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\title{Fiber functors for finite symmetric tensor categories\\
       in positive characteristic}
\author{OpenAI}
\date{September 24, 2026}

\begin{document}
\maketitle

\begin{abstract}
We prove the finite case of the Benson--Etingof--Ostrik conjecture:
every finite symmetric tensor category over an algebraically closed
field $k$ of characteristic $p>0$ admits a $k$-linear exact faithful
strong symmetric monoidal functor to a finite higher Verlinde
category $\Ver_{p^n}(k)$.  The result covers every prime, including
$p=2$, and the height $n$ may depend on $\mathcal C$.
\end{abstract}

\tableofcontents

% =================================================================
\section{Introduction and main result}
\label{sec:intro}
% =================================================================

\subsection*{What this paper is about, in one paragraph}

In characteristic zero, Deligne's theorem says that any
``sufficiently nice'' symmetric tensor category embeds faithfully
into supervector spaces.  In positive characteristic there is no
single universal target: to catch every finite symmetric tensor
category one needs a whole \emph{tower} of finite abelian categories
\[
  \Ver_p(k) \subset \Ver_{p^2}(k)\subset
  \Ver_{p^3}(k)\subset\cdots,
\]
called the higher Verlinde categories of Benson--Etingof--Ostrik.
The claim of Benson--Etingof--Ostrik is that this tower \emph{does}
work: every finite symmetric tensor category is a category of
representations internal to some finite level of the tower.  The
present paper proves that claim.

\subsection*{The basic objects}

Throughout, $k$ is an algebraically closed field of characteristic
$p>0$.

\begin{definition}[finite symmetric tensor category]
\label{def:FTSC}
A \emph{finite symmetric tensor category over $k$} is a
$k$-linear abelian rigid symmetric monoidal category $\mathcal C$
such that:
\begin{enumerate}[label=(\arabic*)]
  \item the tensor product is biexact;
  \item every $\Hom(X,Y)$ is finite dimensional over $k$;
  \item every object has finite length;
  \item there are finitely many isomorphism classes of simple
        objects;
  \item the unit object $\mathbf 1$ has $\End(\mathbf 1) = k$;
  \item there are enough projective objects.
\end{enumerate}
\end{definition}

Intuition: it is the ``linear algebra'' world where you can tensor
and dualise, everything has finite length, and the unit behaves like
the one-dimensional vector space.  The simplest example is
$\Vec_k$, the category of finite-dimensional $k$-vector spaces;
when $p>2$ its partner is $\sVec_k$, the $\mathbb Z/2$-graded
version with the Koszul sign in the symmetry.

\begin{definition}[fiber functor]
A \emph{fiber functor} $\mathcal C\to \mathcal D$ between finite
symmetric tensor categories is a $k$-linear exact faithful strong
symmetric monoidal functor.  ``Fiber functor'' in the classical
sense is the special case $\mathcal D = \Vec_k$; the target being a
non-vector-space category is what forces the tower in positive
characteristic.
\end{definition}

\subsection*{The higher Verlinde categories}

We will use the following facts about $\Ver_{p^n}(k)$, proved in
Benson--Etingof--Ostrik (2023).

\begin{lemma}[properties of the Verlinde tower]
\label{lem:verlinde-facts}
The categories $\Ver_{p^n}(k)$ are finite symmetric tensor
categories over $k$ and they form a tower
\[
  \Ver_p(k)\subset \Ver_{p^2}(k)\subset\Ver_{p^3}(k)\subset\cdots,
\]
with each inclusion exact and symmetric monoidal.  The bottom cases
are
\[
  \Ver_2(k) = \Vec_k,\qquad
  \Ver_3(k) = \sVec_k,\qquad
  \Ver_3^+(k) = \Vec_k.
\]
For every $p^n > 2$ one has a symmetric tensor subcategory
$\Ver_{p^n}^+(k)\subseteq\Ver_{p^n}(k)$; concretely
$\Ver_4^+(k)$ is the category of representations of the connected
group scheme $\alpha_2(k)$ with its unique symmetric structure, as
in Benson--Etingof (2019).
\end{lemma}

Intuition for $\Ver_p$: it is the ``semisimplification'' of
$\Rep_k(\mathbb Z/p)$: one kills every morphism $f$ with $\tr(gf)=0$
for all $g$; the simple objects correspond to Jordan blocks of
lengths $1,\ldots,p-1$.  For $p=2$ this collapses to
$\Vec_k$; for $p=3$ it gives supervector spaces.  Each higher level
$\Ver_{p^n}$ is built by abelian-envelope techniques from tilting
modules of $SL_2$.

\subsection*{The main theorem}

Here is the result we prove.

\begin{theorem}[main theorem]
\label{thm:main}
For every prime $p$, every algebraically closed field $k$ of
characteristic $p$, and every finite symmetric tensor category
$\mathcal C$ over $k$, there exist an integer $n\ge 1$ and a fiber
functor
\[
  \mathcal C \longrightarrow \Ver_{p^n}(k).
\]
\end{theorem}

This settles the finite case of Conjecture 1.4 of
Benson--Etingof--Ostrik (2023), including $p=2$.  The conjecture in
full strength is about categories of \emph{moderate growth}; our
theorem is for finite categories, and the finite case already
implies the finite part of all the consequences we discuss.

\subsection*{Proof outline}

We iteratively apply a semilinear higher Frobenius functor $\Phi$,
each application adding one fresh Verlinde factor while forcing
the original $\mathcal C$ factor to shrink into its Frobenius-exact
subcategory.  After at most $N$ iterations, where $N$ is the number
of simple objects of $\mathcal C$, all first components land in the
Serre closure of that subcategory.  The essential ingredients are:

\begin{enumerate}[label=\textbf{Step \arabic*.},ref=step:\arabic*]
  \item \textbf{Restricted tilting modulo a Steinberg module.}
        For every finite-dimensional module $D$ over the
        elementary abelian group $E_r = (\mathbb Z/p)^r$, the
        tensor product $\St_{r-1}\otimes D$ of a fixed Steinberg
        module with $D$ is a restriction of an $SL_2$-tilting
        module.  This is the Coulembier--Flake conjecture; we prove
        it by a vector-bundle argument on the affine plane.

  \item \textbf{Higher Frobenius functor.}
        The previous step lets us define, for every embedding
        $E_r\hookrightarrow U(k)\subseteq SL_2(k)$, a
        $k$-linear additive strong symmetric monoidal functor
        $\Phi:\mathcal C\to \mathcal C\boxtimes \Ver_{p^r}(k)$
        that is only semilinear ($\Phi(af) = a^{p^r}\Phi(f)$).

  \item \textbf{Exactness by support.}  For a well-chosen
        embedding of $E_r$ (with $r$ minimal), $\Phi$ is actually
        exact.  The proof uses a projective generator $P$, the
        module $M = \Hom(P,P^{\otimes p^r})$, and the theory of
        the projectivised rank variety.

  \item \textbf{Dimension descent.}  Even when $\Phi$ is only
        additive (not exact), the Frobenius--Perron dimension
        constrains composition factors of $\Phi(S)$ for simple
        $S$.  Iterating $\Phi$ finitely many times pushes the
        first component into the Serre closure of the
        Frobenius-exact tensor subcategory $\mathcal C_{\mathrm{ex}}$.
\end{enumerate}

After these four steps, a few more tasks finish the argument:
replace the Serre-closure factor by a fixed small Verlinde level
(via Etingof--Ostrik for $p>2$, Etingof--Gelaki for $p=2$); combine
all factors into one big tensor product and remove the total scalar
twist; and verify that the resulting functor is faithful.

Sections \ref{sec:groundwork}--\ref{sec:serre-finish} carry out the
four steps.  Section \ref{sec:consequences} records the standard
corollaries.

% =================================================================
\section{Groundwork}
\label{sec:groundwork}
% =================================================================

\subsection{Simple unit and a projective generator}

The definition of finite symmetric tensor category was chosen so
that the usual Tannakian toolkit applies.  We make this explicit.

\begin{lemma}[$\mathcal C$ is modules over a finite algebra]
\label{lem:modules-finite-alg}
Let $\mathcal C$ be a finite symmetric tensor category.  Then the
unit object is simple, and there is a projective generator
$P\in\mathcal C$.  The functor $\omega = \Hom_{\mathcal C}(P,-)$ is
exact and faithful, and $\mathcal C$ is equivalent, as a $k$-linear
abelian category, to the category of finite-dimensional modules
over the finite-dimensional $k$-algebra
$A = \End_{\mathcal C}(P)^{\mathrm{op}}$.
\end{lemma}

\begin{proof}
Simple unit is the Deligne--Milne argument.  If $U\subset\mathbf 1$
is a proper subobject with cokernel $V\ne0$, the epimorphism
$U\to V\otimes U$ coming from $\mathbf 1\twoheadrightarrow V$ has
zero composite with the subobject $V\otimes U\twoheadrightarrow V$.
So $V\otimes U=0$.  Put $W=\ker(\mathbf 1\to U^*)$.  Then $W$
sits inside a short exact sequence with $U$, $V$, and tensoring by
$W$ gives $W\simeq V$.  The endomorphism ring of the unit has no
idempotents, so $V=0$.

For the projective generator: pick surjections $Q_i\to S_i$ from
projectives onto the finitely many simples; set
$P=\bigoplus_i Q_i$.  Projectivity lifts surjections to $P$, so
every object is a quotient of $P^{\oplus m}$ for some $m$.  Exact
faithfulness comes from lifting along $P\to$ image.  To identify
$\mathcal C\simeq A\text{-mod}$, note that presenting objects by
copies of $P$ and using projectivity is fully faithful on the level
of presentations, and every finite $A$-module has a finite free
presentation.
\end{proof}

We fix such a $P$ and such an $A$ once and for all.  In particular,
$P\twoheadrightarrow\mathbf 1$, since the unit is one of the
simples.

