\documentclass[11pt]{amsart}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{bookmark}
\hypersetup{pdftitle={The Benson--Etingof--Ostrik conjecture}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newcommand{\one}{\mathbf1}
\newcommand{\C}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\Q}{\mathcal Q}
\newcommand{\T}{\mathcal T}
\newcommand{\V}{\mathcal V}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Rep}{Rep}
\DeclareMathOperator{\Ver}{Ver}
\renewcommand{\Vec}{\operatorname{Vec}}
\DeclareMathOperator{\sVec}{sVec}
\DeclareMathOperator{\Mod}{Mod}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\coev}{coev}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\gr}{gr}
\numberwithin{equation}{section}
\title[The Benson--Etingof--Ostrik conjecture]
{The Benson--Etingof--Ostrik conjecture}
\date{10 November 2026}

\begin{document}
\begin{abstract}
We prove the conjecture of Benson--Etingof--Ostrik (2023): every symmetric
tensor category of moderate growth over an algebraically closed field of
positive characteristic admits a symmetric tensor functor to a higher
Verlinde category, possibly of infinite height.
Using maximal nilpotence and geometric reductivity, we reduce the problem
to proving that finitely generated maximally nilpotent tensor categories
are finite.
The main ingredients are the construction of generic points using
alternating norms and a Frobenius reduction of the fundamental coend to
an ordinary Hopf subalgebra.
\end{abstract}
\maketitle

\part{All positive characteristics}
\label{part:all}

\section{Main theorem and preliminaries}\label{sec:inputs}

We follow \cite{EGNO} for the basic conventions on tensor categories,
duality, ind-completions, Deligne products, and extension groups.
All tensor categories are symmetric, with \(\End(\one)=k\);
tensor functors are exact, symmetric, and linear over the indicated
ground field.
Unless a different hypothesis is stated, the ground field \(k\) is
algebraically closed of characteristic \(p>0\).

The main theorem resolves the conjecture of Benson, Etingof, and Ostrik
\cite[Conjecture 1.4]{BEO} in full generality.

\begin{theorem}[Benson--Etingof--Ostrik conjecture]\label{thm:main}
Let \(k\) be an algebraically closed field of characteristic \(p>0\).
For every symmetric tensor category \(\C\) over \(k\) of moderate
growth, there exists an exact faithful \(k\)-linear symmetric monoidal
functor
\[
                         F:\C\longrightarrow\Ver_{p^\infty,k}.
\]
If \(\C\) is finitely generated, \(F\) factors through
\(\Ver_{p^r,k}\) for some finite \(r\ge1\); in particular, \(\C\)
fibres over a finite tensor category.
\end{theorem}

Here \(\Ver_{p^r,k}\) is the finite higher Verlinde category of
height \(r\) \cite{BEO}. We omit \(k\) when it is fixed, and put
\(\Ver_{p^\infty,k}=\bigcup_{r\ge1}\Ver_{p^r,k}\) under the standard
tensor embeddings. A fibre functor with target \(\D\) means a tensor
functor to \(\D\).

Theorem~\ref{thm:mn-finite} proves the intermediate assertion that
finitely generated maximally nilpotent (MN) categories are finite.
Moderate growth is automatic for MN categories by
Lemma~\ref{lem:mn-consequences}(4).
The finiteness proof uses the ordinary-coend results established
in Parts~\ref{part:odd} and~\ref{part:two}.
Proposition~\ref{prop:mn-embedding} then passes from finite MN
categories to arbitrary MN categories by constructing a countable
cofinal chain and gluing its embeddings.
Section~\ref{sec:extensions} also proves that every tensor subcategory
of an MN category in odd characteristic is Serre.

\subsection*{Notation specific to this paper}
A tensor subcategory is full and closed under subquotients, finite sums,
tensor products, and duals; it need not be closed under extensions.
Write \(\langle G\rangle\) for the tensor subcategory generated by \(G\).
Thus \emph{finitely generated} means \(\C=\langle G\rangle\), whereas
\emph{finite} has the usual meaning of \cite[Section 1.8]{EGNO}:
equivalence as an abelian category to finite-dimensional modules over a
finite-dimensional algebra. Write \(\ell(X)\) for composition length.
Moderate growth means that for every \(X\) some \(b_X>0\) satisfies
\(\ell(X^{\otimes n})\le b_X^n\) for all \(n\ge1\).

We use \(\one\), \(X^*\), \(\ev_X\), \(\coev_X\), and \(c_{X,Y}\)
for the unit, dual, evaluation, coevaluation, and symmetry.
An invertible object is also called a \emph{line}; it is \emph{even}
if its self-symmetry is the identity. The categories of vector spaces
and super vector spaces are denoted by \(\Vec_k\) and \(\sVec_k\).
An \emph{ordinary} ind-object is a direct sum of copies of \(\one\).
A \emph{finite object} is an object of \(\C\), as opposed to an
arbitrary object of \(\Ind\C\).

Algebras are commutative and unital ind-algebras unless stated otherwise.
We write \(\Sym^nX=(X^{\otimes n})_{S_n}\) for ordinary symmetric
powers and \(\Sym X=\bigoplus_{n\ge0}\Sym^nX\).
An algebra is finitely generated if it is a quotient of \(\Sym X\)
for a finite \(X\); an \(A\)-module is finitely generated if it is
a quotient of \(A\otimes X\) for such an \(X\).
We write \(\Mod_A(\Ind\C)\) for all \(A\)-modules in \(\Ind\C\),
with relative tensor product \(\otimes_A\).

\subsection*{Alternating norms}
For the action of \(S_n\) through the symmetry, put
\[
 a_n^X=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\sigma,
 \qquad A_n(X)=\im a_n^X,\qquad
 \ad(X)=\sup\{n\ge0:A_n(X)\ne0\}.
\]
We call \(a_n^X\) the antisymmetrizer or alternating norm, and
\(\ad(X)\) its top nonzero degree. We use \(A_0(X)=\one\) and
\(\ad(0)=0\). There is no factorial division; in characteristic two
all the signs are \(1\). In positive characteristic, over any field,
moderate growth is equivalent to finiteness of these degrees, and
\[
 \ell(X)\le\ad(X),\qquad
 \ad(X\otimes Y)\le\ad(X)\ad(Y)
\]
\cite[Proposition 4.7(4), Lemma 4.8, and Corollary 4.11]{FE}.
The top nonzero norm image is an even line
(Lemma~\ref{lem:top-line}).

\subsection*{Existence and descent}
A category is \emph{incompressible} if every outgoing tensor functor
embeds it as a tensor subcategory. A tensor functor is \emph{surjective}
if every target object is a subquotient of an image, as in
\cite[Definition 3.1.4]{CEO}. We use the following established results.
\begin{enumerate}
\item Every category of moderate growth over an algebraically closed
field admits a surjective tensor functor to an incompressible category
of moderate growth \cite[Theorem 5.2.1]{CEO}. Its target is finitely
generated if the source is.
\item Every finite tensor category over \(k\) admits a tensor functor
to a finite higher Verlinde category \cite[Theorem B]{CF}, the finite
Benson--Etingof--Ostrik theorem. The Verlinde categories are defined
over the prime field \cite[Section 4.9]{BEO}; their scalar extension
is made explicit in the proof of Theorem~\ref{thm:main}.
\item If \(\V\) is MN and geometrically reductive, then an exact
\(k\)-linear symmetric monoidal functor
\(\C\to\Mod_R(\Ind\V)\), for a nonzero commutative ind-algebra
\(R\), implies the existence of a tensor functor \(\C\to\V\)
\cite[Appendix B, Theorem B.1]{AF}.
\end{enumerate}
MN is defined in Section~\ref{sec:mn}. Geometric reductivity (GR)
means that every epimorphism \(X\twoheadrightarrow\one\) has a split
positive symmetric power. We prove MN implies GR in
Section~\ref{sec:gr}. The Frobenius reduction used for finiteness is
stated at its point of use in Section~\ref{sec:coends}.

We also use Gabriel localization: quotienting a Grothendieck category
by a Serre subcategory closed under coproducts gives an exact quotient
functor with a fully faithful right adjoint
\cite[Section 1.2]{Gabriel}. The next two sections establish the tensor
structure and finiteness properties of our particular quotient.

\section{A highest-norm identity and flatness}

This section and Section~\ref{sec:generic} allow an arbitrary ground field
of positive characteristic, except where stated otherwise.

\begin{lemma}[Parametrized inner inverse]\label{lem:inner-inverse}
Let \(\mathcal M\) be an additive symmetric monoidal category with
additive tensor product. Let \(P,Q\) be dualizable, and
\(0\ne f:P\to Q\). Suppose either \(a_n^P=0\) for all sufficiently
large \(n\), or \(a_n^Q=0\) for all sufficiently large \(n\).
There are a dualizable object \(T\), a nonzero map
\(t:T\to\one\), and \(g:T\otimes Q\to P\) such that
\begin{equation}\label{eq:inner-inverse}
                  fg(\id_T\otimes f)=f(t\otimes\id_P).
\end{equation}
If \(\mathcal M\) is abelian with simple unit, \(t\) is epic.
\end{lemma}
\begin{proof}
Choose the largest \(n\ge1\) for which
\[
 F_n=a_n^Q f^{\otimes n}=f^{\otimes n}a_n^P\ne0.
\]
Put \(T=(Q^{\otimes n})^*\otimes P^{\otimes n}\), and define
\[
 t=\ev_{Q^{\otimes n}}(\id\otimes F_n):T\longrightarrow\one.
\]
The duality bijection
\(\Hom(P^{\otimes n},Q^{\otimes n})
 \simeq\Hom(T,\one)\)
makes \(t\ne0\). This is a matrix-coefficient map, not a trace.

The coset decomposition of \(S_{n+1}\) gives the following integral
identity, with object superscripts suppressed and \((i,n+1)\)
denoting the transposition of positions \(i\) and \(n+1\):
\begin{equation}\label{eq:cosets}
 a_{n+1}=(a_n\otimes\id)
                 \left(\id-\sum_{i=1}^n(i,n+1)\right).
\end{equation}
Apply it to \(f^{\otimes(n+1)}\) and evaluate its first \(n\)
outputs against \((Q^{\otimes n})^*\).
Its left side is zero. The identity term on the right is
\(f(t\otimes\id_P)\).

