64c3814
==== TABLE OF CONTENTS ====
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//consequences.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//descent-graph.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//descent.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//higher.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//introduction.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//main.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//preamble.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//preliminaries.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//projectivity.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//restricted.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//support.tex
Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//references.bib
===========================
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//consequences.tex ====
\section{Classification, uniqueness, and reconstruction}
\label{sec:structural-consequences}
We combine Theorem~\ref{thm:main} with the established structure of the
higher Verlinde categories and generalized Tannaka duality. The resulting
statements concern finite symmetric tensor categories; the uniqueness
assertions apply in characteristic two.
\subsection{Finite incompressible symmetric tensor categories}
For a finite symmetric tensor category $\mathcal C$ over $k$,
\emph{incompressibility} means that every $k$-linear exact strong symmetric
monoidal functor from $\mathcal C$ to a symmetric tensor category over $k$
is fully faithful. This is the finite-source form of incompressibility in
\cite[Section~3.2]{CoulembierEtingofOstrikIncompressible}. The symmetry
condition is part of the definition here.
Write $\operatorname{Vec}_k$ for finite-dimensional vector spaces and,
when $p>2$, $\operatorname{sVec}_k$ for finite-dimensional supervector
spaces with the sign symmetry. For $p^m>2$,
$\operatorname{Ver}_{p^m}^+(k)$ denotes the neutral component of the
$\mathbb Z/2$-grading constructed from the even highest-weight tilting
modules in \cite[Section~4.1]{BEO2023}. The low levels satisfy
\[
\operatorname{Ver}_2(k)\simeq\operatorname{Vec}_k,\qquad
\operatorname{Ver}_3(k)\simeq\operatorname{sVec}_k,\qquad
\operatorname{Ver}_3^+(k)\simeq\operatorname{Vec}_k;
\]
see \cite[Theorem~4.5(iii), Corollary~4.11, and Example~4.52]{BEO2023}.
In particular, the plus notation is used only when $p^m>2$.
\begin{corollary}[Finite incompressible classification]
\label{cor:incompressible-classification}
Let $k$ be algebraically closed of characteristic $p>0$. A finite
symmetric tensor category over $k$ is incompressible if and only if it is
equivalent as a $k$-linear symmetric tensor category to a category in
the corresponding row:
\[
\begin{array}{c|l}
p=2&
\operatorname{Ver}_{2^m}(k)\ (m\ge1),\quad
\operatorname{Ver}_{2^m}^+(k)\ (m\ge2),\\[2pt]
p=3&
\operatorname{Ver}_{3^m}(k),\
\operatorname{Ver}_{3^m}^+(k)\quad(m\ge1),\\[2pt]
p\ge5&
\operatorname{Vec}_k,\ \operatorname{sVec}_k,\quad
\operatorname{Ver}_{p^m}(k),\
\operatorname{Ver}_{p^m}^+(k)\quad(m\ge1).
\end{array}
\]
\end{corollary}
\begin{proof}
Let $\mathcal C$ be incompressible and choose a fiber functor
$F:\mathcal C\to\operatorname{Ver}_{p^n}(k)$ by
Theorem~\ref{thm:main}. Let $\mathcal I$ be the full subcategory of
subquotients of objects $F(X)$, $X\in\mathcal C$. Because $\mathcal C$ is
finite, $\mathcal I$ is a finite tensor subcategory and the corestriction
$F:\mathcal C\to\mathcal I$ is surjective in the tensor-category sense
\cite[Definition~1.8.18, Proposition~1.8.19, and Section~6.3]{EGNO2015}.
Incompressibility makes this corestriction fully faithful. The injective
and surjective Frobenius--Perron-dimension inequalities for finite tensor
categories give, respectively,
\[
\operatorname{FPdim}(\mathcal C)\leq\operatorname{FPdim}(\mathcal I),
\qquad
\operatorname{FPdim}(\mathcal C)\geq\operatorname{FPdim}(\mathcal I).
\]
Equality in either inequality makes the corestriction an equivalence
\cite[Propositions~6.3.3 and~6.3.4]{EGNO2015}.
Corollary~4.61 of \cite{BEO2023} now restricts $\mathcal I$ to the full
and plus higher Verlinde subcategories at heights $m\le n$. For odd $p$,
the bottom cases are included using the explicit convention
$\operatorname{Ver}_1=\operatorname{sVec}_k$ and
$\operatorname{Ver}_1^+=\operatorname{Vec}_k$ of
\cite[Section~1.3]{CoulembierEtingofOstrikIncompressible}. With the low-level
identifications above, these are exactly the displayed possibilities.
Conversely, the full and plus categories in the list are incompressible
for symmetric tensor functors by \cite[Theorem~4.71 and
Remark~4.72(iii)]{BEO2023}. For $\operatorname{Vec}_k$, fullness follows
from $\operatorname{End}(\mathbf1)=k$. For
$\operatorname{sVec}_k$ in odd characteristic, a symmetric tensor functor
sends the odd invertible generator to an odd invertible object. It cannot
be isomorphic to the unit, since their self-symmetries are respectively
$-1$ and $1$. The functor is therefore fully faithful on the two simple
objects, and hence on all their finite direct sums.
\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. Thus the corollary is the finite case of
\cite[Conjecture~B]{CoulembierEtingofOstrikIncompressible}.
\subsection{Uniqueness in characteristic two}
For $p=2$, set
\[
\operatorname{Ver}_{2^\infty}(k)
:=\bigcup_{n\ge1}\operatorname{Ver}_{2^n}(k),
\]
using the exact symmetric tower embeddings. A fiber functor to this union
means a $k$-linear exact faithful strong symmetric monoidal functor, just
as for a finite target.
\begin{corollary}[Uniqueness into the characteristic-two union]
\label{cor:characteristic-two-uniqueness}
Let $k$ be algebraically closed of characteristic two and let
$\mathcal C$ be a finite symmetric tensor category over $k$. There is
exactly one fiber functor
\[
\mathcal C\longrightarrow\operatorname{Ver}_{2^\infty}(k)
\]
up to monoidal natural isomorphism. Its isomorphism class contains a
representative that factors through $\operatorname{Ver}_{2^n}(k)$ for
some finite $n\ge1$. For each fixed $n\ge1$, there is at most one
fiber functor $\mathcal C\to\operatorname{Ver}_{2^n}(k)$ up to
monoidal natural isomorphism.
\end{corollary}
\begin{proof}
Theorem~\ref{thm:main} supplies a fiber functor to a finite stage, and
composition with the tower inclusion supplies a fiber functor to the
union. By \cite[Theorem~9.4.1]{CoulembierEtingofOstrikIncompressible},
an arbitrary tensor category in characteristic two has at most one
$k$-linear exact symmetric tensor functor to
$\operatorname{Ver}_{2^\infty}(k)$ up to monoidal natural isomorphism.
Applying this to $\mathcal C$ proves uniqueness in the union.
For two fiber functors to the same finite stage, this isomorphism lifts
through the fully faithful tower inclusion; its naturality and tensor
compatibility hold at that stage by faithfulness. This proves the
fixed-stage assertion as well.
\end{proof}
\subsection{Generalized Tannaka reconstruction}
Let $\mathcal V$ be a finite symmetric tensor category over $k$.
An affine group scheme internal to $\mathcal V$ is specified by a
commutative Hopf algebra $\mathcal O(G)$ in
$\operatorname{Ind}(\mathcal V)$. The fundamental group
\[
\pi(\mathcal V)
:=\underline{\mathrm{Aut}}^{\otimes}(\operatorname{Id}_{\mathcal V})
\]
acts canonically on every object of $\operatorname{Ind}(\mathcal V)$.
Following \cite[Sections~4.1.1--4.1.4]{CoulembierEtingofOstrikIncompressible},
a \emph{$\mathcal V$-group} is an internal affine group scheme $G$ with
a homomorphism $\phi:\pi(\mathcal V)\to G$ such that the coaction on
$\mathcal O(G)$ induced by conjugation through $\phi$ equals its canonical
$\pi(\mathcal V)$-coaction. The category $\operatorname{Rep}(G,\phi)$
consists of $G$-representations in $\mathcal V$ whose restriction along
$\phi$ is the canonical $\pi(\mathcal V)$-action. Write
$\omega:\operatorname{Rep}(G,\phi)\to\mathcal V$ for the forgetful functor.
\begin{corollary}[Reconstruction over a finite higher Verlinde level]
\label{cor:generalized-tannaka}
Let $k$ be algebraically closed of characteristic $p>0$, and let
$\mathcal C$ be a finite symmetric tensor category over $k$. Choose
$n\ge1$ and a fiber functor
$F:\mathcal C\to\mathcal V=\operatorname{Ver}_{p^n}(k)$ as in
Theorem~\ref{thm:main}. There is a $\mathcal V$-group $(G,\phi)$ and a
$k$-linear symmetric tensor equivalence
\[
E:\mathcal C\xrightarrow{\sim}\operatorname{Rep}(G,\phi)
\]
with a monoidal natural isomorphism $\omega\circ E\cong F$. One may take
\[
G=\underline{\mathrm{Aut}}^{\otimes}(F),\qquad
\phi:\pi(\mathcal V)\longrightarrow
\underline{\mathrm{Aut}}^{\otimes}(F)
\]
to be the canonical homomorphism induced by $F$.
