1-\varepsilon)|V|\ell(X)$. Choose a subset
$V_{1}\subseteq V^{\prime\prime}\subseteq V$ as in Lemma 2.2 (this is possible
because of the choice of $\varepsilon\leq 1/2$ that implies that more than
half of the elements of $V$ actually belong in $V_{0}$). By the definition of
$\bar{\phi}$, for all $v\in V_{1}$, there is a commutative diagram of the form

$\textstyle{X^{(N_{r_{0}}(v))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\cong}$$\textstyle{X^{(N_{2r_{0}}(v))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{X^{(N_{r_{0}}(B))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{X^{(N_{2r_{0}}(B))}.}$

By the choice of $r_{0}$, the restriction of $\phi$ to $X^{(N_{r_{0}}(B))}$ is
not a monomorphism, and so the restriction of $\bar{\phi}$ to
$X^{(N_{r_{0}}(v))}$ cannot be monic for any $v\in V_{1}$. In particular,

(3)
$\ell\left(\bar{\phi}\left(X^{(N_{r_{0}}(v))}\right)\right)\leq\ell\left(X^{(N_{r_{0}}(v))}\right)-1=|N_{r_{0}}\left(B\right)|\ell(X)-1.$

Let $N_{r_{0}}(V_{1}):=\bigcup_{v\in V_{1}}N_{r_{0}}(v)\subseteq V^{\prime}$,
so that $X^{(V^{\prime})}\cong X^{(N_{r_{0}}(V_{1}))}\oplus
X^{(V^{\prime}\setminus N_{r_{0}}(V_{1}))}$. Thus,

$\displaystyle\ell\left(\mathrm{Im}\left(\bar{\phi}\right)\right)$
$\displaystyle=\ell\left(\bar{\phi}\left(X^{(V^{\prime})}\right)\right)\leq\ell\left(\bar{\phi}\left(X^{(N_{r_{0}}(V_{1}))}\right)\right)+\ell\left(\bar{\phi}\left(X^{(V^{\prime}\setminus
N_{r_{0}}(V_{1}))}\right)\right)$ $\displaystyle\leq\sum_{v\in
V_{1}}\ell\left(\bar{\phi}\left(X^{(N_{r_{0}}(v))}\right)\right)+\ell\left(X\right)\left(|V^{\prime}|-|V_{1}||N_{r_{0}}\left(B\right)|\right)$
$\displaystyle\leq|V_{1}|\left(|N_{r_{0}}(B)|\ell(X)-1\right)+\ell(X)\left(|V^{\prime}|-|V_{1}||N_{r_{0}}\left(B\right)|\right)$
$\displaystyle\leq\ell(X)|V|-|V|/(2|N_{2r_{0}+1}(B)|)$
$\displaystyle\lneq|V|\ell(X)\left(1-\varepsilon\right),$

where the second line comes by the fact that $|V^{\prime}\setminus
N_{r_{0}}(V_{1})|=|V^{\prime}|-|V_{1}||N_{r_{0}}(B)|$, the third line is
obtained applying (3) and the fourth line follows by recalling that
$|V^{\prime}|\leq|V|$ and the estimate in Lemma 2.2(1). The final strict
inequality then comes from the choice of $\varepsilon$ and it is the
contradiction we were looking for. ∎

###  3.2. The general result for Grothendieck categories

At this point, we know that, given a Grothendieck category $\mathcal{G}$, a
finitely generated sofic group $G$, and an object $N\in\mathcal{G}$, the
endomorphism ring of $\uparrow_{\mathbf{1}}^{G}N$ is directly finite as far as
$N$ is of finite length. In the following corollary we show that, in fact,
this last hypothesis can be weakened a lot, as it is enough to assume that $N$
is Noetherian.

######  Corollary 3.2.

Let $\mathcal{G}$ be a Grothendieck category, $G$ a finitely generated sofic
group, $N\in\mathcal{G}$ a Noetherian object and
${}_{G}M:=\uparrow^{G}_{\mathbf{1}}N$. Then,
$\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ is directly finite.

###### Proof.

Let $\phi$, $\psi\in\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ such that
$\phi\circ\psi=\mathrm{id}_{M}$, let ${}_{G}K:=\ker(\phi)\in\mathcal{G}^{G}$
and suppose, looking for a contradiction, that $K\neq 0$. By Corollary 2.9,
there is an ordinal $\alpha$ such that
$\bar{K}:=\mathbf{Q}^{G}_{\alpha}(\mathbf{T}^{G}_{\alpha+1}(K))\neq 0$. Let
now $\bar{\mathcal{G}}:=\mathcal{G}_{\alpha+1}/\mathcal{G}_{\alpha}$, that is
semi-Artinian,
$\bar{N}:=\mathbf{Q}_{\alpha}(\mathbf{T}_{\alpha+1}(N))\in\bar{\mathcal{G}}$,
that is of finite length (being both Noetherian and semi-Artinian) and
$\bar{M}:=\mathbf{Q}_{\alpha}^{G}(\mathbf{T}^{G}_{\alpha+1}(M))\cong\uparrow^{G}_{\mathbf{1}}\bar{N}\in\bar{\mathcal{G}}^{G}$.
Consider also the two morphisms
$\bar{\phi}:=\mathbf{Q}_{\alpha}^{G}(\mathbf{T}^{G}_{\alpha+1}(\phi))$ and
$\bar{\psi}:=\mathbf{Q}_{\alpha}^{G}(\mathbf{T}^{G}_{\alpha+1}(\psi))\in\mathrm{End}_{\bar{\mathcal{G}}^{G}}({}_{G}\bar{M})$
and note that ${}_{G}\bar{K}=\ker(\bar{\phi})$ in $\bar{\mathcal{G}}$. Since
$\bar{\mathcal{G}}$ is semi-Artinian, Proposition 3.1 tell us that
$\bar{K}=0$, which is a contradiction. ∎

In the above corollary we have weakened the hypotheses on the base object $N$.
In the following corollary we show that also the hypotheses on the acting
group $G$ can be weakened, in fact, we can take $G$ to be an arbitrary sofic
group.