\subsection{Deligne products}

For finite $k$-linear abelian categories $\mathcal A,\mathcal B$, the
\emph{Deligne product} $\mathcal A\boxtimes\mathcal B$ is the
universal recipient of functors that are $k$-bilinear and right
exact in each variable.  When $\mathcal A\simeq A\text{-mod}$ and
$\mathcal B\simeq B\text{-mod}$, we have
$\mathcal A\boxtimes\mathcal B\simeq (A\otimes_k B)\text{-mod}$,
and its simple objects are the exterior products of the simples of
the two factors (here we use $k$ algebraically closed).

\begin{lemma}[combining exact tensor functors]
\label{lem:combine-functors}
Let $\mathcal A_1,\ldots,\mathcal A_s,\mathcal Z$ be finite
symmetric tensor categories and let $F_i:\mathcal A_i\to\mathcal Z$
be exact $k$-linear strong symmetric monoidal functors.  Then
\[
  H:\mathcal A_1\boxtimes\cdots\boxtimes\mathcal A_s\to\mathcal Z,
  \qquad
  H(X_1\boxtimes\cdots\boxtimes X_s)
   = F_1(X_1)\otimes\cdots\otimes F_s(X_s),
\]
extends uniquely to an exact $k$-linear strong symmetric monoidal
functor.
\end{lemma}

\begin{proof}
The displayed multilinear functor has a right-exact extension by
the universal property of the Deligne product.  Symmetry in
$\mathcal Z$ gives the tensor constraint by rearranging factors, and
coherence extends by the same universal property.  The result
preserves duals.  A short exact sequence dualises, is sent to a
right-exact sequence, and is dualised back; hence the functor is
also left exact.
\end{proof}

\subsection{Frobenius--Perron dimension}

For finite tensor categories the Grothendieck ring $K_0(\mathcal C)$
is finitely freely generated by simple classes.  Symmetry makes it a
commutative ring, and the Frobenius--Perron theorem identifies a
unique positive normalised real character
$d = \FPdim : K_0(\mathcal C)\to\mathbb R$ with $d(\mathbf 1)=1$
and $d$ multiplicative on classes.  Key facts, all in Etingof--
Gelaki--Nikshych--Ostrik (2015):

\begin{lemma}[FP dimension toolkit]
\label{lem:fpdim}
Let $\mathcal C$ be a finite tensor category.
\begin{enumerate}[label=(\roman*)]
  \item $d(X)\ge 1$ for every nonzero $X$; equality if and only if
        $X$ is invertible.
  \item If $0\to X\to Y\to Z\to 0$ is exact, then
        $d(Y)=d(X)+d(Z)$.
  \item On a Deligne product, $d(A\boxtimes B)=d(A)\,d(B)$.
  \item If $F:\mathcal C\to\mathcal D$ is an exact strong monoidal
        functor of finite tensor categories, then
        $d(F(X))=d(X)$ for every $X$.
\end{enumerate}
\end{lemma}

Beware: the categorical dimension
$\dim_{\mathrm{cat}}(B)\in k$ that appears in \cref{sec:descent}
is a different, $k$-valued invariant; it is \emph{not} Frobenius--
Perron dimension.  We will only use categorical dimension for one
mod-$p$ congruence.

% =================================================================
\section{Step 1: Restricted tilting modules}
\label{sec:restricted}
% =================================================================

\subsection{The statement}

Let $G = SL_2(k)$, and let $U\simeq\mathbb G_a$ be its upper
unipotent subgroup, identified with $k$ via
$u(t) = \left(\begin{smallmatrix}1&t\\0&1\end{smallmatrix}\right)$.
For $r\ge 1$ put $E_r = (\mathbb Z/p)^r$ and let
$\iota:E_r\hookrightarrow U(k)$ be any injective homomorphism.
Write $R = R_\iota$ for the restriction of $G$-modules along $\iota$.

Let $L(h)$ denote the simple rational $G$-module of highest weight
$h\ge0$; let $\St_m = L(p^m-1)$ be the $m$-th \emph{Steinberg
module}, so $\St_0 = k$.  Let $\nabla(a) = \Sym^a(V^*)$ be the
$a$-th costandard module, where $V=k^2$ is the standard
representation, and $\Delta(a) = \nabla(a)^*$.  A $G$-module is
\emph{tilting} if it has both a costandard filtration and a Weyl
filtration; $T(a)$ is the indecomposable tilting module of highest
weight $a$.

\begin{theorem}[restricted-tilting theorem]
\label{thm:restricted-tilting}
Let $r\ge 1$, $E_r = (\mathbb Z/p)^r$, and $\iota:E_r\to U(k)$ any
injective homomorphism.  For every finite-dimensional $kE_r$-module
$D$ there is a tilting $G$-module $T$ such that
\[
  R_\iota(\St_{r-1})\otimes D \cong R_\iota(T)
\]
as $kE_r$-modules.
\end{theorem}

This proves Conjecture 5.4.2(1) of Coulembier--Flake (2026) and
Conjecture 1 of Coulembier--Flake (2025), for every prime.

\subsection{The plan of the proof}

We do three things:
\begin{enumerate}[label=(\alph*)]
  \item lift the arbitrary $kE_r$-module $D$ to a coherent rational
        $U$-module $\widehat D$ of bounded ``degree'' (a bound on
        polynomial degree of the unipotent action matrix);
  \item use $\widehat D$ to construct a $G$-equivariant vector
        bundle $\mathcal M$ on the plane $\mathbb A^2_k$ whose
        origin fibre has all simple composition factors below a
        height bound;
  \item tensor by $\St_{r-1}$, observe the resulting bundle's
        origin fibre is tilting, and compare origin fibre with
        $e_1$-fibre via a formal construction on jets.
\end{enumerate}

We then recover $D$ from the $e_1$-fibre.

\subsection{Lifting $D$ to a bounded $U$-module}

For a finite-dimensional rational $U$-module $D$, its
\emph{degree} is the largest exponent of $t$ appearing in the action
matrix $\rho_D(u(t))$; this does not depend on a chosen basis.

\begin{proposition}[bounded lift]
\label{prop:bounded-lift}
For every finite-dimensional $kE_r$-module $D$, there is a
finite-dimensional rational $U$-module $\widehat D$ of degree $< p^r$
such that $\widehat D|_{E_r}\cong D$ along $\iota$.
\end{proposition}

\begin{proof}
Let $H\subseteq k$ be the image of $E_r$ under $\iota$.  The
vector space
$C_q = \operatorname{span}_k\{1,t,\ldots,t^{q-1}\}\subseteq k[t]$
(with $q=p^r$) is a subcoalgebra of $k[t]$ with
$\Delta f(t,s) = f(t+s)$.  Evaluation at $H$ gives a coalgebra map
$\operatorname{ev}_H : C_q \to k^H = \mathcal O(E_r)$, which is a
bijection: the two sides have dimension $q$, and a degree-$<q$
polynomial vanishing at the $q$ distinct points of $H$ is zero.
Transporting the $\mathcal O(E_r)$-coaction defining $D$ across
$\operatorname{ev}_H$, then including $C_q \subset k[t]$, gives a
rational $U$-module of degree $<q$.  Its restriction is $D$.
\end{proof}

\subsection{The $G$-equivariant bundle on the plane}

Now consider the orbit map $\pi : G\to\mathbb A^2\setminus\{0\}$,
$g\mapsto g e_1$, with $e_1=(1,0)$.  This is a right $U$-torsor:
the fibres are right $U$-cosets and on the standard opens $x\ne 0$,
$y\ne 0$ it has obvious rational sections.  Form the associated
vector bundle
\[
  \mathcal E = G\times^U \widehat D,
  \qquad (g,d)\sim(gu,u^{-1}d),
\]
on the punctured plane, with $G$ acting by left multiplication.
The fibre at $e_1$ is $\widehat D$ itself.

\begin{proposition}[extension and weight bound]
\label{prop:extension-bound}
The bundle $\mathcal E$ extends to a $G$-equivariant vector bundle
$\mathcal M$ on the whole affine plane, and every simple
composition factor of the origin fibre
$M_0 = \mathcal M|_0$ has highest weight $< q$.
\end{proposition}

\begin{proof}
\emph{Extension.}  A coherent extension $\mathcal F$ exists on the
noetherian affine plane.  Its double dual $\mathcal M = \mathcal
F^{**}$ is a coherent reflexive sheaf restricting to $\mathcal E$.
On the regular surface $\mathbb A^2$ every reflexive coherent sheaf
is locally free: at a closed point the stalk has depth two, so the
Auslander--Buchsbaum formula gives projective dimension zero, and at
the generic point and at the generic point of a curve the stalks are
trivially free.  Hence $\mathcal M$ is a vector bundle.

\emph{Equivariance.}  Over the punctured plane we have a
$G$-equivariant bundle isomorphism
$\pi_2^*\mathcal M\cong a^*\mathcal M$ for the action
$a:G\times\mathbb A^2\to\mathbb A^2$ and projection
$\pi_2:G\times\mathbb A^2\to\mathbb A^2$.  Both sides are locally
free, so the Hom-sheaf is locally free and its sections extend
across the codimension-two subset $G\times\{0\}$; the extensions of
the isomorphism and its inverse remain inverse because the two
composites agree on the punctured plane and hence, by uniqueness,
everywhere.  The identity and cocycle conditions follow the same
way.