\end{corollary}
\begin{proof}
The finite-category conventions, including the simple unit supplied by
Lemma~\ref{lem:finite-conventions}, make $\mathcal C$ and $\mathcal V$
pretannakian categories in the terminology of
\cite{CoulembierEtingofOstrikIncompressible}. The chosen $F$ is a tensor
functor in that paper's sense. Its Theorem~4.2.1 assigns to $F$ the
$\mathcal V$-group displayed above, and the inverse assignment is the
forgetful functor from $\operatorname{Rep}(G,\phi)$. The resulting
equivalence over $\mathcal V$ is precisely the one stated here.
\end{proof}
The classification and existence assertions here remain confined to
finite categories. Uniqueness is asserted in characteristic two, both
in the tower union and, whenever a fiber functor exists, at each fixed
finite stage. The existence theorem supplies some finite height depending
on $\mathcal C$; it does not supply existence at every prescribed height.
In the reconstruction, $G$ is asserted to be affine; no
finiteness or finite-type property of this group scheme is used or claimed.
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//consequences.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//descent-graph.tex ====
\begin{figure}[t]
\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,
Lemma~\ref{lem:equality-descent} 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}
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//descent-graph.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//descent.tex ====
\section{Removing the original category from the target}
\label{sec:dimension-descent}
Choose an exact higher Frobenius functor by
Theorem~\ref{thm:exact-higher-frobenius}. Write
\(q=p^r\), \(\mathcal W=\operatorname{Ver}_{q}(k)\), and retain
\[
\Phi:\mathcal C\longrightarrow\mathcal C\boxtimes\mathcal W.
\]
The functor is \(q\)-power semilinear, exact, and strong symmetric monoidal.
Its evaluation property says that, for every projective \(Q\in\mathcal C\),
\begin{equation}\label{eq:descent-evaluation}
(\operatorname{Hom}_{\mathcal C}(Q,-)\boxtimes\operatorname{Id})(\Phi(X))
\simeq
\Theta\bigl(\operatorname{Hom}_{\mathcal C}(Q,X^{\otimes q})\bigr).
\end{equation}
Here \(\Theta:\operatorname{Rep}_k(E_r)\to\mathcal W\) is the
\(k\)-linear strong symmetric monoidal functor used in its construction.
Our objective is to show that sufficiently many iterations of \(\Phi\)
place the first factor in the Serre closure of a Frobenius-exact subcategory.
The Serre closure is the smallest full subcategory closed under
subobjects, quotients, and extensions that contains it.
\subsection{Ordinary Frobenius and a Grothendieck-ring identity}
For an object \(Y\in\mathcal C\), let \(c\) denote the cyclic permutation of
the \(p\) factors of \(Y^{\otimes p}\), and set \(D=1-c\). Thus \(D^p=0\).
The components of the ordinary Frobenius functor are
\[
F_i(Y)=
\frac{\ker D\cap\operatorname{im}D^{i-1}}
{\ker D\cap\operatorname{im}D^i},
\qquad 1\leq i<p.
\]
If \(\ell_i\) is the simple object of \(\operatorname{Ver}_p(k)\)
represented by a Jordan block of length \(i\), the functor is
\[
F(Y)=\bigoplus_{i=1}^{p-1}F_i(Y)\boxtimes\ell_i.
\]
We use the semilinear convention, so the objects \(F_i(Y)\) belong to
\(\mathcal C\). This construction and its monoidal structure are due to
Etingof--Ostrik \cite[Sections~3--5]{EtingofOstrikFrobenius}.
It need not be exact.
Let $K_0(\mathcal C)$ be the Grothendieck ring, with a basis given by
simple classes, addition induced by short exact sequences, and
multiplication induced by tensor product. Symmetry makes it commutative.
Frobenius--Perron dimension $d$ is its unique positive normalized real
character: it is additive on exact sequences, multiplicative on tensor
products, and equals one on the unit. Each simple object has dimension
at least one. On a Deligne product it is the product of the two dimensions;
see \cite[Sections~3.3 and~4.5]{EGNO2015}. We need two results from
\cite[Propositions~6.3 and~7.4]{EtingofOstrikFrobenius}:
\begin{equation}\label{eq:ordinary-dimension}
d(F(Y))\leq d(Y),\qquad
\mathcal C_{\mathrm{ex}}=
\{Y\in\mathcal C:d(F(Y))=d(Y)\}.
\end{equation}
The ordinary Frobenius functor is exact on $\mathcal C_{\mathrm{ex}}$,
which is a tensor subcategory. Such a category is called
\emph{Frobenius exact}. These statements require only finitely many
simple isomorphism classes.
The tensor subcategory is closed under subquotients and duals; extension
closure is not asserted here.
The next identity will connect the two Frobenius constructions: a
composition factor of \(Y^{\otimes p^r}\) whose multiplicity is nonzero
modulo \(p\) must occur in an iterated ordinary Frobenius component.
\begin{lemma}\label{lem:ordinary-congruence}
In the commutative ring \(K_0(\mathcal C)/pK_0(\mathcal C)\), one has
\[
[Y]^p=\sum_{i=1}^{p-1}i[F_i(Y)].
\]
Consequently, for every \(r\geq1\),
\begin{equation}\label{eq:iterated-ordinary-congruence}
[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 \(0\leq j\leq p\).
The exact sequences
\[
0\longrightarrow K_j\longrightarrow\operatorname{im}D^j
\xrightarrow{D}\operatorname{im}D^{j+1}\longrightarrow0
\qquad(0\leq j<p)
\]
give \([Z]=\sum_{j=0}^{p-1}[K_j]\).
Since \(K_0\supseteq\cdots\supseteq K_p=0\), summing the successive
quotients in these filtrations gives the integral identity
\[
[Z]=\sum_{i=1}^{p-1}i[F_i(Y)]+p[K_{p-1}].
\]
Reduction modulo \(p\) proves the first assertion.
For the second, raise this identity to successive \(p\)-th powers in
the commutative ring \(K_0(\mathcal C)/p\), and apply the first assertion
to each concrete object \(F_{i_s}\cdots F_{i_1}(Y)\). This argument
does not apply a nonexact functor to a Grothendieck-group relation.
\end{proof}
\subsection{Equality in dimension forces Frobenius exactness}
An exact strong monoidal functor between finite tensor categories preserves
Frobenius--Perron dimension, also when its semilinearity is a field
automorphism. Indeed, its map on Grothendieck rings preserves the unit,
products, and positive classes. It sends nonzero objects to nonzero
objects: apply exactness to the monomorphism
\(\mathbf1\to X\otimes X^*\) for \(X\ne0\).
Composing with dimension therefore gives a positive normalized character;
the Perron--Frobenius characterization makes it the dimension character
of the source \cite[Propositions~3.3.6, 3.3.13, and~4.5.7]{EGNO2015}. In particular,
\[
d(\Phi(X))=d(X).
\]
The simple objects of \(\mathcal C\boxtimes\mathcal W\) are precisely
\(A\boxtimes B\), with \(A,B\) simple, and their dimensions are
\(d(A)d(B)\).
\begin{lemma}\label{lem:equality-descent}
Let \(S\) be a simple object of \(\mathcal C\), and let
\(A\boxtimes B\) be a composition factor of \(\Phi(S)\).
Then \(d(A)\leq d(S)\). If equality holds, then
\[
\Phi(S)\simeq A\boxtimes B,\qquad B\text{ is invertible},
\qquad S\in\mathcal C_{\mathrm{ex}}.
\]
In this case \(A\) is the unique first component of a composition factor
of \(\Phi(S)\).
\end{lemma}
\begin{proof}
Every nonzero object has dimension at least one. Thus
\(d(S)=d(\Phi(S))\geq d(A)d(B)\geq d(A)\).