######  Corollary 3.3.

Let $\mathcal{G}$ be a Grothendieck category, $G$ an arbitrary sofic group
(not necessarily finitely generated), $N\in\mathcal{G}$ a Noetherian object
and ${}_{G}M:=\uparrow^{G}_{\mathbf{1}}N$. Then,
$\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ is directly finite.

###### Proof.

Suppose that we have two maps
$\phi,\,\psi\in\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ such that
$\phi\circ\psi=\mathrm{id}_{M}$. We suppose, looking for a contradiction, that
$\ker(\phi)\neq 0$. Since $\mathcal{G}$ is a Grothendieck category,
$\ker(\phi)=\sum\\{\ker(\phi)\cap N^{(F)}:F\subseteq G\text{ finite}\\}$, so
we can find a finite subset $F\subseteq G$ such that $\ker(\phi)\cap
N^{(F)}\neq 0$. Consider the following finite subset of $G$:

$S:=F\cup\\{g\in G:\phi_{g,1}\neq 0\\}\cup\\{g\in G:\psi_{g,1}\neq 0\\},$

and let $H\leq G$ be the subgroup generated by $S$, which is clearly finitely
generated and sofic. We claim that $N^{(H)}$ is both a $\phi$\- and a
$\psi$-invariant subobject of $M$; to see this, consider $h\in H$, $g\in G$,
and suppose that $\phi_{g,h}\neq 0$. Since $\phi$ is $G$-equivariant,
$\phi_{g,h}=\phi_{h^{-1}g,1}$, so $h^{-1}g\in S\subseteq H$ that, together
with the fact that $h\in H$, implies that $g=h(h^{-1}g)\in H$; the argument
for $\psi$ is completely analogous. We can then consider
$M^{\prime}:=\uparrow_{\mathbf{1}}^{H}N$ and the two maps
$\phi^{\prime}:=\phi_{\restriction M^{\prime}}$ and
$\psi^{\prime}:=\psi_{\restriction M^{\prime}}$, that are clearly
$H$-equivariant, that is, they are morphisms in $\mathcal{G}^{H}$. Finally,
note that $\phi^{\prime}\circ\psi^{\prime}=\mathrm{id}_{M^{\prime}}$ and
$0\neq\ker(\phi)\cap N^{(F)}\leq\ker(\phi^{\prime})$, and this is absurd by
Corollary 3.2. ∎

Finally, let us show that the above statements about direct finiteness can be
easily reformulated in terms of stable finiteness.

######  Corollary 3.4.

Let $\mathcal{G}$ be a Grothendieck category, $G$ a sofic group,
$N\in\mathcal{G}$ a Noetherian object and
${}_{G}M:=\uparrow^{G}_{\mathbf{1}}N$. Then,
$R:=\mathrm{End}_{\mathcal{G}^{G}}({}_{G}M)$ is stably finite.

###### Proof.

Let $k\in\mathbb{N}_{>0}$ and consider the following isomorphism:

$\mathrm{Mat}_{k}(R)\cong\mathrm{End}_{\mathcal{G}^{G}}(\uparrow_{\mathbf{1}}^{G}(N^{(k)})),$

where $N^{k}$ is still a Noetherian object. Hence, to verify that $R$ is
stably finite, that is, that $\mathrm{Mat}_{k}(R)$ is directly finite for each
$k\in\mathbb{N}_{>0}$, it is enough to verify that
$\mathrm{End}_{\mathcal{G}^{G}}(\uparrow_{\mathbf{1}}^{G}X)$ is directly
finite for each Noetherian object $X\in\mathcal{G}$, and we already know this
by Corollary 3.3. ∎

To conclude, let us reformulate the above results in the special case when the
category $\mathcal{G}=R\text{-}\mathrm{Mod}$ is the category of left
$R$-modules for some ring $R$.

######  Corollary 3.5.

Let $R$ be a ring, $G$ a sofic group and ${}_{R}N$ a Noetherian left
$R$-module. Then, $\mathrm{End}_{R[G]}(R[G]\otimes_{R}N)$ is a stably finite
ring. In particular, if $R$ is a left Noetherian ring, then
$R[G]\cong\mathrm{End}_{R[G]}(R[G]\otimes_{R}R)$ is stably finite.

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  * [13] S. Virili, “Algebraic entropy of amenable group actions”, Mathematische Zeitschrift 291 n.3 (2019): 1389–1417.
  * [14] B. Weiss, “Sofic groups and dynamical systems”, Sankhyā: The Indian Journal of Statistics, Series A (2000): 350–359.