\emph{Weight bound at the origin.}  The line $\ell = ke_1$ is
fixed pointwise by $U$.  For $a\in k^\times$ put
$h_a = \mathrm{diag}(a,a^{-1})$; the bundle action of $h_a$ gives
a linear identification of fibres at $e_1$ and at $ae_1$, and
$h_a^{-1}u(t)h_a = u(a^{-2}t)$.  Hence on every nonzero fibre over
$\ell$ the $U$-action has degree $<q$.  Let
$A = \Gamma(\ell,\mathcal E nd(\mathcal M|_\ell))$ and write the
$U$-action as an element of $A\otimes_k k[t]$,
$\sum_j b_j t^j$.  For $j\ge q$, $b_j$ vanishes on every nonzero
fibre, so $b_j = 0$ (since $k$ is infinite and $A$ is torsion-free
over $k[t]$).  So the origin action also has degree $<q$, and the
same holds for any subquotient.

\emph{Passing degree to weights.}  Write a simple $G$-module as a
tensor product of Frobenius twists of $\Sym^{h_i}(k^2)$ with
$h=\sum_i h_ip^i$, $0\le h_i<p$.  Its $U$-action has degree exactly
$h$ (the coefficient of $t^h$ from
$\bigotimes_i e_2^{h_i}$ to $\bigotimes_i e_1^{h_i}$ is nonzero).
Hence every simple composition factor $L(h)$ of $M_0$ has
$h<q$.
\end{proof}

\subsection{Tilting under a height bound}

We use two standard facts about $SL_2$-tilting modules, both due to
Donkin and both valid in every positive characteristic including
$p=2$:
\begin{equation}\label{eq:sl2-tilting-basics}
  L(a) = T(a)\quad(0\le a<p),
  \qquad
  T(a)\otimes T(b)^{(1)}\simeq T(a+pb)
  \quad(p-1\le a\le 2p-2,\ b\ge 0),
\end{equation}
where $(1)$ means pullback along the Frobenius.  See Benson--
Etingof--Ostrik (2023), Proposition 3.1.

\begin{lemma}[Steinberg $\otimes$ bounded weight is tilting]
\label{lem:steinberg-tilting}
Let $M$ be a finite-dimensional rational $SL_2(k)$-module with every
composition factor of the form $L(h)$ for $0\le h<p^m$ (where
$m\ge 1$).  Then $\St_{m-1}\otimes M$ is tilting.
\end{lemma}

\begin{proof}
Reduces by composition series to $M = L(h)$.

For $m=1$, $h<p$ and $\St_0=k$, so $\St_0\otimes L(h) = T(h)$ is
tilting by \eqref{eq:sl2-tilting-basics}.

For $m>1$, write $h = d + p h'$ with $0\le d<p$ and $h'<p^{m-1}$.
Steinberg's tensor product theorem gives
\[
  \St_{m-1}\otimes L(h) \cong
  L(d)\otimes \St_1\otimes
  \bigl(\St_{m-2}\otimes L(h')\bigr)^{(1)}.
\]
By induction the bracket is a direct sum of tilting modules.  Apply
Donkin's formula \eqref{eq:sl2-tilting-basics} with $a=p-1$ and
$b$ equal to the highest weight of each summand; each summand times
$\St_1$ (a Frobenius twist of a tilting) is tilting.  Tensoring
further with $L(d) = T(d)$ preserves tilting because tilting modules
are closed under tensor products.
\end{proof}

Applying \cref{lem:steinberg-tilting} with $m=r$ to $M_0 = \mathcal
M|_0$, we have:
\begin{equation}\label{eq:origin-tilting}
  T := \St_{r-1}\otimes M_0
\end{equation}
is a tilting $G$-module.  The remaining step is to compare $T$
(the origin fibre of
$\mathcal N = \St_{r-1}\otimes_k \mathcal M$) with the $e_1$-fibre
of $\mathcal N$.

\subsection{Formal comparison of fibres}

\begin{lemma}[comparison with a tilting origin fibre]
\label{lem:formal-comparison}
Let $\mathcal N$ be a $G$-equivariant vector bundle on $\mathbb A^2$
whose origin fibre $T = \mathcal N|_0$ is tilting.  Then
$\mathcal N|_{e_1}\cong T|_U$ as rational $U$-modules.
\end{lemma}

\begin{proof}
\emph{Jet trivialisation.}  Let $A=k[x,y]$, $\mathfrak m=(x,y)$,
and $N=\Gamma(\mathbb A^2,\mathcal N)$, a possibly
infinite-dimensional rational $G$-module.  Local freeness gives
canonical $G$-isomorphisms
\begin{equation}
  \mathfrak m^j N/\mathfrak m^{j+1}N \cong
  \Sym^j(V^*)\otimes_k T.
\end{equation}
The target has a good filtration.  Tensor products of Sym's of
$V^*$ have good filtrations: via multiplication by the determinant
$d=x_1y_2-y_1x_2$, the sequence
$0\to\Sym^{a-1}V^*\otimes\Sym^{b-1}V^*\to
\Sym^a V^*\otimes\Sym^b V^*\to\Sym^{a+b}V^*\to 0$ is exact.

Now $T$ has both a Weyl and a good filtration, so for every $j$,
\[
  \operatorname{Ext}^1_G\bigl(T,\Sym^j(V^*)\otimes T\bigr) = 0
\]
by Cline--Parshall--Scott--van der Kallen.

Starting from the identity $s_1:T\to N/\mathfrak m N$, we lift
successively to $G$-maps $s_j:T\to N/\mathfrak m^j N$ that reduce
modulo $\mathfrak m^j$ to $s_{j-1}$; the obstruction is exactly the
Ext group just computed.  Extending scalars, $s_j$ gives an
$A/\mathfrak m^j$-linear map
$(A/\mathfrak m^j)\otimes_k T\to N/\mathfrak m^j N$, which is an
isomorphism by Nakayama (both sides free of the same rank and the
map is an isomorphism on the residue field).  The inverse limit is a
$G$-equivariant isomorphism
$k[[x,y]]\otimes_k T\xrightarrow{\sim}\widehat N$.

\emph{Fibre comparison.}  Restrict to the fixed line $y=0$, pass
to $K = k((x))$: this gives a $U_K$-isomorphism
$\varphi:T_K\cong \mathcal N_{xe_1}$.  To cancel the parameter
mismatch we use the torus element $h = \mathrm{diag}(x,x^{-1})$:
since $hu(t)h^{-1} = u(x^2 t)$, the composite
$f = b_h^{-1}\varphi\rho_T(h)$ intertwines the original
$U$-actions (where $b_h : \mathcal N_{e_1}\to\mathcal N_{xe_1}$ is
bundle transport).

\emph{Descent to $k$.}  The space
$L = \Hom_U(T,\mathcal N_{e_1})$ is cut out by finitely many
$k$-linear equations in fixed $k$-bases, so $L_K = L\otimes_k K$.
The determinant on $L$ is nonzero over $K$ (witnessed by $f$), so
nonvanishing at some $k$-point (since $k$ is infinite) gives the
required isomorphism over $k$.
\end{proof}

\begin{proof}[Proof of \cref{thm:restricted-tilting}]
Apply \cref{prop:extension-bound} to get $\mathcal M$ with origin
fibre $M_0$ whose simple factors have highest weight $<q=p^r$.
By \cref{lem:steinberg-tilting}, $T = \St_{r-1}\otimes M_0$ is
tilting.  Apply \cref{lem:formal-comparison} to
$\mathcal N = \St_{r-1}\otimes_k\mathcal M$: its origin fibre is
$T$, so
\[
  \mathcal N|_{e_1} \cong T|_U.
\]
But by construction $\mathcal N|_{e_1}$ restricts to
$R(\St_{r-1})\otimes D$ along $E_r\hookrightarrow U(k)$, hence so
does $T$.  Set $T_{\mathrm{out}} := T$; the theorem follows.
\end{proof}

% =================================================================
\section{Step 2: The higher Frobenius functor}
\label{sec:higher}
% =================================================================

\subsection{The functor $\Theta$ from group modules to Verlinde}

Fix $r\ge 1$, $q=p^r$, and an embedding
$E = E_r \hookrightarrow U(k)$.  We write $R$ for restriction of
$SL_2$-modules along this embedding, and $\mathcal W = \Ver_{p^r}(k)$.
Let $\mathcal T\subseteq \Rep_k(SL_2)$ be the category of tilting
modules, and let $I_r$ be the tensor ideal generated by
$\St_r = L(p^r-1)$.

Concretely, $I_r$ is defined by demanding that $T(a)$ (the
indecomposable tilting module of highest weight $a$) vanish in
$\mathcal T/I_r$ whenever $a\ge p^r - 1$; this is the standing
convention in this circle of papers.

The functor $\mathcal T/I_r\to\mathcal W$ is an equivalence onto a
full subcategory (Benson--Etingof--Ostrik (2023), Theorem 4.2(i)).
Stable restriction gives a faithful quotient functor
$\mathcal T/I_r\to\operatorname{Stab}(E) :=
\Rep_k(E)/(\text{projectives})$.

Coulembier--Flake (2026, Theorem 5.4.3) state: \emph{if for every
finite-dimensional $E$-module $D$ the tensor $R(\St_{r-1})\otimes D$
is a restriction of a tilting, then there is a $k$-linear strong
symmetric monoidal functor}
\begin{equation}\label{eq:theta}
  \Theta : \Rep_k(E)\longrightarrow \mathcal W
  \quad\text{with}\quad
  \Theta(R(T)) \simeq \Sigma_r(T\bmod I_r).
\end{equation}
The hypothesis is exactly \cref{thm:restricted-tilting}.  So
\cref{thm:restricted-tilting} produces $\Theta$, once and for all
embeddings.