If the two ends are equal, positivity of the dimensions of all
composition factors shows that \(\Phi(S)\) has length one and
\(d(B)=1\). A simple object of dimension one is invertible, since
\(B\otimes B^*\) has dimension one and contains the unit.
Let \(Q_A\) be the projective cover of \(A\). Over the algebraically
closed field \(k\),
\[
\dim_k\operatorname{Hom}_{\mathcal C}(Q_A,X)=[X:A]
\]
for every \(X\): the functor is exact and has dimension one on \(A\)
and zero on every other simple object. Applying
\eqref{eq:descent-evaluation} to \(S\) gives
\[
\Theta\bigl(\operatorname{Hom}_{\mathcal C}(Q_A,S^{\otimes q})\bigr)
\simeq B.
\]
A symmetric monoidal functor preserves categorical dimension. This
dimension of the representation on the left is its vector-space dimension
viewed in \(k\), whereas the categorical dimension of the invertible
object \(B\) is nonzero: its product with the dimension of \(B^*\) is one. Consequently,
\begin{equation}\label{eq:multiplicity-nonzero}
[S^{\otimes q}:A]\not\equiv0\pmod p.
\end{equation}
This uses categorical dimension only to establish the congruence;
the real-valued dimension \(d\) remains Frobenius--Perron dimension.
By \eqref{eq:iterated-ordinary-congruence} and
\eqref{eq:multiplicity-nonzero}, there are indices
\(i_1,\ldots,i_r\in\{1,\ldots,p-1\}\) such that \(A\) is a composition
factor of \(Y_r\), where \(Y_0=S\) and \(Y_t=F_{i_t}(Y_{t-1})\).
For every \(t\), Equation~\eqref{eq:ordinary-dimension} and
\(d(\ell_{i_t})\geq1\) give
\[
d(Y_{t-1})\geq d(F(Y_{t-1}))\geq d(Y_t).
\]
Since \(d(Y_r)\geq d(A)=d(S)\), all inequalities are equalities.
In particular \(d(F(S))=d(S)\), so
\(S\in\mathcal C_{\mathrm{ex}}\).
\end{proof}
\paragraph{The finite graph.}
We adapt the finite-simple dimension descent used by Etingof--Ostrik
for ordinary Frobenius iteration
\cite[Proposition~10.1]{EtingofOstrikFrobenius} to the exact functor
\(\Phi\).
Let \(N\) be the number of simple isomorphism classes in \(\mathcal C\).
Form a directed graph on these classes, putting an arrow \(S\to A\)
when \(A\boxtimes B\) occurs in \(\Phi(S)\) for some simple \(B\).
Every vertex has an outgoing arrow. By Lemma~\ref{lem:equality-descent},
dimension is nonincreasing along arrows. On a directed cycle it is
constant, every vertex belongs to \(\mathcal C_{\mathrm{ex}}\), and every
vertex has just one outgoing neighbor. Thus a path entering a cycle
cannot leave it. A path of \(N\) edges visits \(N+1\) vertices, so it
contains such a cycle. Its endpoint therefore belongs to
\(\mathcal C_{\mathrm{ex}}\).
Figure~\ref{fig:dimension-descent} illustrates this trapping mechanism:
the unique outgoing neighbor at each cycle vertex prevents a path from
leaving the cycle, even though dimension need not decrease strictly
before the cycle.
\input{figures/descent-graph}
The graph controls the possible first components under repeated
application of \(\Phi\). To turn its path bound into a statement about
an actual tensor functor, we must iterate \(\Phi\) with compatible
scalar actions on all the factors.
\subsection{Scalar twists and iteration}
\label{subsec:scalar-iteration}
Put $\sigma(a)=a^q$ for $a\in k$, where $q=p^r$ is the power
chosen above. Since $k$ is algebraically closed, $\sigma$ is an
automorphism. A functor is $\sigma$-semilinear if
$F(af)=\sigma(a)F(f)$ for every scalar $a$ and morphism $f$.
We now define the iteration of $\Phi$ without mixing different
scalar actions.
For a $k$-linear category $\mathcal A$, let $\mathcal A^{[\sigma]}$
have the same objects and additive morphism groups, with scalar action
\[
a\mathbin{\cdot_{[\sigma]}}f=\sigma^{-1}(a)f.
\]
Thus a $\sigma$-semilinear functor $\mathcal A\to\mathcal B$ is
precisely a $k$-linear functor $\mathcal A^{[\sigma]}\to\mathcal B$.
For finite abelian categories the universal property of the Deligne
product gives
\[
(\mathcal A\boxtimes\mathcal B)^{[\sigma]}
\simeq\mathcal A^{[\sigma]}\boxtimes\mathcal B^{[\sigma]}.
\]
Consequently two functors with the same semilinearity have a well-defined
Deligne product with that semilinearity: apply the ordinary Deligne
product to their $k$-linear versions after this identification.
Every higher Verlinde category is defined over the prime field
\cite[Section~4.9]{BEO2023}; the resulting scalar equivalences are also
recorded in \cite[Remark~2.3.8]{CoulembierFlakeHigher}.
Hence, for every integer $a$,
scalar transport along $b\mapsto b^{p^a}$ gives an exact strong symmetric
semilinear autoequivalence of $\operatorname{Ver}_{p^s}(k)$.
Here negative powers denote inverse Frobenius automorphisms of $k$.
Indeed, scalar transport acts on a presentation of the category and
its tensor structure over $\mathbb F_p$; all structure maps are
preserved because the automorphism fixes $\mathbb F_p$.
Its inverse is transport along the inverse field automorphism.
These autoequivalences are scalar transports, distinct from the ordinary
Frobenius functor, which need not be exact.
Let $J_\sigma$ denote such an autoequivalence of
$\mathcal W=\operatorname{Ver}_{q}(k)$ with semilinearity $\sigma$.
Set $\Psi_0=\operatorname{Id}_{\mathcal C}$ and $\Psi_1=\Phi$.
For $j\geq1$, define
\begin{equation}\label{eq:scalar-iteration}
\Psi_{j+1}
=\bigl(\Phi\boxtimes J_\sigma^{\boxtimes j}\bigr)\circ\Psi_j:
\mathcal C\longrightarrow\mathcal C\boxtimes\mathcal W^{\boxtimes(j+1)}.
\end{equation}
Each factor in the parenthesized functor has semilinearity $\sigma$,
so the product is defined as above. By induction $\Psi_j$ is exact,
strong symmetric monoidal, and $\sigma^j$-semilinear.
The old $\mathcal W$-factors are only transported by equivalences.
We now apply the graph argument to the first components of its
composition factors.
\subsection{The image after iteration}
The scalar-compatible iteration \eqref{eq:scalar-iteration} gives an exact functor
\[
\Psi_N:\mathcal C\longrightarrow
\mathcal C\boxtimes\mathcal W^{\boxtimes N}.
\]
Exactness is essential here: applying the next functor to a composition
series shows that every first component of every composition factor
follows a path of one additional edge. The autoequivalences on the other
factors send simple objects to simple objects and do not change the first
component. Induction, first for simple inputs and then for a composition
series of an arbitrary input, shows that all first components of
\(\Psi_N(X)\) belong to \(\mathcal C_{\mathrm{ex}}\).
Let \(\mathcal D\) be the full subcategory of \(\mathcal C\) consisting
of objects whose simple composition factors belong to
\(\mathcal C_{\mathrm{ex}}\). This is the Serre closure of
\(\mathcal C_{\mathrm{ex}}\). It is closed under tensor products, because
biexactness gives tensor-product filtrations and
\(\mathcal C_{\mathrm{ex}}\) is a tensor subcategory closed under
subquotients. Exact duality also preserves it. Hence \(\mathcal D\)
inherits the rigid symmetric tensor structure and the simple unit.
For completeness, the relevant abelian and Deligne-product assertions
have a concrete finite-algebra description. Choose a basic finite
dimensional algebra \(H\) with \(\mathcal C\simeq H\text{-}\mathrm{mod}\),
and let \(e\) be the sum of the primitive idempotents corresponding to the
simple objects outside \(\mathcal C_{\mathrm{ex}}\). The functor
\(M\mapsto eM\) is exact, and vanishes on exactly the allowed simples.
Thus
\[
\mathcal D\simeq(H/HeH)\text{-}\mathrm{mod}.
\]
In particular \(\mathcal D\) is finite and has enough projectives. If
\(\mathcal W^{\boxtimes N}\simeq J\text{-}\mathrm{mod}\), then
\[
\mathcal D\boxtimes\mathcal W^{\boxtimes N}
\simeq\bigl((H/HeH)\otimes_kJ\bigr)\text{-}\mathrm{mod}
\]
is the full Serre subcategory of
\((H\otimes_kJ)\text{-}\mathrm{mod}\) annihilated by \(e\otimes1\).