††institutetext: Department of Physics, National University of Singapore,
2 Science Drive 3, 117551, Singapore

# Comment on “Flavor invariants and renormalization- group equations in the
leptonic sector with massive Majorana neutrinos”

Jianlong Lu<EMAIL_ADDRESS>
###### Abstract

Recently in [_JHEP_ 09 (2021) 053], Wang et al. discussed the polynomial ring
formed by flavor invariants in the leptonic sector with massive Majorana
neutrinos. They have explicitly constructed the finite generating sets of the
polynomial rings for both two-generation scenario and three-generation
scenario. However, Wang et al.’s claim of the finiteness of the generating
sets of the polynomial rings and their calculation by the approach of Hilbert
series with generalized Molien-Weyl formula are both based on their assertion
that the unitary group ${\rm U}(n,\mathbb{C})$ is reductive, which is
unfortunately incorrect. The property of being reductive is only applicable to
linear algebraic groups. And it is well-known that the unitary group ${\rm
U}(n,\mathbb{C})$ is not even a linear algebraic group. In this paper, we
point out the above issue and provide a solution to fill in the accompanying
logical gaps in [_JHEP_ 09 (2021) 053]. Some important results from the theory
of linear algebraic group, the invariant theory of square matrices and group
theory are needed in the analysis. We also clarify some somewhat misleading or
vague statements in [_JHEP_ 09 (2021) 053] about the scope of flavor
invariants. Note that, although built from incorrect assertion, Wang et al.’s
calculation results in [_JHEP_ 09 (2021) 053] are nonetheless correct, which
is ultimately because the ring of invariants of ${\rm U}(n,\mathbb{C})$ is
isomorphic to that of ${\rm GL}(n,\mathbb{C})$ which is itself reductive.

###### Keywords:

Flavor Invariants, Linear Algebraic Groups, Invariant Theory, Majorana
Neutrinos

††arxiv: 2110.08210

##  1 Introduction

The physical framework of our discussion is encoded in the following effective
Lagrangian density for leptons,

$\displaystyle\mathcal{L}_{{\rm lepton}}=-\overline{l_{{\rm L}}}M_{l}l_{{\rm
R}}-\frac{1}{2}\overline{\nu_{{\rm L}}}M_{\nu}\nu_{{\rm L}}^{{\rm
C}}+\frac{g}{\sqrt{2}}\overline{l_{{\rm L}}}\gamma^{\mu}\nu_{{\rm
L}}W_{\mu}^{-}+{{\rm h.c.}}\ ,$ (1)

where $M_{l}$ and $M_{\nu}$ are the charged lepton mass matrix and the
Majorana neutrino mass matrix respectively. These two mass matrices can be
diagonalized by three unitary matrices $\\{V_{l},V_{l}^{\prime},V_{\nu}\\}$
via

$\displaystyle\hat{M}_{l}$
$\displaystyle=V_{l}^{\dagger}M_{l}V_{l}^{\prime}={\rm
Diag}\\{m_{e},m_{\mu},m_{\tau}\\},$ (2) $\displaystyle\hat{M}_{\nu}$
$\displaystyle=V_{\nu}^{\dagger}M_{\nu}V_{\nu}^{*}={\rm
Diag}\\{m_{1},m_{2},m_{3}\\}.$ (3)

From $V_{l}$ and $V_{\nu}$, one has the Pontecorvo-Maki-Nakagawa-Sakata matrix
pon ; mns :

$\displaystyle V=V_{l}^{\dagger}V_{\nu}.$ (4)

The two mass matrices $M_{l}$ and $M_{\nu}$ are not entirely physical, in the
sense that one can perform the following transformations without altering the
lepton Lagrangian density $\mathcal{L}_{{\rm lepton}}$,

$\displaystyle l_{{\rm L}}\rightarrow l_{{\rm L}}^{\prime}=U_{{\rm L}}l_{{\rm
L}},\quad\nu_{{\rm L}}\rightarrow\nu_{{\rm L}}^{\prime}=U_{{\rm L}}l_{{\rm
L}},\quad l_{{\rm R}}\rightarrow l_{{\rm R}}^{\prime}=U_{{\rm R}}l_{{\rm R}},$
(5) $\displaystyle M_{l}\rightarrow M_{l}^{\prime}=U_{{\rm L}}M_{l}U_{{\rm
R}}^{\dagger},\quad M_{\nu}\rightarrow M_{\nu}^{\prime}=U_{{\rm
L}}M_{\nu}U_{{\rm L}}^{{\rm T}},$ (6)

where $U_{{\rm L}}$ and $U_{{\rm R}}$ are arbitrary $3\times 3$ unitary
matrices. One can easily eliminate the troublesome factors $U_{{\rm
R}}^{\dagger}$ and $U_{{\rm L}}^{{\rm T}}$ by defining the following Hermitian
matrices (with $n\in\mathbb{N}^{+}$)

$\displaystyle H_{l}\equiv M_{l}M_{l}^{\dagger},\quad H_{\nu}\equiv
M_{\nu}M_{\nu}^{\dagger},\quad G_{l\nu}^{(n)}\equiv
M_{\nu}(H_{l}^{*})^{n}M_{\nu}^{\dagger},$ (7)

whose transformation rules are easily obtained:

$\displaystyle H_{l}\rightarrow H_{l}^{\prime}=U_{{\rm L}}H_{l}U_{{\rm
L}}^{\dagger},\quad H_{\nu}\rightarrow H_{\nu}^{\prime}=U_{{\rm
L}}H_{\nu}U_{{\rm L}}^{\dagger},\quad G_{l\nu}^{(n)}\rightarrow
G_{l\nu}^{(n)}\ {}^{\prime}=U_{{\rm L}}G_{l\nu}^{(n)}U_{{\rm L}}^{\dagger}.$
(8)

The satisfying layout of $U_{{\rm L}}$ and $U_{{\rm L}}^{\dagger}$ in the
above transformation rules naturally motivates us to write down a class of
flavor invariants, ${\rm Tr}(A_{1}A_{2}...A_{k})$, in which
$A_{1},A_{2},...,A_{k}\in\\{H_{l},H_{\nu},G_{l\nu}^{(n)}\\}$ with some
positive integer $k$. Of course, such form is not a compulsory requirement for
flavor invariants of $\\{H_{l},H_{\nu},G_{l\nu}^{(n)}\\}$. For example, it is
straightforward to prove that ${\rm Det}(G_{l\nu}^{(1)}+H_{l}H_{\nu})$ is also
a flavor invariant. Although $n$ can be any positive integer, the Cayley-
Hamilton theorem tells us that we only need to focus on $n<N$ when dealing
with $N$-generation scenario.
In jar1 ; jar2 , Jarlskog has given an invariants named after her, which in
three-generation scenario can be written as
$\mathcal{J}=\cos\theta_{12}\sin\theta_{12}\cos^{2}\theta_{13}\sin\theta_{13}\cos\theta_{23}\sin\theta_{23}\sin\delta$
with parameters in the so-called standard parametrization of mixing matrix.
One can use invariants under flavor basis transformations to construct
sufficient and necessary conditions for CP conservation in the leptonic sector
within specific ranges of parameters branco ; yz1 ; lu ; yz2 . In jm , Jenkins
and Manohar analyzed the flavor invariants from the perspective of invariant
theory. In wyz , Wang et al. claimed that the rings of flavor invariants in
the leptonic sector with two and three generations of charged leptons and
massive Majorana neutrinos are both finitely generated. For both two-
generation and three-generation scenarios, they calculated the numbers of
elements in the generating sets, which were also explicitly constructed in wyz
. However, Wang et al.’s claim of the finiteness of the generating sets and
their calculation by the approach of Hilbert series with generalized Molien-
Weyl formula are both based on their assertion that the unitary group ${\rm
U}(n,\mathbb{C})$ is reductive. This assertion is obviously incorrect, since
${\rm U}(n,\mathbb{C})$ is not even a linear algebraic group. The property of
being reductive is only applicable to linear algebraic groups. Such serious
gaps in wyz  imply that the correctness of their results is unjustified. It is
the aim of this paper to fill in those gaps of wyz .
The remaining part of this paper is organized as follows. In section 2, we
briefly talk about the notions of linear algebraic group and reductive group.
In section 3, we present some relevant results in the invariant theory of
square matrices, especially for the unitary group ${\rm U}(n,\mathbb{C})$ and
the general linear group ${\rm GL}(n,\mathbb{C})$. Eventually, in section 4,
we fill in the logical gaps of wyz . In section 5, we clarify some somewhat
misleading or vague statements in wyz  about the scope of flavor invariants.
In section 6, we give some remarks on the relation between ${\rm
U}(n,\mathbb{C})$ and ${\rm GL}(n,\mathbb{C})$. The readers who are familiar
with basic knowledges of algebraic geometry, linear algebraic groups and
invariant theory may skip directly to section 4 and section 5.

##  2 Linear algebraic groups and reductive groups

To fully understand the notion of linear algebraic group requires some
knowledges of algebraic geometry. However, it is practically impossible and
unnecessary to make this section a comprehensive survey of algebraic geometry
or even if only linear algebraic group. Thus only the results needed for our
reasoning are presented. For more systematic discussion of linear algebraic
group, interested readers are recommended to refer to borel ; gtm21 ; springer
and the third part of mac .
First we would like to give the definition of linear algebraic group (see for
example section 1 in chapter I of borel , section 7 in chapter II of gtm21 ,
chapter 2 of springer  and appendix A of derksen ):

###### Definition 1

An algebraic group is an algebraic variety $G$ equipped with morphisms of
varieties $\mu:G\times G\rightarrow G$ ($\mu(x,y)=xy$) and $\iota:G\rightarrow
G$ ($\iota(x)=x^{-1}$) that give $G$ the structure of a group. If the
underlying variety is affine, then $G$ is called a linear algebraic group.

To be more accessible, a linear algebraic group is a subgroup of the general
linear group ${\rm GL}(n,K)$ over some algebraically closed field $K$, defined
by polynomial equations. Here we would like to emphasize the keyword
“polynomial”. Some common examples of linear algebraic group are as follows:

  * •

General linear group ${\rm GL}(n,K)$: the space of all $n\times n$ matrices
with entries in $K$ and nonzero determinant. This example is not totally
trivial, since the definition of ${\rm GL}(n,K)$ is usually formulated with
inequality instead of polynomial equation. This issue can be easily dealt with
by matching the element $(x_{ij})$ of ${\rm GL}(n,K)$ with the point
$(x_{ij},z)\in K^{n^{2}+1}$ satisfying $z\det(x_{ij})=1$ mac . Note that here
we have employed the existence and uniqueness of multiplicative inverse of any
element in the field $K$.