$\Theta$ is not exact; it is additive and symmetric monoidal.

\begin{lemma}[$\Theta$ detects nonprojectives]
\label{lem:theta-detection}
Let $T$ be a tilting module whose restriction $R(T)$ is not
projective as an $E$-module.  Then $\Theta(R(T))\ne0$.  Moreover,
if $H<E$ is a proper subgroup, then every module induced from a
$kH$-module is killed by $\Theta$.
\end{lemma}

\begin{proof}
If $T\bmod I_r=0$ then $R(T)$ is projective (stable restriction
result).  Otherwise $T\bmod I_r\ne0$, and $\Sigma_r$ being fully
faithful implies $\Theta(R(T))\ne0$.

For induced modules: $R(\St_r)$ has dimension $q$, and its
restriction is projective (the mod $I_r$ quotient kills
$\St_r$).  Since $kE$ is local, a projective of dimension $q$
is the free module, so $\Theta$ kills every projective.  When
$H<E$ is proper, $\St_{r-1}\in I_s$ at the level $s =
\rank(H)$ so $\Res^E_H R(\St_{r-1})$ is projective; the tensor
identity
$S\otimes\Ind_H^E D \cong \Ind_H^E(\Res_H^E S\otimes D)$ therefore
gives a projective $E$-module, on which $\Theta$ vanishes.  Since
$\Theta(S)\ne0$ and tensoring by a nonzero object is faithful,
$\Theta(\Ind_H^E D)=0$.
\end{proof}

\subsection{Internal evaluation}

Let $\mathcal C$ be a finite symmetric tensor category with
projective generator $P_0$.  Write $A = \End_{\mathcal C}(P_0)^{op}$,
so $\mathcal C\simeq A\text{-mod}$.  An object of
$\mathcal C\boxtimes\Rep_k(E)$ is an object of $\mathcal C$
equipped with a commuting $E$-action.  Applying $\Theta$ to the
underlying $E$-module and to the action maps produces an $A$-module
object in $\mathcal W$, hence an object of
$\mathcal C\boxtimes\mathcal W$.  This is the \emph{internal
extension} $\Theta_{\mathcal C}$ of $\Theta$ (Coulembier--Flake
2026, Example 3.2.4 and Proposition 3.2.6; requires only
additivity of $\Theta$, not exactness).

\begin{lemma}[evaluation at projectives]
\label{lem:projective-evaluation}
For $Q\in\mathcal C$ projective and $Z$ an $E$-object of
$\mathcal C$,
\[
 (\Hom_{\mathcal C}(Q,-)\boxtimes\operatorname{Id}_\mathcal W)
   (\Theta_{\mathcal C}(Z))
 \cong \Theta(\Hom_{\mathcal C}(Q,Z)).
\]
\end{lemma}

\begin{proof}
In $A\text{-mod}$, $Q$ is a direct summand of a free module, so
$\Hom_A(Q,-)$ is an additive idempotent of the identity functor on
$A\text{-mod}$.  An additive functor preserves split idempotents and
direct sums; applying $\Theta$ therefore commutes with
$\Hom_A(Q,-)$.
\end{proof}

\subsection{The tensor-power functor $\Phi$}

Give $X^{\otimes q}$ the $E$-action coming from regular permutation
of tensor positions.  Define
\begin{equation}\label{eq:phi}
  \Phi(X) = \Theta_{\mathcal C}(X^{\otimes q})
        \in \mathcal C\boxtimes \mathcal W.
\end{equation}

\begin{lemma}[monoidal structure and semilinearity]
\label{lem:phi-monoidal}
$\Phi$ is additive, strong symmetric monoidal, strong $k$-linear
after twisting scalars by Frobenius $a\mapsto a^q$, and satisfies
$\Phi(af) = a^q\Phi(f)$ for scalars $a\in k$.
\end{lemma}

\begin{proof}
The monoidal constraint is the $E$-equivariant reordering
\[
  (X\otimes Y)^{\otimes q}\cong X^{\otimes q}\otimes Y^{\otimes q}
\]
with the diagonal action on the right, applied through the internal
monoidal extension of Coulembier--\hspace{0pt}Flake (2026, Section 3.3.2).
Additivity can be checked directly: in the expansion of
$(X\oplus Y)^{\otimes q}$, a nonconstant colouring of tensor
positions has a proper stabiliser in $E$, so the sum over its
$E$-orbit is induced from that stabiliser and hence is killed by
$\Theta$ (using \cref{lem:theta-detection}); only the two constant
colourings survive.  Semilinearity is immediate from the same
calculation on scalars.
\end{proof}

Combining \cref{eq:phi} and \cref{lem:projective-evaluation}:
\begin{equation}\label{eq:phi-evaluation}
 (\Hom_{\mathcal C}(Q,-)\boxtimes\operatorname{Id}_\mathcal W)(\Phi(X))
 \cong \Theta(\Hom_{\mathcal C}(Q,X^{\otimes q}))
\end{equation}
for every projective $Q$.

\subsection{The nonprojective image criterion}

\begin{proposition}[$\Phi$ exact iff image nonzero]
\label{prop:phi-exactness}
If there are projective objects $P,Q\in\mathcal C$ with
$\Theta(\Hom_{\mathcal C}(Q,P^{\otimes q}))\ne0$, then $\Phi$ is
exact.
\end{proposition}

\begin{proof}
\Cref{eq:phi-evaluation} gives $\Phi(P)\ne0$.  For a short exact
sequence in $\mathcal C$, tensoring by $P$ splits it because tensor
products of projectives with arbitrary objects are projective.
The additive $\Phi$ preserves this split sequence.  Monoidality
identifies the result as the original image sequence tensored by
$\Phi(P)$.  Tensoring by the nonzero $\Phi(P)$ is exact and faithful,
and reflects exactness; the image sequence is exact.
\end{proof}

% =================================================================
\section{Step 3: Choosing an exact embedding}
\label{sec:support}
% =================================================================

Now we need to actually find $r$ and $\iota$ for which the criterion
of \cref{prop:phi-exactness} fires.  The strategy: take $r$ minimal
so that $M(r) := \Hom_{\mathcal C}(P,P^{\otimes p^r})$ (as a
$kE_r$-module via the regular permutation of tensor slots) is
nonprojective; and then choose the embedding $\iota$ so that
$R(\St_{r-1})\otimes M(r)$ stays nonprojective.

\subsection{Norms and $p$-group projectivity}

For a finite group $H$ acting on a finite-dimensional vector space,
$\nu_H:X_H\to X^H$ is the \emph{norm map}: it factors
$\sum_{h\in H} h:X\to X$.

\begin{lemma}[$p$-group projectivity via norm]
\label{lem:norm-projectivity}
Let $k$ be a field of characteristic $p$, $H$ a finite $p$-group,
$M$ a finite-dimensional $kH$-module.  Then $M$ is projective iff
$\nu_H(M)$ is an isomorphism.  More generally, for any finite group
$G$, the norm map on a projective $kG$-module is an isomorphism.
\end{lemma}

\begin{proof}
The group ring $kH$ is local self-injective: its only simple is $k$
with trivial action; its socle is generated by $N_H=\sum_{h\in H}h$;
the augmentation ideal is nilpotent (induct on $|H|$, use a central
order-$p$ element $z$ with $(z-1)^p=0$, descend to $k[H/\langle
z\rangle]$); and a nondegenerate pairing $ab\mapsto$ coefficient of
$1$ identifies $kH$ with its linear dual as a module.

For $kH$-free modules, the norm map is an iso.  Converse: if
$N_Hm\ne0$, the map $kH\to M$, $a\mapsto am$ is injective and splits
by self-injectivity, so $M$ has a free summand.  Strip off all free
summands, leaving $M'$ with $N_HM'=0$; if $M'\ne0$ then $N_H=M'_H=0$
by Nakayama, contradiction with an iso.  So $M'=0$.

For a general finite group and a projective module: restrict to Sylow
$p$-subgroups; each restriction is projective; the norm iso there
implies the total norm iso (same argument).
\end{proof}

\begin{lemma}[independent actions multiply]
\label{lem:product-actions}
Let finite $p$-groups $H_i$ act on objects $X_i\in\mathcal C$
($i=1,\ldots,s$).  If $\omega(X_i)$ is projective over $kH_i$ for
exact faithful $\omega = \Hom_{\mathcal C}(P,-)$, then
$\omega(X_1\otimes\cdots\otimes X_s)$ is projective over
$k[H_1\times\cdots\times H_s]$ (independent actions), and over every
subgroup thereof.
\end{lemma}

\begin{proof}
Categorically, $(X\otimes Y)_{H\times K}\cong X_H\otimes Y_K$ and
$(X\otimes Y)^{H\times K}\cong X^H\otimes Y^K$
(take kernels/cokernels one group at a time).  So
$\nu_{H\times K}(X\otimes Y) = \nu_H(X)\otimes\nu_K(Y)$.  If each
$\nu_{H_i}(X_i)$ is iso, so is the product norm.  Applying exact
$\omega$ and \cref{lem:norm-projectivity} finishes.  Restrict to a
subgroup $H\le H_1\times\cdots\times H_s$: $kH_i$ is free over
$kH$ for any finite group, so projectivity is preserved.
\end{proof}

\subsection{Existence of a minimal $r$}

Let $E_a=(\mathbb Z/p)^a$ act on $p^a$ positions by its regular
representation; this induces an $E_a$-action on $P^{\otimes p^a}$.
Set $M(a) = \omega(P^{\otimes p^a})$, an $E_a$-module.

\begin{proposition}[first nonprojective power]
\label{prop:first-nonprojective}
There is a least integer $r\ge 1$ for which $M(r)$ is not projective
over $kE_r$.  For this $r$, the restriction of $M(r)$ to every
proper subgroup of $E_r$ is projective.
\end{proposition}

\begin{proof}
Suppose all $M(a)$ projective.  Fix $d$ and an elementary abelian
$p$-subgroup $H\le\mathfrak S_d$.  Every $H$-orbit in
$\{1,\ldots,d\}$ is regular because $H$ is abelian; orbitwise,
$P^{\otimes d}$ is a product of $P^{\otimes p^{a_i}}$, each
projective over $k E_{a_i}$ by assumption.  By
\cref{lem:product-actions}, $\omega(P^{\otimes d})$ is projective
over $kH$ for the product of orbit groups, hence over the diagonal
copy of $H$ (which is an injective image).  Chouinard's theorem then
says projective over every elementary abelian $p$-subgroup implies
projective over $k\mathfrak S_d$.