It consists exactly of objects with allowed first components.
We have therefore obtained an exact strong symmetric monoidal functor
\begin{equation}\label{eq:descent-result}
\Psi_N:\mathcal C\longrightarrow
\mathcal D\boxtimes\mathcal W^{\boxtimes N}.
\end{equation}
The remaining step is to fiber \(\mathcal D\) over a small Verlinde
category and to remove the total scalar twist.
\subsection{A fiber functor on the Serre closure}
\label{subsec:serre-fiber}
It remains to replace $\mathcal D$ in Equation~\eqref{eq:descent-result}
by a fixed low Verlinde level and correct the scalar action.
We first check finiteness for the subcategories in the fiber-functor
theorem. More generally, let $\mathcal B$ be a full additive
subcategory of $A\text{-}\mathrm{mod}$ closed under subobjects and
quotients, where $A$ is finite dimensional. Put
$I=\bigcap_{M\in\mathcal B}\operatorname{Ann}_A(M)$.
Since $A$ is finite dimensional, this intersection is already a finite
intersection. A finite direct sum $M\in\mathcal B$ therefore satisfies
$\operatorname{Ann}_A(M)=I$.
Give $\operatorname{End}_k(M)$ the left $A$-action by postcomposition:
$a\cdot u=\rho_M(a)\circ u$. The action map embeds the left regular
$A/I$-module in $\operatorname{End}_k(M)\cong M^{\oplus\dim_k M}$,
so $A/I\in\mathcal B$.
Every finite $A/I$-module is a quotient of a finite direct sum of
$A/I$. Thus $\mathcal B=(A/I)\text{-}\mathrm{mod}$ and, in particular,
$\mathcal B$ is finite abelian and has enough projectives.
This applies to $\mathcal C_{\mathrm{ex}}$, which is a tensor
subcategory closed under subquotients
\cite[Proposition~7.4]{EtingofOstrikFrobenius}.
The inherited tensor product and duality therefore make
$\mathcal C_{\mathrm{ex}}$ a finite symmetric tensor category; these
properties for $\mathcal D$ were established above.
By \cite[Theorem~8.1]{EtingofOstrikFrobenius}, the Frobenius-exact
category $\mathcal C_{\mathrm{ex}}$ has a $k$-linear fiber functor
to $\operatorname{Ver}_p(k)$. If $p>2$, its Serre closure equals
$\mathcal C_{\mathrm{ex}}$
\cite[Corollary~9.5]{EtingofOstrikFrobenius}; hence the same is true
of $\mathcal D$.
If $p=2$, the preceding fiber functor has target
$\operatorname{Ver}_2(k)=\operatorname{Vec}_k$. The finite tensor
subcategory $\mathcal C_{\mathrm{ex}}\subseteq\mathcal D$ contains
every simple object of $\mathcal D$, by the definition of $\mathcal D$.
Thus the extension theorem of Etingof--Gelaki
\cite[Theorem~2.21(2)]{EtingofGelaki2021} applies to this inclusion and
gives a $k$-linear fiber functor $\mathcal D\to\mathcal V$, exactly as
in \cite[Corollary~9.8]{EtingofOstrikFrobenius}. Here
\[
\mathcal V=\operatorname{Rep}_k\bigl(k[d]/(d^2),\,
\Delta(d)=d\otimes1+1\otimes d,\,R=1\otimes1+d\otimes d\bigr).
\]
In that result, ``almost Frobenius exact'' means precisely the existence
of this exact faithful symmetric tensor functor
\cite[Definition~8.13]{EtingofOstrikFrobenius}.
The category $\mathcal V$ is the category $\mathcal C_1$ of
\cite[Introduction]{BensonEtingof2019}; by
\cite[Theorem~4.5(iii)]{BEO2023},
\[
\mathcal V\simeq\operatorname{Ver}_4^+(k)
\subset\operatorname{Ver}_4(k)
\]
as symmetric tensor categories. This is the only additional level
needed in characteristic two.
\subsection{Completion of the fiber functor}
\label{subsec:completion}
Set $n=\max(2,r)$ and $\mathcal Z=\operatorname{Ver}_{p^n}(k)$.
The exact symmetric tower inclusions
\cite[Theorem~4.34]{BEO2023}, together with the preceding
fiber functor, give $k$-linear exact strong symmetric functors
\[
E:\mathcal D\longrightarrow\mathcal Z,
\qquad I:\mathcal W\longrightarrow\mathcal Z.
\]
They combine to a $k$-linear exact strong symmetric functor
\begin{equation}\label{eq:combined-functor}
H:\mathcal D\boxtimes\mathcal W^{\boxtimes N}\longrightarrow\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).
\end{equation}
Exactness and the symmetric tensor structure follow from
Lemma~\ref{lem:combine-functors}. The composite $H\Psi_N$ has semilinearity $\sigma^N$.
Let $J_{\sigma^{-N}}$ be scalar transport on $\mathcal Z$ with
semilinearity $\sigma^{-N}$. Then
\[
\mathcal F=J_{\sigma^{-N}}\circ H\circ\Psi_N:
\mathcal C\longrightarrow\operatorname{Ver}_{p^n}(k)
\]
is $k$-linear, exact, and strong symmetric monoidal.
Finally, an exact strong monoidal functor between tensor categories
with simple units is faithful. Indeed, if $X\ne0$, the coevaluation
$\mathbf1\to X\otimes X^*$ is nonzero by a triangle identity, hence
injective by simplicity of $\mathbf1$. Its exact image shows that
$\mathcal F(X)\ne0$. For a nonzero morphism $f$, exactness gives
$\operatorname{im}\mathcal F(f)\cong \mathcal F(\operatorname{im}f)\ne0$.
Thus $\mathcal F$ is the required fiber functor, proving Theorem~\ref{thm:main}.
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//descent.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//higher.tex ====
\section{From restricted tiltings to a higher Frobenius functor}
\label{sec:higher}
Fix $r\geq1$, put $q=p^r$, and choose an injective homomorphism
$E=(\mathbb Z/p)^r\hookrightarrow U(k)$. Write $R$ for restriction
from $SL_2$, and put $\mathcal W=\operatorname{Ver}_{p^r}(k)$.
Theorem~\ref{thm:restricted-tilting} supplies the
representation-theoretic input for the higher Frobenius construction of
Coulembier and Flake. This section records that construction, explains
how to evaluate it at projectives, and gives the exactness criterion we
will use.
We will repeatedly use the following elementary fact. In a rigid
abelian tensor category with simple unit, tensoring by a nonzero
object $Y$ is exact and faithful. Indeed, the coevaluation
$\mathbf1\to Y\otimes Y^*$ is a nonzero monomorphism. Tensoring
it by $X$ shows that $X\otimes Y=0$ forces $X=0$. Exactness then
shows that tensoring by $Y$ detects nonzero morphisms through their
images. An exact faithful functor reflects exactness as well.
\subsection{The functor on group representations}
Let $\mathcal T$ be the category of tilting $SL_2$-modules, and let
$I_r$ be its tensor ideal generated by $\operatorname{St}_r$.
The canonical functor
\[
\Sigma_r:\mathcal T/I_r\lhook\joinrel\longrightarrow\mathcal W
\]
is fully faithful \cite[Theorem~4.2(i)]{BEO2023}.
For every $s\geq1$, $T(a)$ vanishes modulo $I_s$ exactly when
$a\geq p^s-1$ \cite[Section~3.1]{BEO2023}.
Stable restriction factors through a faithful
functor $\mathcal T/I_r\to\operatorname{Stab}(E)$
\cite[Theorem~5.3.5]{CoulembierFlakeHigher}.
Here $\operatorname{Stab}(E)$ denotes the quotient of
$\operatorname{Rep}_k(E)$ by morphisms factoring through projective
modules.
The assertion that
$R(\operatorname{St}_{r-1})\otimes D$ is a restricted tilting module
for every $D\in\operatorname{Rep}_k(E)$ is precisely the hypothesis
of \cite[Conjecture~5.4.2(1), Theorem~5.4.3]{CoulembierFlakeHigher}.
Since we have proved that assertion, the cited implication produces a
$k$-linear strong symmetric monoidal functor
\begin{equation}\label{eq:theta}
\Theta:\operatorname{Rep}_k(E)\longrightarrow\mathcal W,
\qquad \Theta(R(T))\simeq\Sigma_r(T\bmod I_r).