  * •

Special linear group ${\rm SL}(n,K)$: the space of all $x$ in ${\rm GL}(n,K)$
satisfying ${\rm Det}(x)=1$.

  * •

Orthogonal group ${\rm O}(n,K)$: the space of all $x$ in ${\rm GL}(n,K)$
satisfying $x^{\rm T}x=I_{n}$.

  * •

Special orthogonal group ${\rm SO}(n,K)$: the intersection of ${\rm SL}(n,K)$
and ${\rm O}(n,K)$.

  * •

Symplectic group ${\rm Sp}(2n,K)$: the space of all $x$ in ${\rm GL}(n,K)$
satisfying $x^{\rm T}jx=j$, in which $j\equiv\begin{pmatrix}0&I_{n}\\\
-I_{n}&0\end{pmatrix}$.

  * •

The group of upper triangular matrices: the space of all $x$ in ${\rm
GL}(n,K)$ satisfying $x_{ij}=0$ for $i>j$.

  * •

The group of upper unipotent matrices: the space of all $x$ in ${\rm GL}(n,K)$
satisfying $x_{ij}=0$ for $i>j$ and $x_{kk}=1$ for $1\leq k\leq n$.

One can see that all of the above examples are defined by polynomial equations
of matrix entries. For comparison, one may think about the unitary group ${\rm
U}(n,\mathbb{C})$, which is a typical non-example of linear algebraic group.
Recall that ${\rm U}(n,\mathbb{C})$ is defined to include any element in ${\rm
GL}(n,\mathbb{C})$ whose inverse is its conjugate transpose. It is
straightforward to see that the unitary group is not a linear algebraic group,
since the definition of unitary group relies on the _complex conjugate_ of
matrix entries, which is not something polynomials can handle. This fact is
also explicitly mentioned by Procesi in section 11 of procesi .
Another concept worth mentioning is the unipotent radical of a linear
algebraic group $G$, which is defined as follows (see for example section 11
in chapter IV of borel , section 19.5 in chapter VII of gtm21 , chapter 6 of
springer  and section 2.2.2 and appendix A.3 of derksen ):

###### Definition 2

The unipotent radical $R_{u}(G)$ of $G$ is the unique maximal connected
unipotent normal subgroup of $G$.

The unipotent radical $R_{u}(G)$ of $G$ contains all unipotent elements in the
radical of $G$. For example, the radical of ${\rm GL}(n,K)$ contains all
$n\times n$ scalar matrices with entries from $K$. Among these scalar
matrices, only the identity matrix is unipotent due to its exclusive
characteristic polynomial $(x-1)^{n}=0$. Therefore the unipotent radical of
${\rm GL}(n,K)$ is said to be trivial.
With the notion of unipotent radical, one is ready to define reductive group
(see for example section 11 in chapter IV of borel , section 19.5 in chapter
VII of gtm21 , chapter 6 of springer  and appendix A of derksen ):

###### Definition 3

A linear algebraic group $G$ is said to be reductive if its unipotent radical
$R_{u}(G)$ is trivial, i.e., $R_{u}(G)=\\{e\\}$.

As mentioned above, ${\rm GL}(n,K)$ has trivial unipotent radical, thus being
reductive. There are some other common linear algebraic groups which are also
reductive, such as special linear group ${\rm SL}(n,K)$, orthogonal group
${\rm O}(n,K)$, special orthogonal group ${\rm SO}(n,K)$ and symplectic group
${\rm Sp}(2n,K)$. One simple non-example of reductive group is the subgroup of
${\rm GL}(n,K)$ consisting of all upper triangular matrices, sometimes denoted
by $B(n,K)$, with $n\geq 2$. The unipotent radical of $B(n,K)$ is one of its
subgroups, whose elements all have diagonal entries equal to $1$.
At the end of this section, we would like to remark that there exist at least
three distinct but related concepts, which are “reductive group” (sometimes
called “group theoretically reductive group”), “linear reductive group” and
“geometrically reductive group”.
Reductive group and linear reductive group are linked by the following theorem
nagata :

###### Theorem 1

If ${\rm char}(K)=0$, then a linear algebraic group is reductive if and only
if it is linearly reductive.

On the other hand, two directions of the following theorem have been proved in
nagata  and haboush  respectively.

###### Theorem 2

For any characteristic of $K$, a group is geometrically reductive if and only
if it is reductive.