Here is an explicit factorization of the other terms.
Let
\[
 b_i:P^{\otimes n}\otimes Q\longrightarrow Q^{\otimes n}\otimes P
\]
interchange the \(i\)-th parameter and the last input, apply \(f\)
to the other \(n-1\) parameter factors, and leave the last output
factor unchanged. Define
\[
 g_i=(\ev_{Q^{\otimes n}}\otimes\id_P)
       (\id\otimes a_n^Q\otimes\id_P)
       (\id\otimes b_i).
\]
Naturality of the symmetry gives the explicit identification
\[
 (\ev_{Q^{\otimes n}}\otimes\id_Q)
 \bigl(\id\otimes[(a_n^Q\otimes\id_Q)
                 f^{\otimes(n+1)}(i,n+1)_P]\bigr)
       =fg_i(\id_T\otimes f).
\]
Thus \(g=\sum_i g_i\) gives
\eqref{eq:inner-inverse}. The construction uses neither scalar
points nor factorial division; all signs come from the symmetry.
If the unit is simple, every nonzero map to it is epic.
\end{proof}

In an abelian monoidal category with right exact tensor product,
an object \(M\) is \emph{flat} if \(M\otimes-\) is exact.
A monomorphism is \emph{pure} if it stays monic after tensoring
with any object.

\begin{lemma}[Purity of an image]\label{lem:purity}
Suppose \(\mathcal M\) is abelian symmetric monoidal, tensor product
is right exact, and the unit is simple.
Let \(f:P\to Q\) satisfy Lemma~\ref{lem:inner-inverse}.
Then \(\im(f)\hookrightarrow Q\) remains monic after tensoring with
any object. If \(Q\) is flat, \(\coker f\) is flat.
\end{lemma}
\begin{proof}
Write \(f=j\pi\), where \(\pi:P\twoheadrightarrow I=\im f\)
and \(j:I\hookrightarrow Q\). Set \(r=\pi g\).
Cancel \(j\) in \eqref{eq:inner-inverse}, and descend along the
epimorphism \(\id_T\otimes\pi\). This gives
\begin{equation}\label{eq:purity}
                  r(\id_T\otimes j)=t\otimes\id_I.
\end{equation}
For any \(M\), let \(J=\ker(j\otimes\id_M)\).
Restrict \eqref{eq:purity}, tensored with \(M\), to \(T\otimes J\).
It says that the composite
\[
       T\otimes J\xrightarrow{\,t\otimes\id_J\,}J
                              \hookrightarrow I\otimes M
\]
is zero. The first arrow is epic by right exactness, so \(J=0\).
This proves purity without assuming that \(M\) is flat.

For completeness, let \(M'\hookrightarrow M\) have cokernel \(M''\).
Purity gives exact rows
\[
 0\longrightarrow I\otimes M'\longrightarrow Q\otimes M'
       \longrightarrow(\coker f)\otimes M'\longrightarrow0
\]
and the analogous row for \(M\). Flatness of \(Q\), together with
the monomorphisms from \(I\), makes both left vertical arrows monic.
The snake lemma identifies the possible kernel on the right with
the kernel of \(I\otimes M''\to Q\otimes M''\).
This is zero by purity. Hence \(\coker f\) is flat.
\end{proof}

\begin{remark}\label{rem:presentations}
For an ind-algebra \(A\) in a tensor category, call \(A\otimes X\),
with \(X\) finite, \emph{finite free}. Such objects need not be
projective. Nevertheless every \(A\)-module is a filtered colimit
of cokernels of maps between finite free modules.

Indeed, take a free cover
\(P=\bigoplus_{\alpha}(A\otimes X_\alpha)\twoheadrightarrow M\).
For each finite subsum \(P_J\), its relation module
\(\ker(P_J\to M)\) is the union of images of finite free modules.
Take the cokernels of all finite sets of these relations in all
such \(P_J\). These form a filtered system with colimit \(M\).
Each is a cokernel of a map \(A\otimes X\to A\otimes Y\).
The relevant compactness is just
\[
 \Hom_A(A\otimes X,-)=\Hom_{\Ind\C}(X,-),
\]
which commutes with filtered colimits. It does not assert
projectivity; on the right, the module's \(A\)-action is forgotten.
\end{remark}

\section{The generic category of a domain}\label{sec:generic}

For subobjects \(E,F\subset A\), their product is the image of
\(E\otimes F\to A\); similarly, \(E^n\) denotes the image of
\(E^{\otimes n}\to A\) under multiplication.
An ideal of \(A\) is an \(A\)-submodule of its regular module.
A nonzero algebra is \emph{simple} if its only ideals are
zero and itself.
The ideal generated by \(E\) is \(AE=\im(A\otimes E\to A)\).
It is \emph{finitely generated} when \(E\) can be chosen finite.
For an ideal \(I\) and an \(A\)-module \(M\), write
\(IM=\im(I\otimes_A M\to M)\); \(I\) annihilates \(M\) if
\(IM=0\). We call \(A\ne0\) a
\emph{domain} if the product of any two nonzero ideals is nonzero.
This definition is internal to the tensor category.

\begin{lemma}[Torsion localization]\label{lem:torsion}
Let \(A\) be a domain in \(\Ind\C\). Let \(\T_A\) consist of the
modules that are unions of submodules annihilated by nonzero
finitely generated ideals of \(A\). These are called \emph{torsion}
modules; \emph{torsion free} means having no nonzero torsion
submodule. Then \(\T_A\) is a localizing Serre tensor ideal,
where tensor ideal means closed under tensoring with arbitrary
\(A\)-modules. The Gabriel quotient
\[
 \Q_A=\Mod_A(\Ind\C)/\T_A,\qquad
 q:\Mod_A(\Ind\C)\longrightarrow\Q_A
\]
is a nonzero Grothendieck category with a right exact symmetric
tensor product that is cocontinuous, meaning colimit-preserving,
in each variable. The functor \(q\) is exact and strong
symmetric monoidal, and \(qA\) is its simple tensor unit.
The tensor product on \(\Q_A\) is induced by \(\otimes_A\);
we write it simply as \(\otimes\).
\end{lemma}
\begin{proof}
Products of nonzero finitely generated ideals are again nonzero
and finitely generated. A finite sum of annihilated submodules
is therefore annihilated by one such ideal. Closure under
subobjects, quotients, and sums follows.

For extension closure, take an extension with torsion endpoints
and a finitely generated submodule \(E\) of its middle term.
Its image in the quotient is killed by a nonzero finitely generated
ideal \(I\). Hence \(IE\) lies in the torsion submodule.
The module \(IE\) is finitely generated: tensor finite free covers
of \(I\) and \(E\), and use right exactness.
Thus \(JIE=0\) for another nonzero finitely generated ideal \(J\).
Since every module is the union of its finitely generated
submodules, extension closure follows.
Tensoring an \(I\)-annihilated module with any module preserves
annihilation; colimits give the tensor-ideal assertion.

Tensor descent requires checking kernels, not just cokernels.
Suppose \(j:M'\hookrightarrow M\) has cokernel killed by a
nonzero ideal \(I\). The action \(I\otimes_A M\to M\) factors
through \(M'\). Its restriction to \(I\otimes_A M'\) agrees with
the action on \(M'\): compose with \(j\) and cancel that
monomorphism \emph{before} tensoring further.
For an arbitrary module \(N\), the action on \(M'\otimes_A N\)
therefore factors as
\[
 I\otimes_A M'\otimes_A N
 \longrightarrow I\otimes_A M\otimes_A N
 \longrightarrow M'\otimes_A N.
\]
It kills \(\ker(j\otimes_A\id_N)\).
The cokernel after tensoring is killed by \(I\) as well.
If \(M/M'\) is only locally torsion, write it as a filtered union
of annihilated submodules and take their inverse images in \(M\).
Exactness of filtered colimits expresses the kernel after
tensoring as a union of the just-considered kernels.
For an epimorphism with torsion kernel, right exactness suffices.
Thus tensoring preserves every morphism with torsion kernel and
cokernel, and descends to the Gabriel quotient.

The usual localization theorem gives the Grothendieck category,
the exact quotient functor, and its fully faithful right adjoint
\(q_*\). Tensor preserves colimits in the quotient: lift a diagram
by \(q_*\), compute its colimit before applying \(q\), and use that
both \(q\) and the original relative tensor product preserve
colimits. The symmetry and associativity constraints descend.

There is no nonzero torsion ideal in \(A\): a nonzero finite
subobject of such an ideal would generate a nonzero ideal whose
product with another nonzero ideal is zero. Consequently \(qA\ne0\).
For every nonzero ideal \(I\), choose a nonzero finitely generated
ideal inside \(I\); it kills \(A/I\). Thus \(qI\simeq qA\).
Every subobject \(U\subset qA\) is the image of an ideal of \(A\).
Explicitly, pull back \(q_*U\subset q_*qA\) along \(A\to q_*qA\),
and apply the exact functor \(q\). Hence \(qA\) is simple.
\end{proof}

\begin{theorem}[The generic tensor category]\label{thm:generic}
Suppose every object of \(\C\) has finite alternating norm degree.
For a domain \(A\), tensor product in \(\Q_A\) is exact.
Every object of \(\Q_A\) is a union of finite-length subobjects.
The full finite-length subcategory \(\D_A\) is a tensor category
over the field
\[
                         K_A=\End_{\Q_A}(qA).
\]
Scalars on the nonzero unit give an embedding \(k\hookrightarrow K_A\).
The functor
\[
 F_A:\C\longrightarrow\D_A,\qquad
 X\longmapsto P_X=q(A\otimes X)
\]
is \(k\)-linear, exact, faithful, and strong symmetric monoidal, with
\begin{equation}\label{eq:length-bound}
                           \ell(P_X)\le\ad(X).
\end{equation}
Every object of \(\D_A\) is a quotient of some \(P_X\).
If \(\C\) has moderate growth in positive characteristic,
then so does \(\D_A\).
The field \(K_A\) need not equal \(k\); equality is proved under
the additional hypotheses of Lemma~\ref{lem:point}.
\end{theorem}
\begin{proof}
The \(P_X\) are dualizable, with dual \(P_{X^*}\), and generate
\(\Q_A\) under sums and quotients. Tensoring by a dualizable
object has both adjoints, so the \(P_X\) are flat.
They inherit vanishing of high alternating norms.
For every nonzero map \(f:P_X\to P_Y\),
Lemmas~\ref{lem:inner-inverse} and \ref{lem:purity} show that
\(\coker f\) is flat; this is also true when \(f=0\).
Apply \(q\) to the filtered presentations in
Remark~\ref{rem:presentations}. Every object of \(\Q_A\) is a
filtered colimit of these flat cokernels.
Filtered colimits are exact and commute with tensor product,
so every object is flat. Tensor product is therefore exact.

Tensor products of nonzero objects are nonzero.
A nonzero map \(P_X\to M\) transposes to a nonzero map
\(qA\to M\otimes P_{X^*}\), which is monic because \(qA\) is
simple. Tensoring it with \(N\ne0\) proves \(M\otimes N\ne0\).

We adapt the filtration argument of \cite[Lemma 4.9]{FE} to this
setting, where finite length is not yet known.
Suppose an object \(P\) has a strict finite filtration with \(d\)
nonzero factors \(Q_1,\ldots,Q_d\). Exact tensor product gives
the usual tensor filtration, whose associated graded is
\((\bigoplus_iQ_i)^{\otimes d}\), compatibly with permutations.
On the ordered summand \(Q_1\otimes\cdots\otimes Q_d\),
the composite of the alternating norm and projection back to
that summand is the identity: every other permutation lands
in a different ordered summand. This summand is nonzero.
Thus \(a_d^P\ne0\).
Applied to \(P_X\), this proves \eqref{eq:length-bound}, without
assuming finite length in advance.