\end{equation}
In particular, $\Theta$ is additive. It need not be exact.
\begin{lemma}\label{lem:theta-detection}
If $T$ is a tilting module and $R(T)$ is not projective, then
$\Theta(R(T))\ne0$. Moreover, $\Theta$ annihilates every
$E$-module induced from a proper subgroup of $E$.
\end{lemma}
\begin{proof}
If $T\bmod I_r=0$, stable restriction shows that $R(T)$ is
projective. Thus nonprojectivity makes $T\bmod I_r$ nonzero;
the fully faithful functor $\Sigma_r$ and \eqref{eq:theta} give the
first assertion.
Stable restriction also shows that $R(\operatorname{St}_r)$ is
projective. Its dimension is $q$, so it is the regular module:
projectives over the local algebra $kE$ are free. Equation
\eqref{eq:theta} therefore shows that $\Theta$ kills all projectives.
Set $S=R(\operatorname{St}_{r-1})$. Its dimension $p^{r-1}$ is
strictly smaller than $q$, so it is nonprojective and $\Theta(S)\ne0$.
Let $H<E$ have rank $s<r$. For $s>0$, stable restriction at level
$s$ shows that $\operatorname{Res}^E_H S$ is projective, because
$\operatorname{St}_{r-1}\in I_s$. The assertion is automatic for
$H=1$. For a $kH$-module $D$, the tensor identity gives
\[
S\otimes\operatorname{Ind}_H^E D
\simeq\operatorname{Ind}_H^E
(\operatorname{Res}^E_H S\otimes D),
\]
which is projective. Applying $\Theta$ gives
$\Theta(S)\otimes\Theta(\operatorname{Ind}_H^E D)=0$.
Tensoring by a nonzero object of a tensor category is faithful, so
$\Theta(\operatorname{Ind}_H^E D)=0$.
\end{proof}
\subsection{Internal evaluation}
Let $\mathcal C$ be a finite symmetric tensor category. An object of
$\mathcal C\boxtimes\operatorname{Rep}_k(E)$ is an object of
$\mathcal C$ equipped with an $E$-action. There is an internal
extension of $\Theta$,
\[
\Theta_{\mathcal C}:\mathcal C\boxtimes\operatorname{Rep}_k(E)
\longrightarrow\mathcal C\boxtimes\mathcal W.
\]
We explain its meaning here because $\Theta$ is not assumed exact.
Choose a projective generator $P_0$ and set
$A=\operatorname{End}_{\mathcal C}(P_0)^{\mathrm{op}}$.
Under $\mathcal C\simeq A\text{-}\mathrm{mod}$, an $E$-object is
an $E$-module with a commuting $A$-action. Applying $\Theta$ to
its 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 internal extension used in
\cite[Example~3.2.4, Proposition~3.2.6, and Section~3.3.2]{CoulembierFlakeHigher}; it does not require
right exactness of $\Theta$.
\begin{lemma}\label{lem:projective-evaluation}
For every projective $Q\in\mathcal C$ and every $E$-object $Z$ of
$\mathcal C$, there is a natural isomorphism
\begin{equation}\label{eq:projective-evaluation}
(\operatorname{Hom}_{\mathcal C}(Q,-)\boxtimes
\operatorname{Id}_{\mathcal W})(\Theta_{\mathcal C}(Z))
\simeq\Theta(\operatorname{Hom}_{\mathcal C}(Q,Z)).
\end{equation}
\end{lemma}
\begin{proof}
The $A$-module corresponding to $Q$ is a direct summand of $A^m$
for some $m$. Thus $\operatorname{Hom}_A(Q,-)$ is obtained from
$m$ copies of the underlying object by applying a split idempotent.
An additive functor preserves finite direct sums and split
idempotents. Applying this observation to $\Theta$ gives
\eqref{eq:projective-evaluation}, including its naturality.
\end{proof}
\subsection{The tensor-power construction and exactness}
Give $X^{\otimes q}$ the $E$-action arising from regular
permutations of its tensor factors. Define
\begin{equation}\label{eq:higher-phi}
\Phi(X)=\Theta_{\mathcal C}(X^{\otimes q})
\in\mathcal C\boxtimes\mathcal W.
\end{equation}
By Lemma~\ref{lem:theta-detection} and
\cite[Theorem~5.2.1]{CoulembierFlakeHigher}, this construction has
coherent strong symmetric monoidal constraints. They come from the
$E$-equivariant reordering
$(X\otimes Y)^{\otimes q}\simeq X^{\otimes q}\otimes Y^{\otimes q}$,
with the diagonal action on the right, through the internal monoidal
extension of \cite[Section~3.3.2]{CoulembierFlakeHigher}.
The resulting functor is additive and satisfies
\[
\Phi(af)=a^q\Phi(f)\qquad(a\in k).
\]
Additivity can also be seen directly from the regular action: in the expansion of
$(X\oplus Y)^{\otimes q}$, every nonconstant coloring of the
tensor positions has a proper stabilizer in $E$. The sum over
its orbit is induced from that stabilizer, so is annihilated.
The two constant colorings give $\Phi(X)$ and $\Phi(Y)$.
The corresponding expansion on morphisms proves additivity,
and scalar multiplication gives the displayed semilinearity.
Combining \eqref{eq:higher-phi} and
Lemma~\ref{lem:projective-evaluation} yields
\begin{equation}\label{eq:phi-evaluation}
(\operatorname{Hom}_{\mathcal C}(Q,-)\boxtimes
\operatorname{Id}_{\mathcal W})(\Phi(X))
\simeq\Theta(\operatorname{Hom}_{\mathcal C}
(Q,X^{\otimes q})).
\end{equation}
\begin{proposition}\label{prop:higher-exactness}
If there are projective objects $P,Q\in\mathcal C$ for which
\[
\Theta(\operatorname{Hom}_{\mathcal C}(Q,P^{\otimes q}))\ne0,
\]
then $\Phi$ is exact.
\end{proposition}
\begin{proof}
Equation~\eqref{eq:phi-evaluation} implies $\Phi(P)\ne0$.
Tensoring a short exact sequence by $P$ makes it split, since
tensor products of projectives with arbitrary objects are
projective. The additive functor $\Phi$ preserves this split
sequence. Its monoidal structure identifies the result with
the image sequence tensored by $\Phi(P)$.
Tensoring by the nonzero object $\Phi(P)$ is exact and faithful,
and therefore reflects exactness. Thus the image sequence is
exact. This argument is the nonzero-projective-image criterion of
\cite[Proposition~7.6 and Remark~7.7(i)]{EtingofOstrikFrobenius},
applied to the higher construction as in
\cite[Proposition~5.2.4]{CoulembierFlakeHigher}.
\end{proof}
The next step is now concrete: we will choose $r$ and the
embedding of $E$ so that
\[
\Theta(\operatorname{Hom}_{\mathcal C}(P,P^{\otimes q}))\ne0
\]
for a projective generator $P$.
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//higher.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//introduction.tex ====
\section{Introduction}
\label{sec:introduction}
Let $k$ be an algebraically closed field of characteristic $p>0$.
A \emph{finite symmetric tensor category over $k$} is an essentially small
$k$-linear abelian rigid symmetric monoidal category with biexact tensor
product, finite-dimensional morphism spaces, finite-length objects,
finitely many isomorphism classes of simple objects, enough projectives,
and $\operatorname{End}(\mathbf1)=k$.
A \emph{fiber functor} between such categories is a $k$-linear exact
faithful strong symmetric monoidal functor.
The higher Verlinde categories of Benson--Etingof--Ostrik form a tower
\[
\operatorname{Ver}_{p}(k)\subset
\operatorname{Ver}_{p^2}(k)\subset
\operatorname{Ver}_{p^3}(k)\subset\cdots
\]
of finite symmetric tensor categories
\cite{BEO2023,Coulembier2021Envelopes}.
The first level is the semisimplification of
$\operatorname{Rep}_k(\mathbb Z/p)$: one quotients by morphisms
$f:X\to Y$ satisfying $\tr(gf)=0$ for all $g:Y\to X$.
Its simple objects are
represented by Jordan blocks of lengths $1,\ldots,p-1$
\cite[Section~2.3]{EtingofOstrikFrobenius}.
Thus $\operatorname{Ver}_2$ is the category of vector spaces and
$\operatorname{Ver}_3$ is the category of supervector spaces with its
sign symmetry. The higher levels need not be semisimple. They are finite
abelian envelopes of quotients of the category of tilting $SL_2$-modules
(modules with both Weyl and dual Weyl filtrations), where the tensor
ideal generated by the Steinberg module of highest weight $p^n-1$
is killed at level $n$ \cite[Sections~3.1 and~4.1, Theorem~4.2]{BEO2023}.