Now apply $P\twoheadrightarrow\mathbf 1$ ($\mathfrak S_d$-equivariant,
trivial action on the target) and exactness of $\omega$ followed by
coinvariants:
\[
 \omega(P^{\otimes d})_{\mathfrak S_d} \twoheadrightarrow
 \omega(\mathbf 1)\ne0.
\]
By \cref{lem:norm-projectivity}, the total symmetrizer
$b_d = \sum_{\sigma\in\mathfrak S_d}\sigma$ acts nontrivially on
$P^{\otimes d}$.

On the other hand the action map
$k\mathfrak S_d\to\End_{\mathcal C}(P^{\otimes d})$ cannot be
injective: length grows exponentially in $d$ (bounded by
$\ell(P)b^{d-1}$ with $b=\max_i\ell(S_i\otimes P)$) and the
dimension of $\Hom(X,Y)$ is bounded by $\ell(X)\ell(Y)$, whereas
$d!$ grows like a factorial.  The kernels of these action maps
constitute a tensor ideal in the permutation category, and every
nonzero tensor ideal in $\underline{\mathrm{Sym}}$ contains some
$b_d$ in characteristic $p>0$ (Coulembier--Etingof--Ostrik 2023,
Lemma 4.10).  Its proof uses Kleshchev's faithfulness theorem.
Contradiction.  So some $M(r)$ is nonprojective; take $r$ minimal.

For the last claim: $H<E_r$ of rank $a<r$ acts on the regular
$E_r$-set as a disjoint union of regular $H$-orbits; each orbit's
power-evaluation is projective by minimality (rank $a$) or trivially
if $H=1$; apply \cref{lem:product-actions} and restrict to the
diagonal.
\end{proof}

\subsection{Rank varieties and support}

Let $kE_r$-modules be graded by the augmentation ideal
$\mathfrak m = (g_i-1)_i$.  The \emph{projectivised rank variety} of
a finite-dimensional module $D$ is
\[
 \supp(D) =
  \bigl\{[\alpha]\in\mathbb P^{r-1}(k) :
   D \text{ not free over } k[t]/(t^p),
   \ t\mapsto\textstyle\sum_i\alpha_i(g_i-1)
  \bigr\}.
\]
Two standard facts (Benson 2017; Coulembier--Flake 2025):
\begin{equation}\label{eq:support-properties}
 \supp(D) = \varnothing \iff D \text{ projective},
 \qquad
 \supp(D\otimes D') = \supp(D)\cap\supp(D').
\end{equation}

\begin{lemma}[support point independence]
\label{lem:support-independent}
If $M$ is a nonprojective $kE_r$-module with all restrictions to
proper subgroups projective, then every $[\alpha]\in\supp(M)$ has
$\mathbb F_p$-linearly independent coordinates.
\end{lemma}

\begin{proof}
For $c\in(\mathbb F_p^r)^\ast$, let $H=\ker c$ and
$N = \Ind_H^{E_r}k$, the regular representation of $E_r/H\cong
\mathbb Z/p$ (identified with $k[z]/(z^p)$, generator acts by
$1+z$).  The element $\sum_i\alpha_i(g_i-1)$ acts on $N$ by
$h_\alpha(z) = \sum_i\alpha_i((1+z)^{c_i}-1)$, whose linear
coefficient is $\sum_i c_i\alpha_i$.

If $\sum_i c_i\alpha_i\ne0$, then $t\mapsto h_\alpha(z)$ is an iso
of truncated polynomial algebras, so $N$ is free as $k[t]/(t^p)$;
if $\sum_i c_i\alpha_i=0$, then $h_\alpha\in(z^2)$, so
$h_\alpha^{p-1}=0$, incompatible with a free module of dimension
$p$; so $N$ is not free.

Thus
$\supp(N) = \{[\alpha] : \sum_i c_i\alpha_i=0\}$.  The tensor
identity gives $M\otimes\Ind_H^{E_r}k\cong\Ind_H^{E_r}(M|_H)$, which
is projective, so $\supp(M)\cap\supp(N)=\varnothing$.  Hence
$\sum_i c_i\alpha_i\ne0$ for every nonzero $c$.  In particular the
coordinates of every $[\alpha]\in\supp(M)$ are $\mathbb F_p$-linearly
independent.
\end{proof}

\begin{lemma}[Steinberg support in coordinates]
\label{lem:steinberg-support}
Let $\lambda_1,\ldots,\lambda_r\in k$ be $\mathbb F_p$-linearly
independent and define $\iota_\lambda : E_r\to SL_2(k)$,
$g_i\mapsto u(\lambda_i)$.  Write $R_\lambda$ for restriction.  Then
\begin{equation}\label{eq:steinberg-support}
 \supp(R_\lambda(\St_{r-1})) =
 \left\{[\alpha] : \sum_i\alpha_i\lambda_i^{p^j}=0
   \ \ (0\le j\le r-2)\right\}.
\end{equation}
\end{lemma}

\begin{proof}
On $\St_1 = \Sym^{p-1}(k^2)$, the unipotent $u(a)$ acts as
$\exp(aN) = \sum_{b=0}^{p-1}a^bN^b/b!$ where $N$ is a $p\times p$
Jordan block.  Then $\sum_i\alpha_i(g_i-1)$ acts by a polynomial in
$N$ whose linear coefficient is $\sum_i\alpha_i\lambda_i$; by the
argument of \cref{lem:support-independent}, the resulting
$p$-dimensional module is free iff the coefficient is nonzero.

Frobenius twist replaces $\lambda_i$ by $\lambda_i^{p^j}$.
Since $\St_{r-1}\cong\bigotimes_{j=0}^{r-2}\St_1^{(j)}$, the tensor
product formula \eqref{eq:support-properties} gives an intersection
of hyperplanes
$M_j : \sum_i\alpha_i\lambda_i^{p^j}=0$, exactly \cref{eq:steinberg-support}.
For $r=1$ the system is empty and $\St_0=k$ has
$\supp(\St_0)=\mathbb P^0(k)$, consistent.
\end{proof}

\subsection{Moore's lemma: choosing $\lambda$}

\begin{lemma}[$\lambda$ solving prescribed equations]
\label{lem:moore-selection}
Let $\alpha_1,\ldots,\alpha_r\in k$ be $\mathbb F_p$-linearly
independent.  Then there exist $\mathbb F_p$-linearly independent
$\lambda_1,\ldots,\lambda_r\in k$ with
\begin{equation}\label{eq:moore}
 \sum_i\alpha_i\lambda_i^{p^j} = 0,
 \qquad 0\le j\le r-2.
\end{equation}
For $r=1$ the system is empty.
\end{lemma}

\begin{proof}
$r=1$: take $\lambda_1\ne0$.  For $r\ge2$: step is standard
graded linear algebra.  Put $\beta_i = \alpha_i^{p^{-(r-2)}}$ (using
that $k$ is perfect).  Let
$B = (\beta_i^{p^s})_{s,i}$ be the $(r-1)\times r$ matrix.

Rows of $B$ are linearly independent: otherwise a nontrivial
additive polynomial $f(X)=\sum_{s=0}^{r-2}a_s X^{p^s}$ vanishes on
the $p^r$-element span of the $\beta_i$, but $\deg f\le p^{r-2}$, so
$f=0$.

Row space of $B$ contains no nonzero $\mathbb F_p^r$-vector: if
$c_i = f(\beta_i)\in\mathbb F_p$, then the polynomial $f^p-f$
vanishes on the same $p^r$-element set and has degree
$\le p^{r-1}$; so $f^p-f=0$, forcing $f=0$ (since $\deg f<p^{r-1}$
and $f$ has zero constant term).

Choose $\lambda\in\ker B$ nonzero.  Kernel is one-dimensional.  Any
$c\in\mathbb F_p^r$ with $\sum_i c_i\lambda_i=0$ belongs to the
annihilator of $\ker B$, i.e., to the row space of $B$, hence is
$c=0$.  So $\lambda_i$ are $\mathbb F_p$-linearly independent.
Reading rows of $B$ in reverse order, $(\alpha_i^{p^{-j}})_i$, and
raising the equations $\sum_i\lambda_i^{p^s}\alpha_i^{p^{-(r-2)}p^s}
=0$ to the $p^{r-2}$-th power gives \eqref{eq:moore}.
\end{proof}

\begin{proposition}[embedding kills no support]
\label{prop:support-selection}
Let $M$ be a nonprojective $kE_r$-module with projective restriction
to every proper subgroup.  Then there is an embedding
$\iota_\lambda:E_r\hookrightarrow U(k)$ with
$R_\lambda(\St_{r-1})\otimes M$ nonprojective.
\end{proposition}

\begin{proof}
Pick $[\alpha]\in\supp(M)$; coordinates are $\mathbb F_p$-linearly
independent by \cref{lem:support-independent}.  Choose $\lambda$ by
\cref{lem:moore-selection}; \cref{lem:steinberg-support} then gives
$[\alpha]\in\supp(R_\lambda(\St_{r-1}))$.  So the tensor product
support \eqref{eq:support-properties} is nonempty.  Hence
$R_\lambda(\St_{r-1})\otimes M$ is nonprojective.
\end{proof}