Images of finite sums of the \(P_X\) exhaust every object.
They have finite length, proving local finite length.
A finite-length object \(M\) is a quotient of one \(P_Y\);
its kernel is again finite-length and is covered by a \(P_X\).
Hence
\[
                         P_X\longrightarrow P_Y\longrightarrow M
                         \longrightarrow0.
\]
The category is closed monoidal: cocontinuous tensor functors
on a Grothendieck category have right adjoints.
The internal Hom \(\underline{\Hom}(M,N)\) is characterized by
\[
 \Hom(U,\underline{\Hom}(M,N))\simeq\Hom(U\otimes M,N).
\]
Set \(M^\vee=\underline{\Hom}(M,qA)
             =\ker(P_{Y^*}\to P_{X^*})\), a finite-length object.
For \(P=P_X,P_Y\), dualizability makes the canonical map
\[
 \underline{\Hom}(P,qA)\otimes N
                  \longrightarrow\underline{\Hom}(P,N)
\]
an isomorphism. Left exactness of internal Hom in its first variable
and exactness of \(-\otimes N\) identify the corresponding map for
\(M\) with the isomorphism
\[
 M^\vee\otimes N
   \xrightarrow{\sim}\ker(P_{Y^*}\otimes N\to P_{X^*}\otimes N)
   =\underline{\Hom}(M,N).
\]
Invertibility of this canonical map for every \(N\) is equivalent
to dualizability of \(M\): its adjunction unit and counit give
coevaluation and evaluation, with the duality identities.
Thus \(M\) is dualizable with dual \(M^\vee\). Tensor products of finite-length
objects are finite-length, since they are quotients of objects
\(P_X\otimes P_Y=P_{X\otimes Y}\).

The endomorphism ring of the simple unit is a division ring,
and is commutative by the monoidal unit axioms. It is the field
\(K_A\). Independent maps \(qA\to V\) give a monomorphism from
the corresponding finite sum of units. Consequently
\[
 \dim_{K_A}\Hom(qA,V)\le\ell(V),\qquad
 \dim_{K_A}\Hom(M,N)\le\ell(M^\vee\otimes N).
\]
This proves Hom-finiteness. Essential smallness follows because
the \(P_X\) form a set and, up to isomorphism, the quotients of
each object in a Grothendieck category form a set. These establish all the
tensor-category axioms for \(\D_A\).

For \(M\) a quotient of \(P_X\), the norm bounds give
\[
 \ell(M^{\otimes n})\le\ell(P_{X^{\otimes n}})
           \le\ad(X^{\otimes n})\le\ad(X)^n.
\]
This proves moderate growth.
Finally \(F_A\) is exact because free modules and \(q\) are exact.
For \(X\ne0\), coevaluation is monic
\(\one\hookrightarrow X\otimes X^*\); its image under \(F_A\)
prevents \(F_A(X)\) from vanishing.
An exact functor that detects nonzero objects is faithful.
In particular it preserves alternating norm degree exactly.
\end{proof}

\begin{corollary}[Simple commutative algebras]\label{cor:simple-flat}
Let \(\C\) have moderate growth over a field of positive
characteristic. Every simple commutative ind-algebra \(A\) is
\emph{absolutely flat}: tensor product \(\otimes_A\) is exact on
\(\Mod_A(\Ind\C)\). Moreover,
\[
                  \ell_A(A\otimes X)\le\ad(X)\qquad(X\in\C),
\]
where \(\ell_A\) denotes length in the module category.
Thus \(A\) is \emph{artinian} in the terminology of \cite{AF}:
every finite free \(A\)-module has finite length.
\end{corollary}
\begin{proof}
A simple algebra is a domain, and its only nonzero ideal is \(A\).
Consequently \(\T_A=0\), so Theorem~\ref{thm:generic} applies
directly to \(\Q_A=\Mod_A(\Ind\C)\).
\end{proof}

This answers \cite[Questions 3.27--3.28]{AF} for moderate-growth
categories in positive characteristic.

\begin{corollary}[Scalar extension]\label{cor:scalar-extension}
Let \(\C\) have moderate growth in positive characteristic over
a field \(k\), and let \(L/k\) be a field extension.
The finite-length \(L\)-modules in \(\Ind\C\) form a
moderate-growth tensor category \(\C_L\) over \(L\).
The functor \(X\mapsto L\otimes_kX\) is exact, faithful,
strong symmetric monoidal, and surjective. It preserves finite
tensor generation. For \(X,Y\in\C\),
\begin{equation}\label{eq:base-change-hom}
 \Hom_{\C_L}(L\otimes X,L\otimes Y)
                   =L\otimes_k\Hom_\C(X,Y).
\end{equation}
\end{corollary}
\begin{proof}
Regard \(L\) as an ordinary commutative ind-algebra.
Every subobject of a sum of units is again a sum of units, so
its internal ideals are precisely its ordinary ideals. Thus \(L\)
is simple and its torsion subcategory is zero.
Apply Theorem~\ref{thm:generic} with \(A=L\).
Since \(\End_L(L)=\Hom(\one,L)=L\), it gives the tensor category
\(\C_L\) over \(L\), its moderate growth, and all the stated
properties of free extension. In particular, every finite-length
\(L\)-module is a quotient of some \(L\otimes X\); this gives
surjectivity and preservation of finite generation.
Finally, free-module adjunction and compactness of \(X\) give
\eqref{eq:base-change-hom}.
\end{proof}

\begin{lemma}[A generic point over the original field]\label{lem:point}
Let \(k\) be uncountable and algebraically closed of positive
characteristic, and let \(\C\) have moderate growth.
Let \(B\) be a finitely generated commutative ind-algebra, and
let \(E\subset B\) be finite with \(E^n\ne0\) for all \(n\ge1\).
There are a moderate-growth tensor category \(\D\) over \(k\),
a tensor functor \(F:\C\to\D\), and a unital algebra morphism
\[
                           \epsilon:F(B)\longrightarrow\one
\]
whose restriction to \(F(E)\) is nonzero.
Here \(F\) on ind-objects means its colimit extension.
\end{lemma}
\begin{proof}
Choose an ideal \(P\) maximal among those containing none of
the \(E^n\). A chain union still avoids them, since each \(E^n\)
is finite, so Zorn's lemma applies.
If ideals \(I,J\) are not contained in \(P\), maximality gives
\(E^a\subset P+I\) and \(E^b\subset P+J\) for some \(a,b\).
Then \(E^{a+b}\subset P+IJ\), so \(IJ\not\subset P\).
Therefore \(A=B/P\) is a domain.

Write \(\overline E\) for the image of \(E\) in \(A\).
Every nonzero ideal of \(A\) contains some \(\overline E^n\),
again by maximality. Thus the nonzero ideals
\[
                             J_n=A\overline E^n
\]
are a decreasing cofinal family among all nonzero ideals.
The module \(A\) is torsion free. The morphism formula for a
Serre quotient therefore gives
\[
 K_A=\varinjlim_{0\ne I\subset A}\Hom_A(I,A)
                          =\varinjlim_n\Hom_A(J_n,A).
\]
The transition maps are restriction along smaller ideals.
Restriction to generators gives an injection
\[
 \Hom_A(J_n,A)\hookrightarrow\Hom_{\Ind\C}(\overline E^n,A).
\]
Because \(B\), and hence \(A\), is finitely generated, it is a
countable union of finite objects.
Each of these Hom spaces is consequently countable-dimensional,
and so is \(K_A\) over \(k\).
If \(t\in K_A\) were transcendental, the elements
\((t-c)^{-1}\), \(c\in k\), would be linearly independent:
clear denominators in a finite relation and evaluate at each
distinct \(c\). This contradicts countable dimension.
Thus \(K_A/k\) is algebraic and \(K_A=k\).

Use \(\D=\D_A\) and \(F=F_A\) from Theorem~\ref{thm:generic}.
The map \(A\otimes B\to A\), induced by multiplication and
\(B\to A\), gives \(\epsilon\) after applying \(q\).
Its image on \(A\otimes E\) is \(J_1\).
Since \(qJ_1\simeq qA\), the restriction to \(F(E)\) is even epic.
\end{proof}

\section{Maximal nilpotence from generic points}\label{sec:mn}

Throughout this section the ground field is algebraically closed.
For \(X\in\C\), let
\[
 \widetilde X=\ker\bigl(X\longrightarrow
                      \Hom(X,\one)^*\otimes\one\bigr).
\]
The displayed map is the maximal quotient that is a sum of units;
it is epic. A \emph{nilpotent subobject} \(E\subset B\) means that
the multiplication map \(E^{\otimes n}\to B\) vanishes for some \(n\ge1\).

\begin{definition}\label{def:mn}
A category is \emph{maximally nilpotent}, abbreviated MN, if
\(\widetilde X\), in degree one of \(\Sym(X)\), is nilpotent for
every \(X\). Equivalently the image of
\(\Sym(\widetilde X)\to\Sym(X)\) is finite.
This is the maximal-unit-quotient formulation of MN
\cite[Theorem 7.2.2]{CEO}.
\end{definition}

Finite MN categories are incompressible by
\cite[Theorem 7.2.3(2)]{CEO}. The next proposition proves a converse
over an uncountable field without assuming finiteness.

\begin{proposition}\label{prop:incompressible-mn}
Over an uncountable algebraically closed field of positive
characteristic, every incompressible tensor category of moderate
growth is MN.
\end{proposition}
\begin{proof}
If \(\widetilde X\) were nonnilpotent in \(B=\Sym(X)\),
Lemma~\ref{lem:point} would give \(F\) and an algebra map
\(F(B)\to\one\) nonzero on \(F(\widetilde X)\).
Its degree-one restriction is a map \(F(X)\to\one\).
Incompressibility makes \(F\) full, so this map is \(F(v)\)
for some \(v:X\to\one\).
Every such \(v\) vanishes on \(\widetilde X\), a contradiction.
\end{proof}