For our discussion of flavor invariants in which $K=\mathbb{C}$, we only need
to focus on reductive group since ${\rm char}(\mathbb{C})=0$. Thus the
definitions of linear reductive group and geometrically reductive group are
omitted in this paper. Interested readers may refer to section 2.2.1 and
section 2.2.2 of derksen .
For the benefit of the readers, we would like to remark that, in the vast
literature of mathematics, the phrase “reductive group” is used by some
mathematicians in different circumstances and contexts with different
meanings. For example, in Dieudonné and Carrell’s carrell , ${\rm
U}(n,\mathbb{C})$, as a compact Lie group with all finite dimensional linear
representations being completely reducible, is said to be “reductive”.
However, in this case, the “reductiveness” of ${\rm U}(n,\mathbb{C})$ is no
longer a sufficient condition for the relevant ring of invariants to be
finitely generated over $\mathbb{C}$ for rational group action. As explicitly
expressed in the corollary of the theorem in section 1 of chapter 3 in carrell
, there is one more important condition that should also be included to ensure
the finiteness. We restate it here for the convenience of interested readers.
Let $R=K[a_{1},...,a_{n}]$ be a finitely generated algebra over an arbitrary
field $K$ (here we are interested in the case $K=\mathbb{C}$) and $\Gamma$ be
a group of algebra automorphisms of $R$ (here we are interested in the case
$\Gamma={\rm U}(n,\mathbb{C})$). Then the condition needed for the subalgebra
$I_{\Gamma}\subset R$ (here we are interested in the ring of ${\rm
U}(n,\mathbb{C})$ invariants of $\\{H_{l},H_{\nu},G_{l\nu}^{(k)}\\}$) to be
finitely generated over $K$ is that the orbit under $\Gamma$ of each $f\in R$
is contained in a finite dimensional subspace of $R$ over $K$ carrell . This
condition is not a trivial one and cannot be directly absorbed in the
condition that every finite dimensional rational representation of the group
of our interests (for example ${\rm U}(n,\mathbb{C})$) is completely
reducible. One can also find the relevant description in section 1 of Nagata’s
famous nagata2 . Needless to say, it is not appropriate to choose $R$ to be
the ring of ${\rm U}(n,\mathbb{C})$ invariants of
$\\{H_{l},H_{\nu},G_{l\nu}^{(k)}\\}$, otherwise there will be a circular
argument. On the other hand, when we work in the context of algebraic group,
Hilbert’s finiteness theorem can be neatly formulated, as explicitly stated in
the next section. Therefore, with rational group action, when we say that the
(linear) reductiveness of a group can alone imply the finiteness of the
generating set of a specific ring of invariants, it is the (linearly)
reductive group in the context of algebraic group that we are talking about.

##  3 Invariant theory of square matrices

Invariant theory has a long history of at least one and a half century, with a
vast ocean of literature. Two of the most important papers are Hilbert’s
hilbert1  and hilbert2 , which eventually lead to Hilbert’s finiteness
theorem. For the development of Hilbert’s finiteness theorem, Popov has made a
concise summary in his popov , which is directly quoted here for the
convenience of interested readers:

> The first fundamental theorem was proved by Gordan in 1868 for invariants of
> binary forms gordan  (and in 1870 for systems of binary forms gordan2 ). In
> 1890 Hilbert gave a general nonconstructive proof for any system of forms in
> any number of variables hilbert1 . Although in his version he deals with
> special representations of GL, the idea of the proof can be carried over
> directly to the general case. H. Weyl weyl2  accomplished this for any
> representation of a complex semisimple group, and Mumford mum  extended this
> result to the general case of a reductive algebraic group acting regularly
> on an affine algebraic variety over a field $k$ of characteristic zero.
> Finally, through the joint efforts of Nagata nagata2 , Haboush haboush , and
> Mumford mum , the restriction on the characteristic of $k$ was removed (in
> the case ${\rm char}\ k\neq 0$ Hilbert’s approach does not work). In this
> final form the theorem is now called Hilbert’s theorem on invariants.

From the above description one can clearly see the special role of algebraic
groups in Hilbert’s finiteness theorem.
We have no plan to give a detailed review for invariant theory in this
section. Only the results needed for our reasoning are presented. Interested
readers are recommended to refer to procesi2  for more systematic discussion
of invariant theory of matrices.
First we would like to introduce the celebrated Hilbert’s finiteness theorem
(see for example section 2.2.1 of derksen  or section 2 of derksen2 ):

###### Theorem 3

If $G$ is a linearly reductive group and $V$ is a rational representation,
then $K[V]^{G}$ is finitely generated over $K$.

Next we would like to draw readers’ attention to the following theorem (see
section 11 of procesi  for more discussions):

###### Theorem 4

The unitary invariants of $j$ complex matrices $\\{x_{1},...,x_{j}\\}$ are
polynomials in the elements ${\rm Tr}(w_{i_{1}}...w_{i_{s}})$ where
$w_{i}=x_{i}$ or $w_{i}=x_{i}^{\dagger}$. As a ring, it is isomorphic to the
ring of ${\rm GL}(n,\mathbb{C})$ invariants of $2j$ matrices
$\\{x_{1},...,x_{j},x_{1}^{\dagger},...,x_{j}^{\dagger}\\}$.

This theorem serves as a bridge between ${\rm U}(n,\mathbb{C})$ invariants and
${\rm GL}(n,\mathbb{C})$ invariants of square matrices. As mentioned in
section 2, it makes no sense to talk about whether ${\rm U}(n,\mathbb{C})$ is
reductive or not. But ${\rm GL}(n,\mathbb{C})$ is indeed (linear) reductive.
Now with theorem 4, we can play a home game with our familiar (linear)
reductive groups instead of an away game with ${\rm U}(n,\mathbb{C})$.
The last gear worth mentioning is the theorem given below (see section 1 of
procesi ), which is applicable to ${\rm GL}(n,\mathbb{C})$ invariants:

###### Theorem 5

Any ${\rm GL}(n,\mathbb{C})$ invariant of $j$ $n\times n$ matrices
$\\{x_{1},...,x_{j}\\}$ is a polynomial in the invariants ${\rm
Tr}(x_{i_{1}}...x_{i_{s}})$, in which $x_{i_{1}}...x_{i_{s}}$ runs over all
possible noncommutative monomials.