Our main result is the following.
\begin{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 are an integer $n\geq1$ and a fiber functor
\[
\mathcal C\longrightarrow\operatorname{Ver}_{p^n}(k).
\]
\end{theorem}
This proves the finite case of the Benson--Etingof--Ostrik conjecture
\cite[Conjecture~1.4]{BEO2023}, including characteristic two.
The full conjecture concerns symmetric tensor categories of
\emph{moderate growth}: for each object, the lengths of its tensor
powers grow at most exponentially. Its target is the union of the
higher Verlinde tower. Here both the projective generator and the
finite set of simple classes are essential. The theorem places the
image in a single finite level, whose height may depend on $\mathcal C$;
it makes no assertion about arbitrary nonfinite categories of moderate
growth.
A finite symmetric tensor category is \emph{incompressible} if every
$k$-linear exact symmetric tensor functor out of it is fully faithful.
Theorem~\ref{thm:main} and the higher Verlinde subcategory classification
of Benson--Etingof--Ostrik \cite{BEO2023} classify these finite categories,
giving the finite case of Coulembier--Etingof--Ostrik's Conjecture~B
\cite{CoulembierEtingofOstrikIncompressible}.
In characteristic two, their subterminality theorem further gives
uniqueness up to tensor isomorphism of the functor into the tower union.
Their generalized Tannaka theorem also realizes every finite category
as representations of an affine group scheme internal to a chosen finite
level, with the canonical fundamental-group compatibility.
Section~\ref{sec:structural-consequences} gives the precise statements
and deductions.
\subsection*{Background and contribution}
In characteristic zero, Deligne's theorem makes supervector spaces a
common target for fiber functors from symmetric tensor categories of
moderate growth \cite{Deligne2002}. In positive characteristic, Ostrik proved that
symmetric fusion categories admit fiber functors to
$\operatorname{Ver}_p$ \cite[Theorem~1.5]{Ostrik2020}.
Etingof--Ostrik constructed the ordinary Frobenius functor for
nonsemisimple categories; Coulembier independently constructed its
underlying additive functor as the external Frobenius twist
\cite[Section~4.3]{Coulembier2020Tannakian}.
Etingof--Ostrik established its symmetric monoidal structure and
characterized the finite categories on which it is exact by the
existence of a fiber functor to $\operatorname{Ver}_p$
\cite[Proposition~5.1 and Theorem~8.1]{EtingofOstrikFrobenius}.
Coulembier--Etingof--Ostrik extended that characterization to categories
of moderate growth \cite{CoulembierEtingofOstrik2023}.
The higher levels are needed because the ordinary Frobenius functor
need not be exact. The characteristic-two examples of Benson--Etingof
\cite{BensonEtingof2019} led to the all-prime tower of
Benson--Etingof--Ostrik \cite{BEO2023}, constructed independently through
abelian envelopes by Coulembier \cite{Coulembier2021Envelopes}.
Coulembier--Flake developed higher Frobenius functors from elementary
abelian group representations \cite{CoulembierFlakeHigher}.
Their construction becomes symmetric monoidal once a specific
restricted-tilting conjecture is established. Besides the rank-one case,
the conjecture was already known for $p=r=2$
\cite[Remark~5.4.4(1) and Section~6]{CoulembierFlakeHigher}.
Their subsequent work
proved that assertion for substantial classes of modules, including
modules of Loewy length at most two and Carlson modules, and for the
subcategory they generate under tensor products, duals, syzygy shifts,
finite sums, and summands \cite[Theorem~2]{CoulembierFlakeSteinberg}.
For rank two they also established the assertion for every cyclic
summand of the relevant Steinberg tensor product
\cite[Theorem~3]{CoulembierFlakeSteinberg}.
Our independent representation-theoretic contribution is the full
restricted-tilting conjecture, proved in
Theorem~\ref{thm:restricted-tilting}.
For $r\geq1$, embed $E_r=(\mathbb Z/p)^r$ in the upper unipotent subgroup
of $SL_2(k)$, and write $R$ for restriction along this embedding.
Let $\operatorname{St}_{r-1}$ be the simple $SL_2$-module of highest
weight $p^{r-1}-1$. For every finite-dimensional $E_r$-module $D$, we
show that
\[
R(\operatorname{St}_{r-1})\otimes D\cong R(T)
\]
for a tilting $SL_2$-module $T$. This is
\cite[Conjecture~5.4.2(1)]{CoulembierFlakeHigher}, also formulated as
\cite[Conjecture~1]{CoulembierFlakeSteinberg}.
The proof lifts $D$ to the additive group, extends the associated
equivariant vector bundle from the punctured plane across the origin,
and uses a formal comparison of fibers after tensoring by the
Steinberg module. The bounded-weight tilting input is the established
\cite[Lemma~3.3]{BEO2023}, in the extension-closed form also used in
\cite[proof of Lemma~3.1.6]{CoulembierFlakeSteinberg}.
The geometric comparison is the additional step that treats arbitrary $D$.
\subsection*{Proof strategy}
The restricted-tilting theorem supplies the hypothesis for the
Coulembier--Flake construction. For each embedding of $E_r$ as above,
Section~\ref{sec:higher} therefore produces an additive strong symmetric
monoidal functor
\[
\Phi:\mathcal C\longrightarrow
\mathcal C\boxtimes\operatorname{Ver}_{p^r}(k).
\]
Here $\boxtimes$ denotes the Deligne tensor product of finite abelian
categories. The functor is initially semilinear:
$\Phi(af)=a^{p^r}\Phi(f)$ for $a\in k$ and a morphism $f$.
Its exactness depends on the embedding.
To select an exact functor, choose a projective generator $P$ of
$\mathcal C$. The regular permutation action of $E_r$ on $p^r$ letters
makes $\operatorname{Hom}(P,P^{\otimes p^r})$ an $E_r$-module.
Section~\ref{sec:projectivity} proves that this module is nonprojective
for some $r$, using projectivity detection and the symmetrizer criterion
of \cite[Lemma~4.10]{CoulembierEtingofOstrik2023}.
At the least such $r$, rank support selects an embedding for which
$\Phi$ is exact (Theorem~\ref{thm:exact-higher-frobenius} in
Section~\ref{sec:support}).
It remains to remove the $\mathcal C$ factor.
Section~\ref{sec:dimension-descent} compares the Frobenius--Perron
dimensions on that side of $\Phi$: if $S$ is simple and
$A\boxtimes B$ is a simple constituent of $\Phi(S)$, then the dimension
of $A$ is at most that of $S$. Equality forces $S$ into the
Frobenius-exact subcategory of $\mathcal C$ and makes $A$ the only
possible first component. In the finite graph with arrows $S\to A$,
every cycle therefore lies in that subcategory and cannot be left.
After finitely many iterations, all first components lie there, so
the image lies in its Serre closure on the first side. This adapts the finite-simple descent of
\cite[Section~10]{EtingofOstrikFrobenius} to the exact higher functor.
The Serre-closure results of that paper provide a fiber functor to
$\operatorname{Ver}_p$ in odd characteristic and to
$\operatorname{Ver}_4^+\subset\operatorname{Ver}_4$ in characteristic two.
Combining the resulting factors in one tower level and correcting the
scalar semilinearity completes Theorem~\ref{thm:main}.
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//introduction.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//main.tex ====
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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
$\operatorname{Ver}_{p^n}(k)$. The result includes characteristic two,
and the level $n$ may depend on the category.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\clearpage
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/restricted}
\input{sections/higher}
\input{sections/projectivity}
\input{sections/support}
\input{sections/descent}
\input{sections/consequences}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//main.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//preamble.tex ====
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==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//preamble.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//preliminaries.tex ====
\section{Finite tensor categories and products}
\label{sec:preliminaries}
Throughout, $k$ is algebraically closed of characteristic $p>0$.
Unless explicitly stated otherwise, group representations and rational
algebraic-group modules are finite dimensional over $k$. A tensor functor is strong monoidal;
when called symmetric it also preserves the given symmetries.
We first relate the hypotheses of Theorem~\ref{thm:main} to the usual
finite-tensor-category conventions.
\subsection{The finite-category conventions}
\begin{lemma}\label{lem:finite-conventions}
Under the conventions of Theorem~\ref{thm:main}, the unit of $\mathcal C$
is simple and $\mathcal C$ has a projective generator $P$. The functor
$\omega=\operatorname{Hom}_{\mathcal C}(P,-)$ is exact and faithful, and
$\mathcal C$ is equivalent as a $k$-linear abelian category to the category
of finite-dimensional modules over a finite-dimensional $k$-algebra.