\begin{lemma}[Consequences of maximal nilpotence]\label{lem:mn-consequences}
In an MN category:
\begin{enumerate}
\item If \(\Hom(X,\one)=0\), then \(\Sym(X)\) is finite.
\item Every tensor functor over the ground field out of the category
is full.
\item Every even invertible object is the unit.
\item The category has moderate growth.
\end{enumerate}
MN passes to tensor subcategories.
\end{lemma}
\begin{proof}
For (1), \(\widetilde X=X\).
For (2), as in \cite[Theorem 7.2.3(1)]{CEO}, let
\(F:\C\to\D\) be a tensor functor and \(\varphi:F(X)\to\one\).
The induced algebra map \(F(\Sym X)\to\one\) shows that the
restriction \(\psi:F(\widetilde X)\to\one\) has a zero tensor
power. A nonzero map to the simple unit is epic and has epic
tensor powers, so \(\psi=0\). Thus \(\varphi\) factors through
\(F(X/\widetilde X)\), a finite sum of units, and lifts to
\(X\to\one\). Duality now gives fullness on all morphism spaces.
For (3), an invertible object is simple. If it is even, every
ordinary symmetric power is its nonzero tensor power.
A nonunit even invertible would contradict (1).
For (4), part (1) gives \(\Sym^nL=0\) for some \(n\) whenever
\(L\) is a nonunit simple. The symmetrizer
\(\sum_{\sigma\in S_n}\sigma\), a nonzero element of \(kS_n\),
then acts by zero on \(L^{\otimes n}\): its action factors through
\(\Sym^nL\). Thus \(kS_n\to\End(L^{\otimes n})\) is not
injective, and the same holds in every larger degree.
By \cite[Proposition 4.7(6)]{FE}, we have \(\ad(L)<\infty\).
Also \(\ad(\one)=1\).
If \(L_1,\ldots,L_m\) are the composition factors of \(X\),
exact tensor product gives a filtration of \(X^{\otimes n}\)
with factors \(L_{i_1}\otimes\cdots\otimes L_{i_n}\).
The norm bounds recalled in Section~\ref{sec:inputs} imply
\[
                  \ell(X^{\otimes n})
                      \le\left(\sum_{i=1}^m\ad(L_i)\right)^n.
\]
This proves moderate growth directly, as in \cite[Lemma 5.4]{AF},
without assuming GR or finiteness.
Inheritance is immediate from fullness of a tensor-subcategory
inclusion and the definition of \(\widetilde X\).
\end{proof}

\begin{theorem}[Maximal nilpotence of higher Verlinde categories]
\label{thm:verlinde-mn}
Over any algebraically closed field of positive characteristic,
\(\Ver_{p^r}\) is MN for every \(1\le r\le\infty\).
\end{theorem}
\begin{proof}
First let \(r<\infty\), and choose an uncountable algebraically
closed extension \(L/k\). The finite category \(\Ver_{p^r,L}\)
has moderate growth and is incompressible by
\cite[Theorem 4.71(ii)]{BEO}. Proposition~\ref{prop:incompressible-mn}
therefore makes it MN.

The scalar-extension functor
\(F:\Ver_{p^r,k}\to\Ver_{p^r,L}\) is exact and faithful.
Here the scalar extension agrees with the higher Verlinde category
over \(L\), since its construction is defined over the prime field
\cite[Section 4.9]{BEO}. Formula~\eqref{eq:base-change-hom} gives
\[
 \Hom(FX,\one)=L\otimes_k\Hom(X,\one),
 \qquad F(\widetilde X)=\widetilde{FX}.
\]
Moreover \(F\) commutes with symmetric powers. For every \(X\),
MN over \(L\) makes the multiplication
\(F(\widetilde X)^{\otimes n}\to\Sym^n(FX)\) zero for some \(n\).
Faithfulness makes \(\widetilde X^{\otimes n}\to\Sym^nX\) zero,
proving MN over \(k\).
Finally, every object of \(\Ver_{p^\infty}\) belongs to a finite
level. The full tensor inclusions preserve the maximal unit quotient
and symmetric powers, so MN passes to the union.
\end{proof}

This result is also contained in \cite[Theorem A, Proposition 4.1]{CFNO}.

\begin{remark}[Finite incompressible categories]
A finite tensor category is MN if and only if it is incompressible,
as also recorded in \cite[Proposition 4.3]{CFNO}.
The forward implication is \cite[Theorem 7.2.3(2)]{CEO}.
For the converse, \cite[Theorem B]{CF} gives a functor to a finite
higher Verlinde category; incompressibility makes it a
tensor-subcategory embedding. Apply Theorem~\ref{thm:verlinde-mn}
and inheritance of MN.
\end{remark}

\section{Nilpotent kernels and geometric reductivity}\label{sec:gr}

The top alternating norm supplies the invertible object used in the
determinant argument. We follow the graded Hopf-algebra argument of
\cite[Section 5, Proposition 5.1]{CFNO}.

\begin{lemma}[The top component of a connected graded Hopf algebra]
\label{lem:hopf-top}
Let \(H=\bigoplus_{i=0}^n H_i\) be a not necessarily commutative
Hopf algebra in \(\C\), with all structure maps respecting the grading,
\(H_0=\one\),
\(Z=H_n\ne0\), and invertible antipode \(S\).
Then \(Z\) is invertible. Moreover, for a scalar \(s\in k\)
with \(s^2=1\),
\[
                   S|_Z=s\id_Z,
             \qquad c_{Z,Z}=s\id_{Z\otimes Z}.
\]
\end{lemma}
\begin{proof}
Write \(\mu\) and \(\Delta\) for multiplication and comultiplication.
The Frobenius theorem for finite Hopf algebras
\cite[Corollary 4.5]{Tak} gives an invertible object \(L\) and
a morphism \(\lambda:H\to L\) such that the multiplication
pairing \(\lambda\mu:H\otimes H\to L\) is nondegenerate.
Thus its adjoint is an isomorphism
\(\phi:H\xrightarrow{\sim}H^*\otimes L\).
Here ``finite'' means that \(H\) is an object of \(\C\);
the ambient category need not be finite.

Multiplication vanishes on \(H_i\otimes H_j\) when \(i+j>n\).
Consequently, without assuming that \(\phi\) preserves degree,
its restriction to \(H_i\) factors as a monomorphism
\[
            H_i\lhook\joinrel\longrightarrow
              \bigoplus_{j=0}^{n-i}H_j^*\otimes L.
\]
For \(i=n\), this is a nonzero monomorphism \(Z\hookrightarrow L\).
Since an invertible object is simple, it is an isomorphism.
Both \(Z\) and \(Z\otimes Z\) therefore have endomorphism ring
\(k\); write \(S|_Z=s\id_Z\) and
\(c_{Z,Z}=t\id_{Z\otimes Z}\), where \(t^2=1\).

Let \(\iota:Z\hookrightarrow H\) be the inclusion.
Multiplicativity of \(\Delta\) and the antipode identity give,
as maps \(H\otimes H\to H\otimes H\),
\begin{equation}\label{eq:hopf-top-identity}
 (\mu\otimes\id_H)(S\otimes(\Delta\mu))(\Delta\otimes\id_H)
  =(\id_H\otimes\mu)(c_{H,H}\otimes\id_H)(\id_H\otimes\Delta).
\end{equation}
Indeed, substitute
\(\Delta\mu=(\mu\otimes\mu)
(\id_H\otimes c_{H,H}\otimes\id_H)(\Delta\otimes\Delta)\)
on the left and cancel the adjacent antipode and multiplication
using \(\mu(S\otimes\id_H)\Delta=\eta\varepsilon\), where
\(\eta\) and \(\varepsilon\) are the unit and counit.
Precompose \eqref{eq:hopf-top-identity} with \(\iota\otimes\iota\).
Products of a top-degree factor with a positive-degree factor vanish.
Since \(H_0=\one\), the two sides reduce respectively to
\((S\iota)\otimes\iota\) and
\(c_{H,H}(\iota\otimes\iota)\). Naturality of the symmetry gives
\[
       (S\iota)\otimes\iota
                 =(\iota\otimes\iota)c_{Z,Z}.
\]
Cancelling the monomorphism \(\iota\otimes\iota\) shows \(s=t\).
\end{proof}

\begin{lemma}[The top alternating norm is an even line]\label{lem:top-line}
Let \(X\ne0\) have finite alternating norm degree \(d=\ad(X)\).
Then \(A_d(X)\) is invertible and its self-symmetry is the identity.
\end{lemma}
\begin{proof}
Work in the tensor category \(\C^{\mathrm{gr}}\) of
\(\mathbb Z/2\mathbb Z\)-graded objects of \(\C\), with symmetry
\[
             c'_{V,W}=(-1)^{ab}c_{V,W}
       \quad\text{for homogeneous parities }a,b.
\]
Put \(U=X\) in odd parity. This construction also applies when
\(p=2\), when all the displayed signs are \(1\).

There are two graded Hopf structures on
\(\bigoplus_{m\ge0}U^{\otimes m}\) in \(\Ind\C^{\mathrm{gr}}\).
The tensor Hopf algebra \(T(U)\) has concatenation product and
coproduct determined by making \(U\) primitive.
The shuffle Hopf algebra \(T^{\mathrm{sh}}(U)\) has
deconcatenation coproduct: its component of bidegree \((i,m-i)\)
is the canonical identification
\(U^{\otimes m}=U^{\otimes i}\otimes U^{\otimes(m-i)}\).
Its product on degrees \(i,j\) sums over all permutations of
\(i+j\) positions that preserve the order within each of the
two input blocks, acting through \(c'\).
Partitioning permutations into such shuffles verifies the
bialgebra identities. The connected grading supplies the antipode.
The shuffle product is commutative for \(c'\).

The algebra map \(T(U)\to T^{\mathrm{sh}}(U)\) that is the
identity on \(U\) is a Hopf map, since \(U\) is primitive
on both sides. In degree \(m\) this map is
\(\sum_{\sigma\in S_m}\sigma\), with the action through \(c'\).
On the underlying object \(X^{\otimes m}\), it is exactly
\(a_m^X\): each transposition contributes the additional sign
from the odd parity of \(U\).
Its image \(H\) is therefore a commutative graded Hopf algebra
with
\[
 H_m=A_m(X)\text{ placed in parity }m\bmod2,
 \qquad H=\bigoplus_{m=0}^d H_m.
\]
In particular, \(H\) is a finite object, \(H_0=\one\), and
\(H_d\ne0\). It is generated by the primitive object \(U\),
so its antipode, an algebra map by commutativity, acts on
\(H_m\) by \((-1)^m\). It is thus invertible.