##  4 Filling in the gaps

In section 2, we have talked about the notions of linear algebraic group and
reductive group. Since the unitary group ${\rm U}(n,\mathbb{C})$ is not even a
linear algebraic group, it makes no sense to say that ${\rm U}(n,\mathbb{C})$
is reductive, as wrongly asserted in wyz . Now with all necessary tools in our
hands, we are well prepared to fill in the gaps in wyz . The following
discussion is for $n=3$, i.e., the three-generation scenario. When $n=2$, the
corresponding discussion is almost the same, thus omitted for the sake of
simplicity.
In our discussion of flavor invariants, the relevant algebraically closed
field $K$ is our familiar $\mathbb{C}$ with characteristic $0$. As mentioned
previously, theorem 1 and theorem 2 ensure that three concepts “reductive
group”, “linear reductive group” and “geometrically reductive group” coincide
when $K=\mathbb{C}$.
Making use of theorem 4, one can go over the obstacle caused by the fact that
${\rm U}(3,\mathbb{C})$ is not a reductive group. The unitary invariants of
the Hermitian matrices $\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$ are
polynomials in the elements ${\rm Tr}(w_{i_{1}}...w_{i_{s}})$ where
$w_{i}\in\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$. These invariants
together form a ring, which is isomorphic to the ring of ${\rm
GL}(3,\mathbb{C})$ invariants of the Hermitian matrices
$\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$. In other words, these two
rings are indistinguishable from each other. This is consistent with theorem 5
with $n=3$.
As mentioned in section 2, ${\rm GL}(3,\mathbb{C})$ is reductive. Then
according to theorem 1 and due to ${\rm char}(\mathbb{C})=0$, we know that
${\rm GL}(3,\mathbb{C})$ is also linear reductive. Therefore, theorem 3
immediately implies that the ring of ${\rm GL}(3,\mathbb{C})$ invariants of
the Hermitian matrices $\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$ is
finitely generated over $\mathbb{C}$. Now we can finally say that the ring of
the ${\rm U}(3,\mathbb{C})$ invariants of the Hermitian matrices
$\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$ is finitely generated over
$\mathbb{C}$.
From the above discussion, we can see that the finiteness of the generating
set of the ring of the ${\rm U}(3,\mathbb{C})$ invariants of the Hermitian
matrices $\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$ is not a trivial
result. Hilbert’s finiteness theorem alone is not enough to imply this
conclusion. Wang et al.’s statement in wyz  that “ _This conclusion is a
simple and direct consequence of the general mathematical theorem in the
invariant theory for the reductive group U(N)_ ” is obviously not correct.
The last point we would like to address in this section is the application of
Molien-Weyl formula molien  to calculate Hilbert series in wyz . As repeatedly
mentioned by Derksen and Kemper in derksen , the approach of calculating
Hilbert series for infinite groups using generalized Molien-Weyl formula is
for (linear) reductive groups. Thus Wang et al.’s calculation in section 3,
section 4 and appendix B.3 of wyz  based on incorrect assumption “${\rm
U}(n,\mathbb{C})$ is reductive” is unjustified. Fortunately, this problem is
not very severe, due to the coincidental fact that ${\rm U}(n,\mathbb{C})$ is
the maximal compact subgroup of ${\rm GL}(n,\mathbb{C})$. In the previous
analysis, we have transferred our focus from ${\rm U}(n,\mathbb{C})$
invariants to ${\rm GL}(n,\mathbb{C})$ invariants. Because ${\rm
GL}(n,\mathbb{C})$ is (linear) reductive, we can calculate Hilbert series
using generalized Molien-Weyl formula (see for example section 4.6 of derksen
)

$\displaystyle\mathscr{H}(q)=\int[\mathrm{d}\mu]_{G}\frac{1}{{\rm
Det}(I-qR(g))},$ (9)

in which $G={\rm GL}(n,\mathbb{C})$. Due to the non-compactness of ${\rm
GL}(n,\mathbb{C})$, the integral in equation 9 should be performed on the
maximal compact subgroup of ${\rm GL}(n,\mathbb{C})$ derksen ; weyl , which
happens to be ${\rm U}(n,\mathbb{C})$.