\end{lemma}
\begin{proof}
We recall the unit argument of Deligne--Milne
\cite[Proposition~1.17]{DeligneMilne1982}. Let $i:U\hookrightarrow\mathbf1$
and $V=\operatorname{coker}i$. The epimorphism
$V\otimes U\leftarrow U$ induced by $\mathbf1\twoheadrightarrow V$
has zero composite with the monomorphism $V\otimes i:V\otimes U\to V$.
Thus $V\otimes U=0$. Put $W=\ker(i^*:\mathbf1\to U^*)$.
For a subobject $j:T\hookrightarrow X$, adjunction and exactness give
\[
T\otimes U=0
\quad\Longleftrightarrow\quad
(i^*\otimes\operatorname{id}_X)j:T\longrightarrow U^*\otimes X
\text{ is zero}.
\]
Consequently $W\otimes X$ is the largest subobject of $X$ annihilated by
tensoring with $U$. Taking $X=\mathbf1,V$ gives
$W\otimes U=0$ and $W\otimes V\simeq V$. Tensoring
$0\to U\to\mathbf1\to V\to0$ with $W$ now shows that the natural map
$W\to V$ is an isomorphism. Hence $\mathbf1=U\oplus W$.
Since $\operatorname{End}(\mathbf1)=k$ has no nontrivial idempotents,
$\mathbf1$ is simple.
Choose representatives $S_1,\dots,S_t$ of the simple objects and projective
objects $Q_i$ with epimorphisms $Q_i\twoheadrightarrow S_i$. Set
$P=\bigoplus_iQ_i$. Induction on length, using projectivity to lift an
epimorphism onto a simple quotient, shows that every object is a quotient
of a finite direct sum of copies of $P$. Thus $P$ is a projective
generator. Evaluation by $P$ is exact; it is faithful because a nonzero
image of a morphism receives a nonzero map from $P$, which lifts across
the epimorphism onto that image. Writing
$A=\operatorname{End}(P)^{\mathrm{op}}$, finite presentations by copies of
$P$ identify $\mathcal C$ with $A$-modules of finite dimension. Indeed,
evaluation is fully faithful on such presentations, and every finite
$A$-module has a presentation by finite free $A$-modules. The algebra
$A$ is finite-dimensional by the Hom-space hypothesis.
\end{proof}
We fix such a $P$. In particular there is an epimorphism
$P\twoheadrightarrow\mathbf1$: the unit is one of the $S_i$.
The finite-algebra description also supplies projective covers of all
simple objects whenever they are needed below.
\subsection{Deligne products}
For finite $k$-linear abelian categories $\mathcal A$ and $\mathcal B$,
their Deligne product $\mathcal A\boxtimes\mathcal B$ represents
$k$-bilinear functors that are right exact in each variable.
If $\mathcal A\simeq A\text{-}\mathrm{mod}$ and
$\mathcal B\simeq B\text{-}\mathrm{mod}$, then
$\mathcal A\boxtimes\mathcal B\simeq(A\otimes_k B)\text{-}\mathrm{mod}$.
Since $k$ is algebraically closed, its simple objects are the exterior
products of simple objects of the two categories. Deligne products
of finite tensor categories carry the componentwise tensor structure.
\begin{lemma}\label{lem:combine-functors}
Let $\mathcal A_1,\ldots,\mathcal A_s,\mathcal Z$ be finite symmetric
tensor categories. Exact $k$-linear symmetric tensor functors
$F_i:\mathcal A_i\to\mathcal Z$ induce an exact $k$-linear symmetric
tensor functor
\[
H:\mathcal A_1\boxtimes\cdots\boxtimes\mathcal A_s\to\mathcal Z,
\qquad H(X_1\boxtimes\cdots\boxtimes X_s)=\bigotimes_i F_i(X_i).
\]
Deligne products of exact tensor functors are exact as well.
\end{lemma}
\begin{proof}
The displayed multilinear functor extends to a right-exact functor by
the universal property. Symmetry in $\mathcal Z$ supplies the tensor
constraint by reordering the factors. The constraint, its inverse,
and the coherence identities extend from exterior products by that
same property. The resulting strong monoidal functor preserves duals.
Dualizing a short exact sequence, applying right exactness, and
dualizing back proves left exactness. The identical argument, with
the componentwise tensor structure, proves the last assertion.
\end{proof}
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//preliminaries.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//projectivity.tex ====
\section{A first nonprojective tensor-power module}
\label{sec:projectivity}
Retain the projective generator $P$ and the exact faithful evaluation
$\omega=\operatorname{Hom}_{\mathcal C}(P,-)$ from
Lemma~\ref{lem:finite-conventions}. We now select the rank of the elementary
abelian group for the higher Frobenius construction. The essential point is that exact
evaluation by a projective generator detects internal norm maps, even
though that evaluation does not preserve tensor products.
\subsection{Norms and independent group actions}
For an object $X$ with an action of a finite group $H$, denote its
invariants and coinvariants by $X^H$ and $X_H$. These are, respectively,
the simultaneous kernel and cokernel of the maps $h-1$. The endomorphism
$\sum_{h\in H}h$ factors through a morphism
\[
\nu_H(X):X_H\longrightarrow X^H,
\]
called the norm map. Exact $k$-linear functors preserve invariants,
coinvariants, and this map.
\begin{lemma}\label{lem:norm-projectivity}
Let $H$ be a finite $p$-group and $M$ a finite-dimensional $kH$-module.
Then $M$ is projective if and only if $\nu_H(M)$ is an isomorphism.
For an arbitrary finite group $G$, the norm map of every projective
$kG$-module is an isomorphism.
\end{lemma}
\begin{proof}
For the regular module $kG$, both coinvariants and invariants are
one-dimensional, generated by $[1]$ and $\sum_{g\in G}g$, respectively;
the norm takes the first generator to the second. The assertion therefore
holds for free modules and their direct summands.
For the converse, $kH$ is a local self-injective algebra. Its unique
simple module is the trivial module, and its socle is the line generated
by $N_H=\sum_{h\in H}h$. The augmentation ideal is nilpotent by induction
on $|H|$: choose a central element $z$ of order $p$, use $(z-1)^p=0$,
and pass to $k[H/\langle z\rangle]$. This proves locality. The
nondegenerate associative pairing that takes $(a,b)$ to the coefficient
of $1$ in $ab$ identifies $kH$ with its vector-space dual as a module,
proving self-injectivity. The socle is the space of left $H$-invariants
in the regular module, hence the stated line.
Every nonzero left ideal meets the socle. Thus, if $N_Hm\ne0$, the map
$kH\to M$, $a\mapsto am$, is injective and splits by self-injectivity.
Removing free summands repeatedly gives $M\simeq(kH)^a\oplus M'$ with
$N_HM'=0$. If $M'\ne0$, then $M'_H\ne0$ by Nakayama's Lemma, so its
zero norm map cannot be an isomorphism. An isomorphic norm therefore
forces $M'=0$.
\end{proof}
\begin{lemma}\label{lem:internal-norm-products}
Let finite $p$-groups $H_1,\dots,H_s$ act on objects
$X_1,\dots,X_s$ of $\mathcal C$, respectively. If each
$\omega(X_i)$ is projective over $kH_i$, then
$\omega(X_1\otimes\cdots\otimes X_s)$ is projective over
$k[H_1\times\cdots\times H_s]$, for the independent actions on the
tensor factors. Its restriction to every subgroup of that product is
projective as well.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:norm-projectivity}, each evaluated norm
$\omega(\nu_{H_i}(X_i))$ is an isomorphism. Exact faithfulness of
$\omega$ implies that each internal norm is an isomorphism.
Exactness of tensor products gives the natural identifications
\[
(X\otimes Y)_{H\times K}\simeq X_H\otimes Y_K,
\qquad
(X\otimes Y)^{H\times K}\simeq X^H\otimes Y^K
\]
when $H$ acts on $X$ and $K$ acts on $Y$. One obtains these identities
by taking kernels or cokernels for one group at a time. Under them,
$\nu_{H\times K}(X\otimes Y)=\nu_H(X)\otimes\nu_K(Y)$.
Iteration proves that the product-group norm is an isomorphism.
Applying $\omega$ and Lemma~\ref{lem:norm-projectivity} proves the first
claim. Restriction preserves projectivity because a finite group algebra
is free over the group algebra of any subgroup.