Lemma~\ref{lem:hopf-top} makes \(H_d\) invertible and gives
\(c'_{H_d,H_d}=(-1)^d\id\).
Forgetting the grading is strong monoidal, so \(A_d(X)\) is
invertible in \(\C\). Comparing the two symmetries gives
\[
            (-1)^{d^2}c_{A_d(X),A_d(X)}=(-1)^d\id.
\]
Since \(d^2\equiv d\pmod2\), the self-symmetry is the identity.
This conclusion does not require \(d\) to be even.
\end{proof}


\begin{lemma}[Divided derivatives across a nilpotent kernel]
\label{lem:hasse}
Let \(\alpha:X\twoheadrightarrow\one\) have kernel \(K\), and put
\(B=\Sym X\). Suppose that the multiplication map
\(K^{\otimes N}\to B\) vanishes. If \(q=p^e\ge N\), with \(e\ge0\), there is a
canonical isomorphism
\[
                    D_q:B_{2q-1}\xrightarrow{\ \sim\ }B_{q-1},
                    \qquad B_n=\Sym^nX.
\]
Here \(D_r\) is the coefficient of \(z^r\) in translation by
\(\alpha\), that is, in the algebra map
\[
 \tau:B\longrightarrow B\otimes k[z],\qquad
 x\longmapsto x+\alpha(x)z.
\]
The formula for generators denotes the morphism
\(X\to B_1\oplus B_0z\), so does not assume vector-space points of \(X\).
\end{lemma}
\begin{proof}
Let \(J=(K)\subset B\). Then \(J^N=0\) and
\(B/J=\Sym(X/K)=k[t]\), with \(t\) of degree one.
Translation fixes \(K\), so every coefficient \(D_r\) preserves
\(J^j\). Write \(M_j=J^j/J^{j+1}\), and
\(R_j=(M_j)_j\). The graded \(k[t]\)-module \(M_j\) is generated
in degree \(j\) by the image of \(K^{\otimes j}\). Hence multiplication
gives epimorphisms
\[
  u_{j,n}:R_j\longrightarrow(M_j)_n,
       \qquad u_{j,n}(v)=t^{n-j}v\quad(n\ge j).
\]
These are isomorphisms: \(D_r(R_j)=0\) for \(r>0\), and translation
on the quotient is \(t\mapsto t+z\); taking the coefficient of
\(z^{n-j}\) gives \(D_{n-j}u_{j,n}=\id_{R_j}\).
These calculations use the induced \(k[t]\)-action on \(M_j\),
not a choice of a lift \(\one\to X\).
More generally,
\begin{equation}\label{eq:layer-derivative}
 D_r u_{j,n}=\binom{n-j}{r}u_{j,n-r}
       \quad\text{when }n-r\ge j.
\end{equation}

For \(0\le j<N\le q\), both \(q-1\) and \(2q-1\) are at least
\(j\), and
\begin{equation}\label{eq:hasse-coefficient}
                   \binom{2q-1-j}{q}=1\quad\text{in }k.
\end{equation}
Indeed, the coefficient of \(u^q\) in
\[
 (1+u)^{2q-1-j}=(1+u^q)(1+u)^{q-1-j}
\]
is \(1\), since \(0\le q-1-j<q\). Thus \(D_q\) induces an
isomorphism on every layer of the finite \(J\)-filtrations of
\(B_{2q-1}\) and \(B_{q-1}\). It is an isomorphism.
\end{proof}

\begin{lemma}[A norm determinant gives a section]\label{lem:nilpotent-gr}
In a moderate-growth category, a morphism \(\alpha\) satisfying
Lemma~\ref{lem:hasse} has a split symmetric power. More precisely,
if \(q=p^e\ge N\) and \(d=\ad(\Sym^{q-1}X)\), there is a morphism
\[
 s:\one\longrightarrow\Sym^{dq}X,
             \qquad \Sym^{dq}(\alpha)s=\id_{\one}.
\]
\end{lemma}
\begin{proof}
Set \(Y=B_{q-1}\) and \(Z=B_{2q-1}\). The symmetric algebra has
its standard Hopf coproduct, primitive on \(X\). Its homogeneous
component \(\Delta_{q-1,q}:Z\to Y\otimes B_q\) satisfies
\[
 (\id_Y\otimes\Sym^q\alpha)\Delta_{q-1,q}=D_q.
\]
Consequently
\[
 f=\Delta_{q-1,q}D_q^{-1}:Y\longrightarrow Y\otimes B_q
 \quad\text{satisfies}\quad
 (\id_Y\otimes\Sym^q\alpha)f=\id_Y.
\]
The map \(\Sym^{q-1}\alpha\) is epic, so \(Y\ne0\).
Its top norm \(L=A_d(Y)\) is invertible by Lemma~\ref{lem:top-line}.

Tensor \(f\) with itself \(d\) times, move its coefficient factors
to the right, and multiply them. This gives
\[
 f^{[d]}:Y^{\otimes d}\longrightarrow Y^{\otimes d}\otimes B_{dq}.
\]
Commutativity of \(B\) makes this map \(S_d\)-equivariant, with
\(S_d\) acting on the \(Y\)-factors. In particular
\[
 f^{[d]}a_d^Y=(a_d^Y\otimes\id)f^{[d]}.
\]
Exact tensor product therefore induces a map
\(L\to L\otimes B_{dq}\). Since \(L\) is invertible, it is
\(\id_L\otimes s\) for a unique \(s:\one\to B_{dq}\).
Evaluating the coefficients by \(\Sym^{dq}\alpha\) turns
\(f^{[d]}\) into \(\id_{Y^{\otimes d}}\), and hence the induced
map into \(\id_L\). Cancelling \(L\) proves
\(\Sym^{dq}(\alpha)s=\id_{\one}\).
\end{proof}

\begin{theorem}[MN implies GR]\label{thm:mn-gr}
Over an algebraically closed field of positive characteristic, every
MN tensor category is geometrically reductive:
for each epimorphism \(\alpha:X\twoheadrightarrow\one\), some
\(\Sym^n\alpha\) splits.
\end{theorem}
\begin{proof}
Moderate growth follows from Lemma~\ref{lem:mn-consequences}(4).
Choose a subobject \(X'\subset X\) of minimal length for which
\(\alpha'=\alpha|_{X'}\) is epic. No proper subobject of \(X'\)
maps onto \(\one\). Thus every proper subobject is contained in
\(K'=\ker\alpha'\). In particular \(K'\) is the unique maximal
subobject, and the head of \(X'\) is the simple object \(\one\).
It follows that
\(\Hom(X',\one)=k\alpha'\) and \(\widetilde {X'}=K'\).
MN makes \(K'\) nilpotent in \(\Sym X'\).
Lemma~\ref{lem:nilpotent-gr} gives a section of a symmetric power
of \(\alpha'\). Composing it with \(\Sym^nX'\to\Sym^nX\)
gives the required section.
\end{proof}

\section{Fundamental coends and finiteness}\label{sec:coends}


For a tensor category \(\mathcal A\), its fundamental coend is
the commutative Hopf ind-algebra
\[
               H_{\mathcal A}=\int^{X\in\mathcal A}X^*\otimes X.
\]
Its coefficient maps are \(c_X:X^*\otimes X\to H_{\mathcal A}\).
Dinaturality means that for \(f:X\to Y\) the two maps
\(Y^*\otimes X\to H_{\mathcal A}\), obtained using \(f\) or \(f^*\),
agree. The coend notation \(\int^X\) expresses universality:
maps \(H_{\mathcal A}\to M\) correspond to families of maps
\(X^*\otimes X\to M\) satisfying these same identities.
Multiplication is induced by tensor products, the unit
by \(X=\one\), and the antipode by duality.
Equivalently, there are universal coactions
\(\rho_X:X\to X\otimes H_{\mathcal A}\), natural in \(X\) and
compatible with tensor products.

\begin{lemma}[Coefficient generation and finiteness]\label{lem:coend}
If \(\mathcal A=\langle G\rangle\), its coend is generated as an
algebra by the coefficient images of \(G\) and \(G^*\).
For every simple \(S\), the map
\(c_S:S^*\otimes S\to H_{\mathcal A}\) is monic.
Moreover \(H_{\mathcal A}\) is a finite object if and only if
\(\mathcal A\) is a finite tensor category.
\end{lemma}
\begin{proof}
Coefficients of tensor products are products of coefficients;
coefficients of direct sums are sums of coefficient images.
For a monomorphism \(U\hookrightarrow X\), dinaturality and
the epimorphism \(X^*\twoheadrightarrow U^*\) show that the
coefficient image of \(U\) is contained in that of \(X\).
The same holds for a quotient, using the original epimorphism.
This proves the generation assertion.

For simple coefficients, we recall the external-coend argument in
\cite[proof of Lemma 8.2]{FE}; the same coend will prove the
finiteness criterion. Consider
\[
                 K=\int^{X\in\mathcal A}X^*\boxtimes X
                      \quad\text{in }\Ind(\mathcal A\boxtimes\mathcal A).
\]
The Deligne product exists, and biexact functors induce exact
functors from it \cite[Proposition 1.11.2]{EGNO}.
Tensor multiplication
\(m:\mathcal A\boxtimes\mathcal A\to\mathcal A\) is therefore exact.
Over our algebraically closed field, every simple in the Deligne product is
\(S\boxtimes T\) for simples \(S,T\), and \(S\otimes T\ne0\).
Thus \(m\) kills no nonzero object and is faithful.
Its ind-extension is exact and preserves colimits.
Thus \(m(K)=H_{\mathcal A}\).
The map \(S^*\boxtimes S\to K\) is nonzero: after applying
\(m\) and the coend counit its composite is \(\ev_S\ne0\).
Its source is simple, so it is monic. Applying \(m\) proves
the assertion about \(c_S\).

Now suppose \(H_{\mathcal A}\) is finite.
Every finite subobject \(K'\subset K\) satisfies
\[
                   \ell(K')\le\ell(mK')\le\ell(H_{\mathcal A}).
\]
A filtered union of finite-length subobjects with a common length
bound stabilizes. Hence \(K\) is finite.

Choose a faithful exact linear functor
\(\omega:\mathcal A\to\Vec_k\).
Such a functor exists by \cite[Theorem 1.9.15]{EGNO}; no monoidal
structure on \(\omega\) is asserted.
Put \(\eta(X)=\omega(X^*)^*\).
The functor \(\eta\boxtimes\omega\) sends \(U\boxtimes V\)
to \(\eta(U)\otimes_k\omega(V)\). Applying its ind-extension
to \(K\) gives the reconstruction coalgebra
\[
                   C_\omega=\int^{X\in\mathcal A}
                                      \omega(X)^*\otimes\omega(X).
\]
It is finite-dimensional because \(K\) is finite.
Coalgebra reconstruction \cite[Theorem 1.10.1]{EGNO} identifies
\(\mathcal A\) with
finite-dimensional \(C_\omega\)-comodules, equivalently with
finite-dimensional modules over the ordinary, possibly
noncommutative algebra \(C_\omega^*\). This proves finiteness.

Conversely, if \(\mathcal A\) is finite, choose a projective
generator \(P\). Every object is a quotient of a finite sum
of copies of \(P\). The coefficient argument at the start of
the proof therefore makes \(P^*\otimes P\to H_{\mathcal A}\)
epic, so the coend is finite.
\end{proof}


\begin{corollary}[An ordinary coend forces pointedness]
\label{lem:simple-coefficient}
If \(H_{\mathcal A}\) is ordinary, every simple of \(\mathcal A\)
is invertible.
\end{corollary}
\begin{proof}
For simple \(S\), the preceding lemma makes \(S^*\otimes S\)
a subobject of \(H_{\mathcal A}\), hence a finite sum of units.
Schur's lemma gives
\[
 \dim_k\Hom(\one,S^*\otimes S)=\dim_k\End(S)=1,
\]
so \(S^*\otimes S\simeq\one\).
\end{proof}

\begin{lemma}[A Hopf subalgebra of the fundamental coend]
\label{lem:hopf-subcategory}
Let \(K\subset H_\C\) be a Hopf ind-subalgebra. There is a tensor
subcategory \(\D\subset\C\) whose coend, viewed in \(\Ind\C\), is
\(K\). It consists of the objects whose canonical coaction factors
through \(X\otimes K\subset X\otimes H_\C\).
\end{lemma}
\begin{proof}
Write \(G=\pi(\C)\) for the fundamental group, so
\(\mathcal O(G)=H_\C\), and
\(Q=\operatorname{Spec}K\). The inclusion defines a map
\(q:G\to Q\). The canonical \(G\)-action on its own coordinate
algebra is the adjoint action
\cite[Sections 4.1.1--4.1.3]{CEO}. Restricting to \(K\), and using its
Hopf structure, shows that the adjoint action on \(Q\), pulled back
along \(q\), agrees with the canonical action on \(\mathcal O(Q)\).
Thus \((Q,q)\) is a \(\C\)-group in the sense of
\cite[Section 4.1]{CEO}.