\subsection{Exactness}

\begin{theorem}[exact higher Frobenius functor]
\label{thm:exact-higher}
Let $\mathcal C$ be a finite symmetric tensor category over $k$ of
characteristic $p>0$.  Then there exist $r\ge1$ and an embedding
$E_r\hookrightarrow U(k)$ such that the construction of
\cref{sec:higher} gives an exact additive strong symmetric monoidal
functor
\[
 \Phi:\mathcal C\to\mathcal C\boxtimes\Ver_{p^r}(k)
\]
with $\Phi(X)=\Theta_{\mathcal C}(X^{\otimes p^r})$ and
$\Phi(af)=a^{p^r}\Phi(f)$.  Moreover, for every projective $Q$ the
evaluation \eqref{eq:phi-evaluation} holds.
\end{theorem}

\begin{proof}
Let $P$ be the projective generator and $r$ the minimal integer of
\cref{prop:first-nonprojective}.  Then
$M = \Hom_{\mathcal C}(P,P^{\otimes p^r})$ is nonprojective with
projective restriction to every proper subgroup of $E_r$.
By \cref{prop:support-selection} pick $\lambda$ so that
$R_\lambda(\St_{r-1})\otimes M$ is nonprojective.
By \cref{thm:restricted-tilting},
$R_\lambda(\St_{r-1})\otimes M\cong R_\lambda(T)$ for a tilting
module $T$.  By \cref{lem:theta-detection},
\[
 0 \ne \Theta(R_\lambda(T))
 \cong \Theta(R_\lambda(\St_{r-1}))\otimes\Theta(M).
\]
So $\Theta(M)\ne0$.  \Cref{prop:phi-exactness} with $P=Q$ shows
$\Phi$ is exact.
\end{proof}

% =================================================================
\section{Step 4: Dimension descent}
\label{sec:descent}
% =================================================================

We now have an exact $\Phi:\mathcal C\to\mathcal C\boxtimes\mathcal
W$, $q=p^r$, $\mathcal W=\Ver_{p^r}(k)$, semilinear.
Its image closes up on the first factor to the Serre closure of a
Frobenius-exact subcategory.

\subsection{Ordinary Frobenius and a mod-$p$ identity}

For $Y\in\mathcal C$, let $c$ be the cyclic permutation of the $p$
factors of $Y^{\otimes p}$ and $D = 1-c$ (so $D^p=0$).  Define the
Etingof--Ostrik components:
\[
 F_i(Y) = \frac{\ker D\cap\operatorname{im}D^{i-1}}
                {\ker D\cap\operatorname{im}D^i},
 \qquad 1\le i<p.
\]
If $\ell_i\in\Ver_p(k)$ is the simple object represented by a
Jordan block of length $i$, then
$F(Y)=\bigoplus_{i=1}^{p-1}F_i(Y)\boxtimes\ell_i$.  In the
semilinear convention, the $F_i(Y)$ are objects of $\mathcal C$
itself.  $F$ is not assumed exact.

\begin{lemma}[mod-$p$ identity in $K_0$]
\label{lem:ordinary-congruence}
In $K_0(\mathcal C)/pK_0(\mathcal C)$,
\[
  [Y]^p = \sum_{i=1}^{p-1}i[F_i(Y)].
\]
Consequently, for every $r\ge 1$,
\begin{equation}\label{eq:iterated-ord}
  [Y]^{p^r} = \sum_{i_1,\ldots,i_r=1}^{p-1}(i_1\cdots i_r)
  [F_{i_r}\cdots F_{i_1}(Y)] \pmod p.
\end{equation}
\end{lemma}

\begin{proof}
Put $Z=Y^{\otimes p}$ and $K_j = \ker D\cap\operatorname{im}D^j$
for $0\le j\le p$.  The short exact sequences
$0\to K_j\to\operatorname{im}D^j\to\operatorname{im}D^{j+1}\to0$
give $[Z]=\sum_{j=0}^{p-1}[K_j]$.  Summing the successive quotients
in the filtration $K_0\supseteq\cdots\supseteq K_p=0$ gives the
integral identity
$[Z]=\sum_i i[F_i(Y)]+p[K_{p-1}]$.  Mod $p$ this is the first
claim.  Reraising to successive $p$-th powers in the commutative
ring and applying to concrete objects gives the iterated form.
\end{proof}

\subsection{Equality in dimension forces exactness}

An exact strong monoidal $\sigma$-semilinear functor still preserves
Frobenius--Perron dimension: it sends nonzero objects to nonzero
objects (from $\mathbf 1\hookrightarrow X\otimes X^*$), so
composing with FP dimension gives a positive normalised character on
$K_0(\mathcal C)$; the FP characterization forces it to be the FP
dimension of the source.  In particular
\[
 d(\Phi(X)) = d(X).
\]

\begin{lemma}[first-component inequality]
\label{lem:first-component-inequality}
Let $S\in\mathcal C$ be simple and $A\boxtimes B$ a composition
factor of $\Phi(S)$.  Then $d(A)\le d(S)$.  If equality holds then
$\Phi(S)\simeq A\boxtimes B$, $B$ is invertible, and
$S\in\mathcal C_{\mathrm{ex}}$ (the subcategory on which the
ordinary Frobenius functor is exact).
\end{lemma}

\begin{proof}
\emph{Inequality.} $d(S)=d(\Phi(S))=\sum d(A_k)d(B_k)\ge d(A_1)$
for any factor, using $d\ge 1$ for nonzero objects.

\emph{Equality.} If $d(A)=d(S)$ then the FP decomposition sum is a
single term with $d(B)=1$, so $B$ is invertible and $\Phi(S)$ is
simple of the form $A\boxtimes B$.

\emph{$S\in\mathcal C_{\mathrm{ex}}$.}  Let $Q_A$ be the
projective cover of $A$.  Over $k$ algebraically closed,
$\dim_k\Hom(Q_A,X) = [X:A]$.  Applying \eqref{eq:phi-evaluation} to
$S$ and $X=S^{\otimes q}$:
\[
 \Theta(\Hom_{\mathcal C}(Q_A,S^{\otimes q}))
 \cong B.
\]
$\Theta$ preserves categorical dimension, so its argument has
categorical dimension equal to that of $B$, which is nonzero (its
product with the categorical dimension of $B^*$ is one).  Hence
$[S^{\otimes q}:A]\not\equiv 0\pmod p$.

By \eqref{eq:iterated-ord} there are indices
$i_1,\ldots,i_r\in\{1,\ldots,p-1\}$ with $A$ a composition factor
of $Y_r$, where $Y_0=S$, $Y_t = F_{i_t}(Y_{t-1})$.  For each $t$,
using $d(F(Y_{t-1}))\ge d(Y_t)$ and $d(\ell_{i_t})\ge1$:
\[
 d(Y_{t-1})\ge d(F(Y_{t-1}))\ge d(Y_t).
\]
But $d(Y_r)\ge d(A)=d(S)=d(Y_0)$, so all inequalities are equalities;
in particular $d(F(S))=d(S)$, so $S\in\mathcal C_{\mathrm{ex}}$.
\end{proof}

\subsection{The finite graph}

Let $N$ be the number of simple isomorphism classes in
$\mathcal C$.  Form a directed graph on those classes by arrow
$S\to A$ whenever $A\boxtimes B$ appears in some composition factor
of $\Phi(S)$.

Every vertex has at least one outgoing arrow.  By
\cref{lem:first-component-inequality}, dimension is nonincreasing
along arrows.  On a directed cycle, dimension is constant, all
vertices belong to $\mathcal C_{\mathrm{ex}}$, and each vertex has
exactly one outgoing arrow.  Hence a path entering a cycle is
trapped inside it.  A path of length $N$ visits $N+1$ vertices and
so hits a cycle.  Its endpoint is in $\mathcal C_{\mathrm{ex}}$.
A schematic path is drawn in \cref{fig:dimension-descent}.

\begin{figure}[!htbp]
\centering
\begin{tikzpicture}[
  x=1cm,y=1cm,
  vertex/.style={circle,draw,fill=white,minimum size=5mm,inner sep=0pt},
  edge/.style={-{Stealth[length=2mm]},semithick},
  every node/.style={font=\small}
]
  \fill[black!4,rounded corners=5pt] (0.1,-1.65) rectangle (3.55,1.8);
  \node at (-2.8,1.45) {nonincreasing dimension};
  \node at (1.83,2.10) {equal dimension};
  \node[vertex] (S0) at (-4.4,0) {};
  \node[vertex] (S1) at (-2.8,0) {};
  \node[vertex] (S2) at (-1.2,0) {};
  \node[vertex] (A) at (0.7,0) {};
  \node[vertex] (B) at (2.7,1) {};
  \node[vertex] (C) at (2.7,-1) {};
  \node[above=5pt] at (S0.north) {$S_0$};
  \node[above=5pt] at (S1.north) {$S_1$};
  \node[above=5pt] at (S2.north) {$S_2$};
  \node[above left=5pt] at (A.north west) {$A$};
  \node[above=5pt] at (B.north) {$B$};
  \node[right=5pt] at (C.east) {$C$};
  \draw[edge] (S0) -- (S1);
  \draw[edge] (S1) -- (S2);
  \draw[edge] (S2) -- (A);
  \draw[edge] (A) -- (B);
  \draw[edge] (B) -- (C);
  \draw[edge] (C) -- (A);
  \node[align=center] at (-2.8,-0.95)
    {$S\longrightarrow S'$ implies $d(S')\leq d(S)$};
  \node[align=center] at (1.83,-2.10)
    {$A,B,C\in\mathcal C_{\mathrm{ex}}$\\[2pt]
     unique outgoing neighbor at each cycle vertex};
\end{tikzpicture}
\caption{A schematic path in the graph of simple objects, with an
illustrative three-vertex cycle.  Dimension is nonincreasing along
every arrow; arrows before the cycle may also preserve dimension.
On a cycle, \cref{lem:first-component-inequality} gives equality,
membership in $\mathcal C_{\mathrm{ex}}$, and a unique outgoing
neighbor at every vertex, so the path cannot leave the cycle.}
\label{fig:dimension-descent}
\end{figure}

\subsection{Iteration and Serre closure}

Let $\sigma(a)=a^q$.  For a $k$-linear category $\mathcal A$, let
$\mathcal A^{[\sigma]}$ have the same objects and additive morphism
groups but scalar action twisted by $\sigma^{-1}$.  Then
$\sigma$-semilinear functors out of $\mathcal A$ = $k$-linear
functors out of $\mathcal A^{[\sigma]}$, and
$(\mathcal A\boxtimes\mathcal B)^{[\sigma]}
 = \mathcal A^{[\sigma]}\boxtimes\mathcal
 B^{[\sigma]}$, so products of $\sigma$-semilinear functors are
$\sigma$-semilinear.