##  5 Some remarks on the scope of flavor invariants

A careful reader may have already noticed a problem: from the dimensionless
nature of the Jarlskog invariant
$\mathcal{J}=\cos\theta_{12}\sin\theta_{12}\cos^{2}\theta_{13}\sin\theta_{13}\cos\theta_{23}\sin\theta_{23}\sin\delta$,
it is obvious that the Jarlskog invariant cannot be written as a polynomial of
the $34$ basic flavor invariants constructed in wyz . Indeed, the Jarlskog
invariant is not the only invariant quantity under the flavor basis
transformations that cannot be expressed as a polynomial of the $34$ basic
flavor invariants constructed in wyz . In the three-generation scenario with
Majorana neutrinos, there are totally $12$ physical observables in the
leptonic sector, including three masses of charged leptons
$\\{m_{e},m_{\mu},m_{\tau}\\}$, three masses of Majorana neutrinos
$\\{m_{1},m_{2},m_{3}\\}$, three mixing angles
$\\{\theta_{12},\theta_{13},\theta_{23}\\}$, one Dirac CP-violating phase
$\\{\delta\\}$ and two Majorana CP-violating phases $\\{\rho,\sigma\\}$. All
of these physical observables are invariants under the flavor basis
transformations. But none of them can be written as a polynomial of the basic
flavor invariants constructed in wyz , which is obvious from their dimensions.
The above observation seems to contradict some relevant statements in wyz  by
Wang et al. themselves, such as “ _Any flavor invariants can be expressed as
the polynomials of those 34 basic invariants in the generating set._ ” This is
due to the sloppy usage of the terminology “flavor invariant” in wyz .
Actually, in the invariant theory of $n\times n$ matrices, the concept of
“invariant” is rigorously defined procesi ; procesi2 . The relation between
the polynomial invariants of a given set of Hermitian matrices under certain
transformation and other quantities invariant under the same transformation is
very easy to understand. In the case of algebraically closed fields with
characteristic $0$ such as $\mathbb{C}$, it has been pointed out as first
fundamental theorem in procesi  and procesi2  that the ring consisting of the
former under ${\rm GL}(n,\mathbb{C})$ is generated by the elements ${\rm
Tr}(M)$ with $M$ being any monomial in the given set of Hermitian matrices.
And we know that the ring consisting of the former under ${\rm
U}(n,\mathbb{C})$ is also generated by the elements ${\rm Tr}(M)$ with $M$
being any monomial in the given set of Hermitian matrices, due to the
isomorphism between the ring of ${\rm U}(n,\mathbb{C})$ invariants of $j$
complex square matrices $\\{x_{1},...,x_{j}\\}$ and the ring of ${\rm
GL}(n,\mathbb{C})$ invariants of $2j$ matrices
$\\{x_{1},...,x_{j},x_{1}^{\dagger},...,x_{j}^{\dagger}\\}$. Then from the
nontrivial mass dimensions of elements in $\\{H_{l},H_{\nu},G_{l\nu}^{(k)}\\}$
and the dimensionless nature of Jarlskog invariant, one can immediately see
that Jarlskog invariant is not a flavor invariants of
$\\{H_{l},H_{\nu},G_{l\nu}^{(k)}\\}$ under unitary transformation and cannot
be expressed as a polynomial of the basic invariants, although Jarlskog
invariant is indeed unchanged under such unitary transformation. Therefore
there exists no contradiction between the fact that the Jarlskog invariant
cannot be written as a polynomial of the $34$ basic flavor invariants
constructed in wyz  and the statement that any flavor invariant can be
expressed as a polynomial of those $34$ basic invariants. That is why we say
that it is somewhat misleading or vague for Wang et al. to call the Jarlskog
invariant a “flavor invariant” in the context of invariant theory of square
matrices. Similar explanations can be applied to those $12$ physical
observables about why they are not invariants of
$\\{H_{l},H_{\nu},G_{l\nu}^{(1)},G_{l\nu}^{(2)}\\}$ under the flavor basis
transformations although being invariant under these transformations.

##  6 Some remarks on the relation between ${\rm U}(n,\mathbb{C})$ and ${\rm
GL}(n,\mathbb{C})$

${\rm U}(n,\mathbb{C})$ is a typical example of Lie groups, and ${\rm
GL}(n,\mathbb{C})$ is a typical example of algebraic groups. Lie groups and
algebraic groups are characterized by different extra structures equipped on
the underlying abstract groups. In lu2 , we have detailedly proved that ${\rm
U}(n,\mathbb{C})$ and ${\rm GL}(n,\mathbb{C})$ are not isomorphic to each
other in both the context of Lie group for arbitrary positive integer $n$ and
the context of abstract group for arbitrary integer $n\geq 3$. There indeed
exist intrinsic differences between ${\rm U}(n,\mathbb{C})$ and ${\rm
GL}(n,\mathbb{C})$. These subtleties set up warning signs when one is trying
to blindly apply conclusions from one to the other.
Nonetheless, one can build a bridge between Lie groups and algebraic groups,
such as complexification. A well-known result is that via complexification of
compact Lie groups there is a one-to-one correspondence between compact Lie
groups (up to differentiable isomorphism) and reductive complex algebraic
groups (up to polynomial isomorphism) (see for example chapter 5 of vinberg ).
One example of such correspondence is ${\rm U}(n,\mathbb{C})\rightarrow{\rm
GL}(n,\mathbb{C})$. It is the complexification of a compact Lie group that
enjoys the nice properties of algebraic groups, instead of the complex Lie
group itself. That is why Bröcker and tom Dieck mentioned in their dieck
about looking at the complexification $G_{\mathbb{C}}$ of a compact Lie group
$G$ as an algebraic group.

##  7 Summary

We point out the logical gaps in [_JHEP_ 09 (2021) 053], which comes from Wang
et al.’s incorrect assertion that ${\rm U}(n,\mathbb{C})$ is reductive. We
detailedly fill in the gaps with some important results from the theory of
linear algebraic group, the invariant theory of square matrices and group
theory. Some remarks on the scope of flavor invariants and the relation
between ${\rm U}(n,\mathbb{C})$ and ${\rm GL}(n,\mathbb{C})$ are also given.

## Acknowledgements

The author would like to thank the anonymous referee for his/her helpful
suggestions.

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