Apply internal Tannaka duality \cite[Theorem 4.2.1]{CEO} to
\(\Rep_\C(Q,q)\), the representations whose restriction along
\(q\) is their canonical \(G\)-action. Forgetting the representation
is exact and symmetric monoidal. It is fully faithful: every
morphism in \(\C\) respects canonical \(H_\C\)-coactions, and
the monomorphism \(K\hookrightarrow H_\C\) then implies that it
respects the \(K\)-coactions. Its image is closed under subquotients.
For example, for \(U\subset X\), exact tensor product gives
\[
 (U\otimes H_\C)\cap(X\otimes K)=U\otimes K
       \quad\text{inside }X\otimes H_\C,
\]
which factors the canonical coaction on \(U\) through \(K\).
Quotients follow by the same exactness. Sums, tensor products, and
duals follow from the Hopf structure. The image is therefore the
asserted tensor subcategory \(\D\).

The inverse direction of \cite[Theorem 4.2.1]{CEO} identifies the
internal group of tensor automorphisms
\(\underline{\operatorname{Aut}}^{\otimes}(\omega)\) with \(Q\),
where \(\omega:\Rep_\C(Q,q)\to\C\) is the forgetful functor.
Equivalently, its coefficient map is an isomorphism
\[
 \int^{X\in\Rep_\C(Q,q)}\omega(X)^*\otimes\omega(X)
                         \xrightarrow{\sim}\mathcal O(Q)=K.
\]
Full faithfulness identifies the source with \(H_\D\) in
\(\Ind\C\), so \(H_\D\simeq K\).
\end{proof}

\subsection*{Frobenius reduction}
For a commutative ind-algebra \(H\), put
\[
 H^{[1]}=\im\bigl((H^{\otimes p})^{S_p}\longrightarrow H\bigr),
 \qquad H^{[r+1]}=(H^{[r]})^{[1]},
\]
where the arrow is multiplication and the superscript \(S_p\)
denotes invariants. These are image subalgebras, not scalar twists;
the displayed description is \cite[Lemma 4.2]{CSH}.
For a finitely generated commutative Hopf ind-algebra \(H\) in an
MN, GR category, the reduction uses the following statements.
\begin{enumerate}
\item \(H\) is finitely generated as a module over \(H^{[1]}\), and
\(H^{[1]}\) is finitely generated
\cite[Lemma 4.4(2),(3)]{CSH}. Iteration makes every \(H^{[r]}\)
finitely generated as an algebra and \(H\) finitely generated as a
module over it: compose finite free module covers at successive stages.
\item Every \(H^{[r]}\) is a Hopf subalgebra of \(H\)
\cite[Lemma 4.17(1)]{CSH}.
\item For sufficiently large \(r\), \(\operatorname{Spec}H^{[r]}\)
is purely even \cite[Proposition 4.19(2)]{CSH}.
\end{enumerate}
Here ``purely even'' means that the ideal generated by the joint
kernel of all maps \(H^{[r]}\to\one\) is zero. Thus every finite
subobject embeds in a finite sum of units, and \(H^{[r]}\) is
ordinary. Conversely an ordinary algebra is purely even.
No finiteness of the ambient category or of its set of simples is
assumed in these results.

\begin{theorem}[Finite generation implies finiteness for MN categories]
\label{thm:mn-finite}
Let \(k\) be algebraically closed of positive characteristic.
Every finitely generated MN tensor category \(\C\) is finite.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:mn-consequences}(4) and
Theorem~\ref{thm:mn-gr}, the category has moderate growth and is
geometrically reductive. By Lemma~\ref{lem:coend}, its fundamental
coend \(H=H_\C\) is a finitely generated commutative Hopf ind-algebra. Apply the
Frobenius reduction just stated. For a sufficiently large \(r\),
\[
                       K=H^{[r]}\subset H
\]
is an ordinary Hopf subalgebra, and \(H\) is finitely generated as
a \(K\)-module.

Let \(\D\subset\C\) be the tensor subcategory from
Lemma~\ref{lem:hopf-subcategory}. Its coend is \(K\).
This coend is ordinary also as an ind-object of \(\D\): the
inclusion is full and closed under subobjects, and every finite
sum of units occurring in \(K\) belongs to \(\D\).
The category \(\D\) inherits MN. If \(p>2\), Proposition~\ref{prop:ordinary-odd}
identifies \(\D\) with \(\Vec\) or \(\sVec\). If \(p=2\),
Proposition~\ref{prop:ordinary-mn} makes \(\D\) finite.

Lemma~\ref{lem:coend} makes \(K=H_\D\) finite. Since \(H\)
is a finitely generated \(K\)-module, it is a quotient of \(K\otimes W\)
for some finite \(W\in\C\), so it too is finite.
Apply the same lemma to conclude that \(\C\) is finite.
\end{proof}

\begin{corollary}\label{cor:finite-mn-embedding}
Every finitely generated MN tensor category embeds as a tensor
subcategory of a finite higher Verlinde category.
\end{corollary}
\begin{proof}
It is finite by Theorem~\ref{thm:mn-finite}, and hence incompressible
by \cite[Theorem 7.2.3(2)]{CEO}. Its functor to a finite higher
Verlinde category supplied by \cite[Theorem B]{CF} is therefore a
tensor-subcategory embedding.
\end{proof}

A tensor category is \emph{subterminal} if any two tensor functors
with the same source and that target are isomorphic as symmetric
monoidal functors.

\begin{corollary}[Subterminality]\label{cor:verlinde-subterminal}
The categories \(\Ver_{p^r}\), for \(1\le r\le\infty\),
are subterminal.
\end{corollary}
\begin{proof}
They are MN by Theorem~\ref{thm:verlinde-mn} and GR by
Theorem~\ref{thm:mn-gr}. Apply \cite[Theorem 7.3.1]{CEO},
which states that every MN and GR category is subterminal.
\end{proof}
This conclusion is also recorded in \cite[Proposition 4.2]{CFNO}.

\begin{proposition}[Embedding an arbitrary MN category]
\label{prop:mn-embedding}
Every MN tensor category over \(k\) embeds as a tensor subcategory
of \(\Ver_{p^\infty,k}\).
\end{proposition}
\begin{proof}
Let \(\mathcal A\) be MN. Its finitely generated tensor
subcategories are finite and embed in finite higher Verlinde
categories by Corollary~\ref{cor:finite-mn-embedding}.
The classification \cite[Corollary 4.61]{BEO} shows that these
subcategories have only countably many possible equivalence types.

Within \(\mathcal A\), two such subcategories of the same type
actually coincide. Indeed, let \(\D_1,\D_2\subset\mathcal A\)
have a tensor equivalence \(E:\D_1\to\D_2\). Their join is
finitely generated and MN, so embeds in \(\Ver_{p^\infty}\).
The two resulting functors from \(\D_1\), one by its inclusion and
the other through \(E\), are isomorphic by subterminality.
Their essential images therefore agree. Full faithfulness of the
embedding of the join gives \(\D_1=\D_2\).

Enumerate these subcategories as \(\D_1,\D_2,\ldots\), repeating
the last one if there are only finitely many, and choose generators
\(G_i\) for them. Then
\[
 \mathcal E_n=\langle G_1\oplus\cdots\oplus G_n\rangle,
 \qquad \mathcal A=\bigcup_{n\ge1}\mathcal E_n
\]
is a cofinal increasing sequence of finite MN subcategories.
Choose embeddings \(F_n:\mathcal E_n\to\Ver_{p^\infty}\).
Subterminality gives monoidal isomorphisms
\(F_{n+1}|_{\mathcal E_n}\simeq F_n\).
Choose these for adjacent terms and compose them for longer
intervals; this gives coherent transition isomorphisms.
The functors therefore glue to \(F:\mathcal A\to\Ver_{p^\infty}\).
Every coherence diagram for the tensor structure and every short
exact sequence is contained in one stage, so \(F\) is symmetric
monoidal and exact.
Every pair of objects lies in one \(\mathcal E_n\), so \(F\) is
fully faithful. Every subquotient of an image object already lies
in the image of that same \(\mathcal E_n\), so \(F\) is a
tensor-subcategory embedding.
\end{proof}

\section{Proof of the main theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
First suppose \(k\) is uncountable.
By \cite[Theorem 5.2.1]{CEO}, choose a surjective tensor functor
\(F:\C\to\mathcal I\) to an incompressible category of moderate
growth.
Proposition~\ref{prop:incompressible-mn} makes \(\mathcal I\) MN,
and Proposition~\ref{prop:mn-embedding} embeds it in
\(\Ver_{p^\infty,k}\). Composition proves the
assertion over \(k\).

For a general algebraically closed \(k\), choose an uncountable
algebraically closed extension \(L\).
Corollary~\ref{cor:scalar-extension} shows that \(\C_L\) has
moderate growth.
The preceding paragraph gives
\(\C_L\to\Ver_{p^\infty,L}\).
For each finite \(r\), the construction over the prime field
\cite[Section 4.9]{BEO} identifies
\((\Ver_{p^r,k})_L\) with \(\Ver_{p^r,L}\).
Concretely, a projective generator realizes the finite category
as modules over a finite-dimensional \(k\)-algebra; scalar
extension replaces this algebra by its tensor product with \(L\).
Thus \(\Ver_{p^r,L}\) is the category of finite-length
\(L\)-modules in \(\Ind\Ver_{p^r,k}\), compatibly with the
standard embeddings as \(r\) varies.
Restricting along \(\C\to\C_L\) yields an exact \(k\)-linear
strong symmetric monoidal functor
\[
                 \C\longrightarrow
                 \Mod_L(\Ind\Ver_{p^\infty,k}).
\]
The category \(\Ver_{p^\infty,k}\) is MN by
Theorem~\ref{thm:verlinde-mn}, and geometrically reductive
by Theorem~\ref{thm:mn-gr}.
The neutrality theorem \cite[Appendix B, Theorem B.1]{AF}, which
does not require finite generation of the source or finiteness of
the target, therefore gives \(\C\to\Ver_{p^\infty,k}\).