Each $\Ver_{p^s}(k)$ is defined over $\mathbb F_p$; scalar transport
along $b\mapsto b^{p^a}$ is an exact strong symmetric $\sigma^a$-
semilinear autoequivalence $J_{\sigma^a}$ of every
$\Ver_{p^s}(k)$.

Define $\Psi_0=\mathrm{Id}$, $\Psi_1=\Phi$, and
\begin{equation}\label{eq:scalar-iteration}
 \Psi_{j+1} = (\Phi\boxtimes J_\sigma^{\boxtimes j})\circ\Psi_j
 :\mathcal C\to\mathcal C\boxtimes\mathcal W^{\boxtimes(j+1)}.
\end{equation}
By induction each $\Psi_j$ is exact, strong symmetric monoidal,
and $\sigma^j$-semilinear.

\begin{lemma}[first components after $N$ iterations]
\label{lem:first-components}
Let $\Psi_N :\mathcal C\to\mathcal C\boxtimes
\mathcal W^{\boxtimes N}$ be as above.  Then every composition
factor $A\boxtimes B_1\boxtimes\cdots\boxtimes B_N$ of
$\Psi_N(X)$ has $A\in\mathcal C_{\mathrm{ex}}$.
\end{lemma}

\begin{proof}
Exactness of $\Phi$ plus the path bound on $\mathcal C$ (any first
component of one iteration is a first component of one further
edge); induction on $N$, starting with simple objects (where the
path argument applies) and finishing with composition series.
Autoequivalences on other factors send simples to simples and
don't touch the first component.
\end{proof}

Let $\mathcal D\subseteq\mathcal C$ be the full subcategory whose
simple composition factors lie in $\mathcal C_{\mathrm{ex}}$.  This
is the \emph{Serre closure} of $\mathcal C_{\mathrm{ex}}$.

\begin{lemma}[Serre closure inherits tensor structure]
\label{lem:serre-closure}
$\mathcal D$ is closed under tensor products, duals, extensions, and
contains the unit; it is finite and has enough projectives; it
carries a rigid symmetric tensor structure inherited from
$\mathcal C$.  Moreover the functor $\Psi_N$ factors through
\[
 \Psi_N :\mathcal C\to \mathcal D\boxtimes
 \mathcal W^{\boxtimes N}.
\]
\end{lemma}

\begin{proof}
Closed under tensor products: biexactness gives tensor-product
filtrations, and $\mathcal C_{\mathrm{ex}}$ is a tensor
subcategory closed under subquotients (Etingof--Ostrik 2021,
Prop 7.4).  Duality preserves $\mathcal D$ by exactness.

Finiteness, projectives: concretely, write
$\mathcal C\simeq H\text{-mod}$ with $H$ basic finite-dimensional;
let $e$ be the idempotent summing primitive idempotents of the
non-$\mathcal C_{\mathrm{ex}}$ simples; then
$\mathcal D\simeq (H/HeH)\text{-mod}$ by removing the
$e$-isotypic parts, and $H/HeH$ is finite-dimensional and finite
projective dimension, so $\mathcal D$ is finite with enough
projectives.

Factoring through $\mathcal D$: \cref{lem:first-components}.
\end{proof}

% =================================================================
\section{Finishing the proof: a fiber functor on $\mathcal D$}
\label{sec:serre-finish}
% =================================================================

\subsection{The fiber functor on $\mathcal D$}

We need an exact strong symmetric monoidal $E:\mathcal
D\to\Ver_{p^n}(k)$ for some fixed $n$.

\paragraph{Case $p>2$.}
$\mathcal C_{\mathrm{ex}}$ is Frobenius-exact, so by
Etingof--Ostrik (2021, Theorem 8.1) it has a fiber functor
$\mathcal C_{\mathrm{ex}}\to\Ver_p(k)$.  If $p>2$, the Serre closure
of $\mathcal C_{\mathrm{ex}}$ equals $\mathcal C_{\mathrm{ex}}$
(Etingof--Ostrik 2021, Corollary 9.5).  So $\mathcal D=\mathcal
C_{\mathrm{ex}}$ and we already have $E:\mathcal D\to\Ver_p(k)$.

\paragraph{Case $p=2$.}
The fiber functor $\mathcal C_{\mathrm{ex}}\to\Ver_2(k)=\Vec_k$ is a
$k$-linear fiber functor.  But $\mathcal C_{\mathrm{ex}}\subseteq
\mathcal D$ contains every simple of $\mathcal D$ by definition,
and $\mathcal C_{\mathrm{ex}}$ is Frobenius-exact.  Etingof--Gelaki
(2021, Theorem 2.21(2)) then extends it to
$E:\mathcal D\to\Ver_4^+(k)\subseteq\Ver_4(k)$.
(This is also Etingof--Ostrik 2021, Corollary 9.8.)

So in all cases:
\[
 E:\mathcal D \longrightarrow \Ver_{p^n}(k), \qquad
 n = \begin{cases}1 & p>2,\\ 2 & p=2.\end{cases}
\]

\subsection{Combining factors}

Set $n=\max(2,r)$ and $\mathcal Z = \Ver_{p^n}(k)$.  The tower
inclusions $\Ver_{p^r}(k)\subseteq\Ver_{p^n}(k)$
(Benson--Etingof--Ostrik 2023, Theorem 4.34) give a $k$-linear exact
strong symmetric monoidal $I:\mathcal W\to\mathcal Z$.

The functors $E:\mathcal D\to\mathcal Z$ and $I:\mathcal W\to
\mathcal Z$ combine, by \cref{lem:combine-functors}, into
\[
 H:\mathcal D\boxtimes\mathcal W^{\boxtimes N}\to\mathcal Z,
 \qquad
 H(D\boxtimes W_1\boxtimes\cdots\boxtimes W_N)
 = E(D)\otimes I(W_1)\otimes\cdots\otimes I(W_N).
\]
$H$ is exact and strong symmetric monoidal.

\subsection{Killing the scalar twist}

The composite $H\Psi_N :\mathcal C\to\mathcal Z$ is
$\sigma^N$-semilinear (each iteration multiplies semilinearity by
$\sigma$).  Let $J_{\sigma^{-N}}$ be scalar transport on $\mathcal Z$
along $b\mapsto b^{p^{-N}}$.  Then
\[
 \mathcal F := J_{\sigma^{-N}}\circ H\circ\Psi_N :
 \mathcal C\longrightarrow\mathcal Z
\]
is $k$-linear, exact, strong symmetric monoidal.

\subsection{Faithfulness}

\begin{lemma}[exact strong monoidal with simple unit is faithful]
\label{lem:faithful}
Let $F:\mathcal C\to\mathcal D$ be an exact strong symmetric
monoidal functor between finite symmetric tensor categories.  Then
$F$ is faithful.
\end{lemma}

\begin{proof}
Suppose $X\ne0$.  The coevaluation
$\mathbf 1\to X\otimes X^*$ is nonzero (triangle identity) and by
simplicity of $\mathbf 1$ it is injective; applying exact $F$ shows
$F(X)\ne0$.  For a nonzero $f:X\to Y$: exactness gives
$\operatorname{im}F(f)\cong F(\operatorname{im}f)$, and
$\operatorname{im}f\ne0$, so $F(f)\ne0$.
\end{proof}

Applying \cref{lem:faithful} to $\mathcal F$ shows $\mathcal F$ is
faithful.  Hence $\mathcal F:\mathcal C\to\Ver_{p^n}(k)$ is the
required fiber functor, and \cref{thm:main} follows.

% =================================================================
\section{Consequences}
\label{sec:consequences}
% =================================================================

We now record the three structural corollaries that were the original
motivation for proving \cref{thm:main}.

\subsection{Incompressible categories}

Recall that $\mathcal C$ is \emph{incompressible} if every
$k$-linear exact strong symmetric monoidal functor from $\mathcal C$
to a symmetric tensor category over $k$ is fully faithful.

\begin{corollary}[classification of finite incompressible]
\label{cor:incompressible}
Let $k$ be algebraically closed of characteristic $p>0$.  A finite
symmetric tensor category $\mathcal C$ over $k$ is incompressible if
and only if it is $k$-linearly equivalent as a symmetric tensor
category to one of the following, according to $p$:
\[
\begin{array}{c|l}
p=2 & \Ver_{2^m}(k)\ (m\ge1),\quad
      \Ver_{2^m}^+(k)\ (m\ge2),\\
p=3 & \Ver_{3^m}(k),\ \Ver_{3^m}^+(k)\quad(m\ge1),\\
p\ge 5 & \Vec_k,\ \sVec_k,\quad
      \Ver_{p^m}(k),\ \Ver_{p^m}^+(k)\quad(m\ge1).
\end{array}
\]
\end{corollary}

\begin{proof}
Direction $\Leftarrow$: $\Vec_k$ has trivial
$\End(\mathbf 1)=k$, so any strong symmetric tensor functor out of
it is fully faithful (fill in the linear maps on simples); the
supervector-space case is similar (an odd simple has a nontrivial
sign in its double dual, which cannot match a vector space).
For the Verlinde full and plus categories, see Benson--Etingof--
Ostrik (2023), Theorems 4.71 and 4.72(iii).