Finally, if \(\C=\langle G\rangle\), the image of \(G\) belongs
to some \(\Ver_{p^r,k}\) with finite \(r\). Since that category
is closed under tensor products, duals, sums, and subquotients,
the entire functor factors through it.
\end{proof}

\begin{remark}[Uniqueness]
The resulting tensor functor
\(\C\to\Ver_{p^\infty,k}\) is unique up to isomorphism:
\(\Ver_{p^\infty,k}\) is subterminal by
Corollary~\ref{cor:verlinde-subterminal}.
The same result gives uniqueness for any fixed finite target
\(\Ver_{p^r,k}\) whenever such a functor exists.
\end{remark}

\begin{remark}[Ground fields and height]
The uncountability assumption in Lemma~\ref{lem:point} ensures
\(K_A=k\). The scalar-extension and neutrality argument above removes
this assumption from Theorem~\ref{thm:main}.
The proof gives no explicit bound on the height \(r\) in terms of
a tensor generator.
\end{remark}

\begin{remark}[The generic quotient]
A domain need not be simple: the domain \(k[x,y]\) in
\(\Rep(\mathrm{SL}_2)\) has the proper stable ideal \((x,y)\).
Passing to \(\Q_A\) makes the unit simple; this is the property
used in the highest-norm argument.
\end{remark}

\part{Odd characteristic}\label{part:odd}

Throughout this part, \(k\) is algebraically closed and \(p>2\).

\section{Ordinary coends}\label{sec:ordinary-odd}

We use the theorem that the maximal Tannakian
and super-Tannakian subcategories are Serre, together with its consequence
for categories whose semisimple objects form an integral fusion category
\cite[Appendix, Theorem 7.1 and Proposition 7.2]{EG}.
Here Tannakian and super-Tannakian mean admitting a tensor functor to
\(\Vec_k\) and \(\sVec_k\), respectively.

\begin{proposition}\label{prop:ordinary-odd}
An MN tensor category \(\mathcal U\) with ordinary fundamental coend
is equivalent to \(\Vec_k\) or \(\sVec_k\).
\end{proposition}
\begin{proof}
By Corollary~\ref{lem:simple-coefficient}, all simples are invertible.
The self-symmetry of an invertible object is a scalar \(\pm1\), and
Lemma~\ref{lem:mn-consequences} makes every even invertible the unit.
The product of two odd invertibles is even. Consequently there is at
most one nonunit simple \(\varepsilon\), and if it exists then
\(\varepsilon^{\otimes2}\simeq\one\) and its self-symmetry is \(-1\).
The semisimple objects therefore form a tensor subcategory with at most
two invertible simples, hence an integral fusion category.
The cited theorem makes \(\mathcal U\) super-Tannakian.
Its tensor functor to \(\sVec\) is full by MN and faithful by
exactness. All short exact sequences split after this functor, so their
splittings lift. Thus \(\mathcal U\) is semisimple, and its fully
faithful image in \(\sVec\) is \(\Vec\) or \(\sVec\).
\end{proof}

\begin{example}[Parity of the top norm]
The parity assertion concerns the norm images \(A_n(X)\);
evenness describes their self-symmetry, while the integer \(d\)
can be odd. For instance, \(\ad(\one)=1\) and
\(A_1(\one)=\one\) is even.
For example, let \(\varepsilon\) be the one-dimensional odd
object of \(\sVec_k\). Each permutation \(\sigma\in S_n\)
acts on \(\varepsilon^{\otimes n}\) by \(\operatorname{sgn}(\sigma)\), so
\[
 a_n^\varepsilon=n!\,\id_{\varepsilon^{\otimes n}},\qquad
 A_n(\varepsilon)=
 \begin{cases}
  \varepsilon^{\otimes n},&0\le n<p,\\
  0,&n\ge p.
 \end{cases}
\]
Thus \(\ad(\varepsilon)=p-1\) and
\(A_{p-1}(\varepsilon)\simeq\one\) is even.
At \(p=3\), \(A_1(\varepsilon)\) is odd but the highest
nonzero term is \(A_2(\varepsilon)=\one\).
The ordinary exterior power \(\bigwedge^n X\) is the largest
quotient of \(X^{\otimes n}\) on which each permutation acts
by its sign. For the odd line,
\(\bigwedge^n\varepsilon\simeq\varepsilon^{\otimes n}\)
is nonzero for every \(n\).

\end{example}

\section{Tensor subcategories are Serre}\label{sec:extensions}

At odd primes, every tensor subcategory of a finite higher Verlinde
category is Serre \cite[Proposition 4.59 and Corollary 4.61]{BEO}.
Together with the common finiteness theorem, this gives a short proof
of the corresponding statement in any MN category.

\begin{proposition}[Tensor subcategories of MN categories are Serre]
\label{prop:mn-serre}
Let \(\mathcal A\) be MN, and let \(\D\subset\mathcal A\) be a
tensor subcategory. Then every extension in \(\mathcal A\)
with endpoints in \(\D\) belongs to \(\D\).
\end{proposition}
\begin{proof}
Let \(0\to A\to X\to B\to0\) have \(A,B\in\D\).
The category \(\mathcal E=\langle X\rangle\subset\mathcal A\)
contains \(A,B\), is finitely generated, and inherits MN. By
Corollary~\ref{cor:finite-mn-embedding}, it embeds in a finite
higher Verlinde category. The image of its tensor subcategory
\(\langle A\oplus B\rangle\) is Serre there and therefore
contains the image of \(X\). Full faithfulness gives
\(X\in\langle A\oplus B\rangle\subset\D\).
\end{proof}

\begin{corollary}\label{lem:unit-ext}
In any MN tensor category in odd characteristic,
\[
                         \Ext^1(\one,\one)=0.
\]
\end{corollary}
\begin{proof}
Apply Proposition~\ref{prop:mn-serre} to the semisimple subcategory
\(\D=\Vec\).
\end{proof}

\part{Characteristic two}\label{part:two}

Throughout this part, \(k\) is algebraically closed of characteristic two,
except for the normalization lemma, which only requires perfection.

\section{The unit extension and the Frobenius functor}

The ordinary-coend case need not be semisimple in this characteristic.
Let
\[
 \V=\Ver_4^+=\Rep\bigl(k[d]/(d^2),\ R=1+d\otimes d\bigr),
\]
where \(d\) is primitive. This is the symmetric category of vector
spaces with a square-zero operator \(d\), with symmetry
\(v\otimes w\mapsto w\otimes v+dw\otimes dv\)
\cite{BE2,EG2}.

\begin{example}[A nonsplit extension of the unit]\label{ex:unit}
In \(\V=\Ver_4^+\), let \(E\) have basis \(x,y\), with
\(d(y)=x\) and \(d(x)=0\). The sequence
\[
 0\longrightarrow\one\xrightarrow{\,1\mapsto x\,}E
      \xrightarrow{\,\alpha\,}\one\longrightarrow0,
 \qquad \alpha(x)=0,\quad\alpha(y)=1,
\]
does not split. As a graded ordinary algebra, \(\Sym E\) is
\(k[x,y]/(x^2)\), with \(d\) acting as a derivation.
Thus \(y^2\) is invariant and gives a section of \(\Sym^2\alpha\).
Here \(N=q=2\), and the map \(D_2:B_3\to B_1\) sends
\(y^3\mapsto y\) and \(xy^2\mapsto x\). This illustrates why the
divided derivative remains useful even though ordinary differentiation
in degree two vanishes on squares.
\end{example}


In characteristic two the Frobenius functor has the description
\[
 \operatorname{Fr}(X)
       =\ker(1+c_{X,X})/\im(1+c_{X,X}).
\]
It is additive and symmetric monoidal, with target the Frobenius scalar
twist of the category \cite[Section 3]{FE}; scalar twisting does not
affect composition length. A category is \emph{Frobenius exact} when
this functor is exact. A Frobenius-exact category of moderate growth
admits a tensor functor to \(\Ver_2=\Vec\)
\cite[Theorem 1.1]{FE}.

An exact faithful quasi-tensor functor is equipped with natural tensor
isomorphisms and an isomorphism between the image of the unit and
the target unit, without imposing associativity coherence.
Such a functor to a semisimple category extends from a tensor subcategory
containing all simples \cite[Appendix, Proposition 7.5]{EG}.
This extension theorem applies without a finiteness hypothesis on the source.

\section{MN categories with an ordinary coend}\label{sec:ordinary}

The following lemma extracts from \cite[Proposition 2.5]{EG2} a
normalization argument for finite-dimensional commutative local algebras.

\begin{lemma}[Normalizing a triangular element]\label{lem:finite-r}
Let \(k\) be a perfect field of characteristic two, and let \(B\)
be a finite-dimensional commutative local \(k\)-algebra with
maximal ideal \(\mathfrak m\) and residue field \(k\). Suppose
\[
 R\in1+\mathfrak m\otimes\mathfrak m,\qquad R_{21}R=1.
\]
There are \(Q\in1+\mathfrak m\otimes\mathfrak m\) and
\(d\in\mathfrak m\), with \(d^2=0\), such that
\[
                         Q_{21}^{-1}RQ=1+d\otimes d.
\]
The subscript \(21\) interchanges the two tensor factors.
\end{lemma}
\begin{proof}
Filter \(B\) by powers of \(\mathfrak m\), and \(B\otimes B\) by
total degree:
\[
 \mathcal F^s=\sum_{i+j\ge s}\mathfrak m^i\otimes\mathfrak m^j
                       \qquad(i,j\ge0).
\]
Write \(\deg b\) for the largest \(j\) with
\(b\in\mathfrak m^j\), and set \(\deg0=\infty\).
Start with \(d=0\) and \(n=2\). Inductively suppose
\(R=1+d\otimes d\) modulo terms of total degree at least \(n\), with
\(d=0\) or \(2\deg d<n\). Let \(e\) be the degree-\(n\) error.
Triangularity gives \(\deg(d^2)>n/2\) and \(e=e_{21}\).
Indeed, \(d^2\otimes d^2\) must vanish below degree \(n\);
in degree \(n\) a possible leading term \(w\otimes w\) would
equal \(e+e_{21}\). The latter has zero diagonal in a homogeneous
basis, forcing \(w=0\), and then \(e=e_{21}\).

Since \(k\) is perfect, write
\[
                         e=v\otimes v+h+h_{21},
\]
where \(v\) lies in the degree-\(n/2\) piece of \(\gr B\),
and \(v=0\) for odd \(n\).
Take square roots of the diagonal entries to obtain \(v\), and
use the remaining off-diagonal entries for \(h\). Both tensor
factors of the error have positive degree, so choose lifts
\(v\in\mathfrak m\) and \(h\in\mathfrak m\otimes\mathfrak m\).
Twisting by \(1+h\) leaves the degree-\(n\)
error \(v\otimes v\). If \(d=0\), replace it by \(v\).
Otherwise twist further by \(1+d\otimes v\) and replace \(d\)
by \(d+v\). More explicitly, modulo \(\mathcal F^{n+1}\),
\[
 \begin{aligned}
 &(1+v\otimes d)^{-1}(1+d\otimes d+v\otimes v)(1+d\otimes v)\\
 &\hspace{3em}\equiv1+(d+v)\otimes(d+v).
 \end{aligned}
\]
Indeed, \(\deg(d^2)>n/2\) and \(\deg v\ge n/2>\deg d\)
make all omitted products have degree greater than \(n\).
For example, \((d\otimes d)(d\otimes v)=d^2\otimes dv\)
has degree greater than \(n+\deg d\), and
\((d\otimes v)(v\otimes d)\) has degree at least
\(n+2\deg d\).
The induction terminates because \(\mathfrak m\) is nilpotent;
the product of its twists gives \(Q\).
Finally \((1+d\otimes d)^2=1\) gives \(d^2\otimes d^2=0\),
hence \(d^2=0\).
\end{proof}

\begin{proposition}\label{prop:ordinary-mn}
Let \(k\) be algebraically closed of characteristic two. An MN tensor
category \(\mathcal U\) with ordinary fundamental coend is finite.
Its underlying abelian category is equivalent to \(\Vec_k\) or to
finite-dimensional modules over \(k[d]/(d^2)\).
\end{proposition}
\begin{proof}
Corollary~\ref{lem:simple-coefficient} and
Lemma~\ref{lem:mn-consequences} show that the only simple of
\(\mathcal U\) is \(\one\): every simple is invertible, and its
self-symmetry is a scalar whose square is \(1\), hence is \(1\)
in characteristic two. Thus length is multiplicative under
tensor products, \(\mathcal U\) has moderate growth, and its
Frobenius--Perron dimension is composition length.
By \cite[Proposition 7.4]{EO},
\[
 \mathcal U_{\mathrm{ex}}
      =\{X\in\mathcal U:\ell(\operatorname{Fr}X)=\ell(X)\}
\]
is a Frobenius-exact tensor subcategory. The cited proposition
requires finitely many simples, not a finite category.
It admits a tensor functor to \(\Vec\) by
\cite[Theorem 1.1]{FE}. It inherits MN, so this functor is full.
Consequently \(\mathcal U_{\mathrm{ex}}=\Vec\) inside \(\mathcal U\).

\emph{Commutative reconstruction.}
Extend the identity on \(\Vec\subset\mathcal U\) to an exact
faithful quasi-tensor functor \(F:\mathcal U\to\Vec\), using
\cite[Appendix, Proposition 7.5]{EG}.
Exactness gives \(\dim F(X)=\ell(X)\).
Coalgebra reconstruction identifies \(\mathcal U\) with the
finite-dimensional continuous modules over the initially possibly
noncommutative algebra
\[
 A=\operatorname{Nat}(F,F),\qquad
 J=\ker(A\longrightarrow\End(F(\one))=k).
\]
Use finite collections of objects containing \(\one\) to define
the action topology. Then \(A\) is an inverse limit of finite-dimensional
algebras. Elements of \(J\) lower composition filtrations, so these
finite action quotients are local with residue field \(k\).
Under reconstruction, \(F\) is the forgetful functor.

Write \(\gamma_{X,Y}:F(X)\otimes F(Y)\to F(X\otimes Y)\)
for its tensor isomorphisms and identify \(F(\one)=k\).
The unit constraints identify \(\gamma_{\one,-}\) and
\(\gamma_{-,\one}\) with units \(a,b\in A\) having the same
value \(\lambda\in k^\times\) on \(\one\).
Replacing \(\gamma_{X,Y}\) by
\[
            \lambda\,\gamma_{X,Y}(b_X^{-1}\otimes a_Y^{-1})
\]
makes both unit isomorphisms the identity. We use this normalized
quasi-tensor structure from now on.

In fact \(A\) is commutative. The pairing
\[
 F(X^*)\otimes F(X)\xrightarrow{\gamma_{X^*,X}}
 F(X^*\otimes X)\xrightarrow{F(\ev_X)}k
\]
defines a natural map \(\delta_X:F(X^*)\to F(X)^*\).
It is an isomorphism on \(\one\); both sides are exact
contravariant functors, so induction on length makes it an
isomorphism for every \(X\). This argument does not require
associativity coherence for \(\gamma\).
Thus, with \(H=H_{\mathcal U}\),
\[
 F(H)\simeq\int^{X\in\mathcal U}F(X)^*\otimes F(X).
\]
These vector-space identifications carry the natural map
\[
 \operatorname{Nat}(\id_{\mathcal U},\id_{\mathcal U})
       \longrightarrow A,\qquad u\longmapsto F(u),
\]
to \(\Hom(H,\one)\to\Hom_k(F(H),k)\): the functional
corresponding to \(u\) restricts on \(X^*\otimes X\) to
\(\ev_X(\id_{X^*}\otimes u_X)\), and naturality of \(\gamma\)
identifies its image with the functional for \(F(u_X)\).
The map is bijective
because \(H\) is ordinary. The map preserves composition, and
natural endomorphisms of the identity commute. Hence \(A\) is commutative.

\emph{Bounding each finite quotient.}
Let \(B\) be a finite action quotient of \(A\), with maximal
ideal \(\mathfrak m\). By reconstruction, its finite-dimensional
modules are objects of \(\mathcal U\), via restriction of scalars.
On pairs of \(B\)-modules, transport the
symmetry through the quasi-tensor isomorphisms and remove the
ordinary flip. This is a natural endomorphism of the ordinary
tensor product of the two underlying vector spaces. Evaluating
on regular modules identifies it with an element \(R_B\in B\otimes B\).
The symmetry and unit axioms give
\[
 R_B\in1+\mathfrak m\otimes\mathfrak m,\qquad (R_B)_{21}R_B=1.
\]
Apply Lemma~\ref{lem:finite-r} to obtain \(Q_B,d_B\), and let
\(X\in\mathcal U\) correspond to the cyclic module \(B/d_BB\).
On \(F(X)\otimes F(X)\), the operator
\(Q_{B,21}^{-1}R_BQ_B\) is the identity. Equivalently the
transported symmetry \(\tau R_B\) is conjugate to the ordinary
flip \(\tau\), since
\[
 \tau Q_{B,21}^{-1}R_BQ_B=Q_B^{-1}(\tau R_B)Q_B.
\]
The cohomology of \(1+\tau\) on \(V\otimes V\) has basis
\([v_i\otimes v_i]\), for any basis \((v_i)\) of \(V\).
Exactness of \(F\) and the Frobenius formula therefore give
\(\ell(\operatorname{Fr}X)=\dim F(X)=\ell(X)\).
Hence \(X\in\mathcal U_{\mathrm{ex}}=\Vec\).

Thus \(B/d_BB\) is a nonzero cyclic trivial \(B\)-module, so is
\(k\). Every \(b\in B\) is therefore \(\lambda+d_Bb'\).
Since \(d_B^2=0\), this gives \(d_BB=kd_B\) and \(\dim_kB\le2\).
Their duals form a directed union equal to the reconstruction
coalgebra, so it too has dimension at most two, as does \(A\).
The local algebra \(A\) is consequently
\(k\) or \(k[d]/(d^2)\), and reconstruction proves the assertion.

Only the action of \(R_B\) on the displayed module was used.
No tensor structure on \(B\)-modules or compatibility between
the choices of \(Q_B\) for different quotients is needed.
\end{proof}



\begin{thebibliography}{EGNO15}
\bibitem[BE19]{BE2}
D.~Benson and P.~Etingof,
\emph{Symmetric tensor categories in characteristic 2},
Adv. Math. 351 (2019), 967--999.
\href{https://arxiv.org/abs/1807.05549}{arXiv:1807.05549}.

\bibitem[BEO23]{BEO}
D.~Benson, P.~Etingof, and V.~Ostrik,
\emph{New incompressible symmetric tensor categories in positive
characteristic}, Duke Math. J. 172 (2023), no.~1, 105--200.
\href{https://arxiv.org/abs/2003.10499}{arXiv:2003.10499}.

\bibitem[CEO23]{FE}
K.~Coulembier, P.~Etingof, and V.~Ostrik,
\emph{On Frobenius exact symmetric tensor categories},
with Appendix A by A.~Kleshchev,
Ann. of Math. (2) 197 (2023), no.~3, 1235--1279.
\href{https://arxiv.org/abs/2107.02372}{arXiv:2107.02372}.

\bibitem[CEO24]{CEO}
K.~Coulembier, P.~Etingof, and V.~Ostrik,
\emph{Incompressible tensor categories},
Adv. Math. 457 (2024), 109935.
\href{https://arxiv.org/abs/2306.09745}{arXiv:2306.09745}.

\bibitem[CF26]{CF}
K.~Coulembier and J.~Flake,
\emph{The Benson--Etingof--Ostrik conjecture for finite tensor
categories}, preprint (2026).
\href{https://arxiv.org/abs/2610.04096}{arXiv:2610.04096}.

\bibitem[CFNO]{CFNO}
K.~Coulembier, J.~Flake, J.~Newton, and V.~Ostrik,
\emph{Maximal nilpotence for higher Verlinde categories},
paper in preparation.

\bibitem[CS25]{CSH}
K.~Coulembier and A.~Sherman,
\emph{Homogeneous spaces in tensor categories},
\href{https://arxiv.org/abs/2505.04848}{arXiv:2505.04848} (2025; revised 2026).

\bibitem[CS26]{AF}
K.~Coulembier and A.~Sherman,
\emph{Absolutely flat algebras in tensor categories},
\href{https://arxiv.org/abs/2608.13137}{arXiv:2608.13137} (2026).

\bibitem[EG21a]{EG}
P.~Etingof and S.~Gelaki,
\emph{Finite symmetric integral tensor categories with the Chevalley
property}, with an appendix by K.~Coulembier and P.~Etingof,
Int. Math. Res. Not. IMRN 2021, no.~12, 9083--9121.
\href{https://arxiv.org/abs/1901.00528}{arXiv:1901.00528}.

\bibitem[EG21b]{EG2}
P.~Etingof and S.~Gelaki,
\emph{Finite symmetric tensor categories with the Chevalley property
in characteristic 2}, J. Algebra Appl. 20 (2021), 2140010.
\href{https://arxiv.org/abs/1904.07576}{arXiv:1904.07576}.

\bibitem[EGNO15]{EGNO}
P.~Etingof, S.~Gelaki, D.~Nikshych, and V.~Ostrik,
\emph{Tensor categories}, Mathematical Surveys and Monographs,
vol.~205, American Mathematical Society, Providence, RI, 2015.
\href{https://math.mit.edu/~etingof/egnobookfinal.pdf}{Authors' final version}.

\bibitem[EO21]{EO}
P.~Etingof and V.~Ostrik,
\emph{On the Frobenius functor for symmetric tensor categories in
positive characteristic}, J. Reine Angew. Math. 773 (2021), 165--198.
\href{https://arxiv.org/abs/1912.12947}{arXiv:1912.12947}.

\bibitem[Gar99]{Gabriel}
G.~Garkusha, \emph{Grothendieck categories}, preprint (1999).
\href{https://arxiv.org/abs/math/9909030}{arXiv:math/9909030}.

\bibitem[Tak99]{Tak}
M.~Takeuchi, \emph{Finite Hopf algebras in braided tensor categories},
J. Pure Appl. Algebra 138 (1999), no.~1, 59--82.
\href{https://doi.org/10.1016/S0022-4049(97)00207-7}{doi:10.1016/S0022-4049(97)00207-7}.
\end{thebibliography}
\end{document}