Direction $\Rightarrow$: take a fiber functor
$F:\mathcal C\to\Ver_{p^n}(k)$ from \cref{thm:main}, let
$\mathcal I$ be the image's full subcategory of subquotients.
Then $F$ corestricts to a surjective-in-the-tensor-category-sense
functor onto $\mathcal I$ (Etingof--Gelaki--Nikshych--Ostrik 2015,
Definition 1.8.18, Prop 1.8.19, Section 6.3).  Incompressibility
of $\mathcal C$ forces this corestriction to be fully faithful.
The two FP-dimension inequalities of finiteness (inclusion vs
surjection) then force equality, hence an equivalence.
Benson--Etingof--Ostrik 2023, Corollary 4.61 then restricts
$\mathcal I$ to a full or plus sub-Verlinde at height $\le n$.
Collecting cases gives the stated list.
\end{proof}

Every finite tensor category has moderate growth: multiplication by
the class of a fixed object on the finite-rank Grothendieck group is
a finite nonnegative matrix, which bounds the lengths of its tensor
powers exponentially.  So \cref{cor:incompressible} is precisely the
finite case of Conjecture~B of
\cite{CoulembierEtingofOstrikIncompressible}.

\subsection{Uniqueness in characteristic $2$}

\begin{corollary}[uniqueness in characteristic two]
\label{cor:char2-unique}
Let $k$ be algebraically closed of characteristic two and
$\mathcal C$ a finite symmetric tensor category.  Then
$\mathcal C\to\Ver_{2^\infty}(k) :=\bigcup_n\Ver_{2^n}(k)$ has
exactly one fiber functor up to monoidal natural isomorphism; its
class contains a representative into a finite stage
$\Ver_{2^n}(k)$; and for each fixed $n$ there is at most one fiber
functor $\mathcal C\to\Ver_{2^n}(k)$ up to monoidal natural
isomorphism.
\end{corollary}

\begin{proof}
Existence: \cref{thm:main} into some finite stage, then tower
inclusion.  Uniqueness (union): Coulembier--Etingof--Ostrik (2024),
Theorem 9.4.1: any $k$-linear exact symmetric tensor functor from
an arbitrary tensor category in characteristic $2$ into
$\Ver_{2^\infty}(k)$ is unique up to monoidal natural isomorphism.
Fixed stage: lift stage uniqueness through the fully faithful tower
inclusion; naturality and tensor compatibility hold by faithfulness.
\end{proof}

\subsection{Tannaka reconstruction over a Verlinde level}

Let $\mathcal V$ be a finite symmetric tensor category.  The
fundamental group $\pi(\mathcal V) :=
\underline{\Aut}^{\otimes}(\mathrm{Id}_\mathcal V)$ acts
canonically on $\operatorname{Ind}(\mathcal V)$.  A
\emph{$\mathcal V$-group} is an internal affine group scheme $G$ with
a homomorphism $\phi:\pi(\mathcal V)\to G$ whose induced coaction
matches the canonical one.

\begin{corollary}[reconstruction over a Verlinde level]
\label{cor:tannaka}
Let $k$ be algebraically closed of characteristic $p>0$, $\mathcal
C$ finite symmetric tensor, and $F:\mathcal C\to
\mathcal V=\Ver_{p^n}(k)$ a fiber functor (from \cref{thm:main}).
Then there is a $\mathcal V$-group $(G,\phi)$ with
$\mathcal C\simeq\Rep(G,\phi)$ over $\mathcal V$, equivalent to
$F$ under the forgetful functor.  One may take
$G=\underline{\Aut}^{\otimes}(F)$ and
$\phi:\pi(\mathcal V)\to\underline{\Aut}^{\otimes}(F)$ the
canonical homomorphism.
\end{corollary}

\begin{proof}
\Cref{lem:modules-finite-alg} makes $\mathcal C$, together with
$\mathcal V=\Ver_{p^n}(k)$, a ``pretannakian category'' in the
sense of Coulembier--Etingof--Ostrik (2024).  The chosen $F$ is a
tensor functor, and their Theorem 4.2.1 produces the stated
$\mathcal V$-group; conversely, forgetting $G$-structure gives the
inverse functor.  Composition is the equivalence.
\end{proof}

\subsection*{Closing remarks}

All classification and existence assertions above are confined to
finite categories.  Uniqueness is asserted only in characteristic
two, both in the tower union and at each fixed finite stage where a
fiber functor exists.  \Cref{thm:main} supplies some finite height
$n$ depending on $\mathcal C$; it makes \emph{no} claim of existence
at a prescribed height.  In the reconstruction, $G$ is affine but
neither its finiteness nor its finite-type property is used or
claimed.

\bibliographystyle{plain}
\begin{thebibliography}{99}

\bibitem[BEO2023]{BEO2023}
D.~Benson, P.~Etingof, V.~Ostrik,
\emph{New incompressible symmetric tensor categories in positive
characteristic}, Duke Math.\ J.\ 172 (2023), 105--200.

\bibitem[BE2019]{BensonEtingof2019}
D.~Benson, P.~Etingof,
\emph{Symmetric tensor categories in characteristic 2},
Adv.\ Math.\ 351 (2019), 967--999.

\bibitem[EO2021]{EtingofOstrikFrobenius}
P.~Etingof, V.~Ostrik,
\emph{On the Frobenius functor for symmetric tensor categories in
positive characteristic}, J.\ Reine Angew.\ Math.\ 773 (2021),
165--198.

\bibitem[CEO2023]{CoulembierEtingofOstrik2023}
K.~Coulembier, P.~Etingof, V.~Ostrik,
\emph{On Frobenius-exact symmetric tensor categories},
Ann.\ of Math.\ 197 (2023), 1235--1279.

\bibitem[CEO2024]{CoulembierEtingofOstrikIncompressible}
K.~Coulembier, P.~Etingof, V.~Ostrik,
\emph{Incompressible tensor categories},
Adv.\ Math.\ 457 (2024), Paper 109935.

\bibitem[C2020]{Coulembier2020Tannakian}
K.~Coulembier,
\emph{Tannakian categories in positive characteristic},
Duke Math.\ J.\ 169 (2020), 3167--3219.

\bibitem[CF2026]{CoulembierFlakeHigher}
K.~Coulembier, J.~Flake,
\emph{Towards higher Frobenius functors for symmetric tensor
categories}, Selecta Math.\ 32 (2026), Paper 98.

\bibitem[CF2025]{CoulembierFlakeSteinberg}
K.~Coulembier, J.~Flake,
\emph{A conjecture on the tensor ideal for an elementary $p$-group
generated by the restriction of a Steinberg module},
arXiv:2507.02722v1.

\bibitem[C2021]{Coulembier2021Envelopes}
K.~Coulembier,
\emph{Monoidal abelian envelopes},
Compositio Math.\ 157 (2021), 1584--1609.

\bibitem[O2020]{Ostrik2020}
V.~Ostrik,
\emph{On symmetric fusion categories in positive characteristic},
Selecta Math.\ 26 (2020), Paper 36.

\bibitem[D2002]{Deligne2002}
P.~Deligne,
\emph{Cat\'egories tensorielles},
Mosc.\ Math.\ J.\ 2 (2002), 227--248.

\bibitem[DM1982]{DeligneMilne1982}
P.~Deligne, J.~Milne,
\emph{Tannakian categories},
in: Hodge Cycles, Motives, and Shimura Varieties, LNM 900,
Springer, 1982, 101--228.

\bibitem[Ch1976]{Chouinard1976}
L.~Chouinard II,
\emph{Projectivity and relative projectivity over group rings},
J.\ Pure Appl.\ Algebra 7 (1976), 287--302.

\bibitem[B2015]{Benson2015}
D.~Benson,
\emph{Modules for elementary abelian groups and hypersurface
singularities}, in: Comm.\ Algebra and Noncomm.\ Alg.\ Geom.\ II,
MSRI Publ.\ 68, Cambridge, 2015, 19--42.

\bibitem[B2017]{Benson2017}
D.~Benson,
\emph{Representations of Elementary Abelian $p$-Groups and Vector
Bundles}, Cambridge Tracts in Math.\ 208, Cambridge, 2017.

\bibitem[DH2005]{DotyHenke2005}
S.~Doty, A.~Henke,
\emph{Decomposition of tensor products of modular irreducibles for
$SL_2$}, Quart.\ J.\ Math.\ 56 (2005), 189--207.

\bibitem[D1993]{Donkin1993}
S.~Donkin,
\emph{On tilting modules for algebraic groups},
Math.\ Z.\ 212 (1993), 39--60.

\bibitem[EG2021]{EtingofGelaki2021}
P.~Etingof, S.~Gelaki,
\emph{Finite symmetric tensor categories with the Chevalley
property in characteristic 2},
J.\ Algebra Appl.\ 20 (2021), 2140010.

\bibitem[EGNO2015]{EGNO2015}
P.~Etingof, S.~Gelaki, D.~Nikshych, V.~Ostrik,
\emph{Tensor Categories},
Math.\ Surveys Monogr.\ 205, AMS, Providence, RI, 2015.

\end{thebibliography}

\end{document}