\end{proof}
\subsection{Existence and minimality}
Let $E_a=(\mathbb Z/p)^a$ act on $p^a$ positions by its regular action,
and hence act on $P^{\otimes p^a}$ through the symmetry of $\mathcal C$.
Define
\[
M(a)=\omega(P^{\otimes p^a})\qquad(a\ge1).
\]
\begin{proposition}\label{prop:first-nonprojective}
There is a least integer $r\ge1$ 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 first that every $M(a)$ is projective. Fix $d\ge1$ and an
elementary abelian $p$-subgroup $H\subseteq\mathfrak S_d$.
On each $H$-orbit, the action is regular for a quotient $H/K_i$, because
$H$ is abelian. Choose identifications $H/K_i\simeq E_{a_i}$, allowing
$a_i=0$ for singleton orbits. Regrouping tensor factors identifies
$P^{\otimes d}$ with the tensor product of the corresponding orbit
powers. Each evaluated orbit power is projective for its quotient
group, by the assumption when $a_i>0$, and trivially when $a_i=0$.
Lemma~\ref{lem:internal-norm-products} applies to the independent
product of the quotient groups. The homomorphism
$H\to\prod_iH/K_i$ is injective, since the permutation action of $H$
is faithful. Restriction along it shows that
$\omega(P^{\otimes d})$ is projective over $kH$.
Chouinard's projectivity theorem now implies that
$\omega(P^{\otimes d})$ is projective over $k\mathfrak S_d$
\cite{Chouinard1976}; see also
\cite[p.~20]{Benson2015}.
The tensor power of $P\twoheadrightarrow\mathbf1$ is an
$\mathfrak S_d$-equivariant epimorphism to the unit with its trivial
action. Exact evaluation, followed by coinvariants, therefore gives an
epimorphism
\[
\omega(P^{\otimes d})_{\mathfrak S_d}
\twoheadrightarrow\omega(\mathbf1)\ne0.
\]
Lemma~\ref{lem:norm-projectivity} shows that the total symmetrizer
$b_d=\sum_{\sigma\in\mathfrak S_d}\sigma$ acts nontrivially on
$\omega(P^{\otimes d})$, and hence on $P^{\otimes d}$.
On the other hand, the action maps
\[
k\mathfrak S_d\longrightarrow
\operatorname{End}_{\mathcal C}(P^{\otimes d})
\]
cannot all be injective. To see this without any dimension theory,
write $\ell(X)$ for the length of $X$ and put
$b=\max_i\ell(S_i\otimes P)$. Exactness gives
$\ell(P^{\otimes d})\le\ell(P)b^{d-1}$.
Since the simples have endomorphism field $k$, induction on two
composition series gives
$\dim_k\operatorname{Hom}(X,Y)\le\ell(X)\ell(Y)$.
Thus $\dim_k\operatorname{End}(P^{\otimes d})$ grows at most
exponentially, whereas $\dim_k k\mathfrak S_d=d!$.
For clarity, the kernels form a tensor ideal in the permutation category
$\mathsf{Sym}$: its objects are the nonnegative integers, its
endomorphism algebra at $d$ is $k\mathfrak S_d$, other morphism spaces
are zero, and tensor product is addition with the usual block inclusions
of symmetric groups. The symmetry of $\mathcal C$ gives a monoidal
functor $\mathsf{Sym}\to\mathcal C$, $d\mapsto P^{\otimes d}$, and
the kernels form its nonzero kernel ideal. By
\cite[Lemma~4.10]{CoulembierEtingofOstrik2023}, every nonzero tensor
ideal in $\mathsf{Sym}$ contains some total symmetrizer $b_d$ in
positive characteristic. Its proof uses Kleshchev's faithfulness result
in the appendix to that paper \cite[Appendix~A, Proposition~A.1]{CoulembierEtingofOstrik2023}.
This contradicts the preceding nonvanishing.
The least $r$ therefore exists.
Finally, let $H<E_r$ be proper, of rank $a<r$. Its action on the
regular $E_r$-set is a disjoint union of regular $H$-orbits. Each orbit
power has projective evaluation by minimality when $a>0$, and trivially
when $a=0$. Apply Lemma~\ref{lem:internal-norm-products} to the
independent orbit groups, then restrict to their diagonal copy of $H$.
This proves the last assertion, including $r=1$.
\end{proof}
The proposition supplies a nonprojective $E_r$-module whose restriction
to every proper subgroup is projective. The next step uses its support
to choose the embedding $E_r\hookrightarrow\mathbb G_a$.
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//projectivity.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//restricted.tex ====
\section{Restricted tilting modules}
\label{sec:restricted}
Let $G=\mathrm{SL}_{2,k}$, let $V=k^2$ be its standard module, and set
\[
u(t)=\begin{pmatrix}1&t\\0&1\end{pmatrix}.
\]
Let $U\simeq\mathbb G_a$ be the upper unitriangular subgroup,
parameterized by these matrices. Write $L(h)$ for the simple rational $G$-module of highest weight
$h\geq0$, and $\operatorname{St}_m=L(p^m-1)$ for the $m$th Steinberg
module; in particular $\operatorname{St}_0=k$.
The costandard and Weyl modules are
$\nabla(a)=\operatorname{Sym}^a(V^*)$ and $\Delta(a)=\nabla(a)^*$.
Here $V\simeq V^*$ by the invariant determinant pairing.
A module is \emph{tilting} if it has both a filtration by costandard
modules and a filtration by Weyl modules. Denote by $T(a)$ the
indecomposable tilting module of highest weight $a$.
\begin{theorem}\label{thm:restricted-tilting}
Let $r\geq1$, let $E_r=(\mathbb Z/p)^r$, and let
$\iota:E_r\hookrightarrow U(k)$ be any embedding. Write $R_\iota$
for restriction along $\iota$. For every finite-dimensional
$kE_r$-module $D$, there is a tilting $G$-module $T$ such that
\[
R_\iota(\operatorname{St}_{r-1})\otimes D\simeq R_\iota(T).
\]
\end{theorem}
This proves the restricted-Steinberg assertion of
\cite[Conjecture~1]{CoulembierFlakeSteinberg}, for every embedding and
every prime. The proof has three steps. We lift $D$ to a vector bundle
on the plane with bounded highest weights at the origin. Tensoring
with $\operatorname{St}_{r-1}$ makes that origin fibre tilting. Finally,
an equivariant formal comparison identifies its restriction to $U$
with the fibre on the punctured plane. Write $R=R_\iota$ within this section.
\subsection{A vector bundle with bounded origin weights}
\label{subsec:geometric-lift}
For a finite-dimensional rational $U$-module $D$, its \emph{degree}
is the largest degree of an entry of its action matrix $\rho_D(u(t))$.
This number is independent of the chosen basis. The zero module will
always satisfy every degree bound below.
Keep the embedding $\iota:E_r\hookrightarrow U(k)$ fixed, and put $q=p^r$. We will replace an arbitrary
$kE_r$-module by a $G$-module at the origin of the standard plane.
The two modules need not be isomorphic on restriction; their comparison
will come after tensoring with a Steinberg module.
\begin{proposition}\label{prop:bounded-origin}
For every finite-dimensional $kE_r$-module $D$, there is a
$G$-equivariant vector bundle $\mathcal M$ on $\mathbb A^2_k$ such that
the stabilizer action on the fibre at $e_1=(1,0)$ restricts to $D$
along $\iota$. Every simple composition factor of the rational
$G$-module $M_0=\mathcal M|_0$ has highest weight less than $q$.
\end{proposition}
\begin{proof}
We first lift $D$ to a rational $U$-module of degree less than $q$.
Let $H\subset k$ be the image of $E_r$ under its additive embedding.
The vector space
\[
C_q=k[t]_{<q}=\operatorname{span}_k\{1,t,\ldots,t^{q-1}\}
\]
is a subcoalgebra of $k[t]$, with comultiplication
$\Delta f(t,s)=f(t+s)$ and counit $f\mapsto f(0)$.
Evaluation induces a coalgebra map
\[
\operatorname{ev}_H:C_q\longrightarrow k^H=\mathcal O(E_r).
\]
It is bijective: a polynomial of degree less than $q$ vanishing at
the $q$ distinct elements of $H$ is zero, and both spaces have
dimension $q$. Transporting the $\mathcal O(E_r)$-coaction of $D$
across this isomorphism and then including $C_q\subset k[t]$
gives a rational $U$-module $\widehat D$ of degree less than $q$.
Its restriction is $D$. This is a lift of individual representations;
we do not require the lifting operation to preserve tensor products.
The orbit map $G\to\mathbb A^2\setminus\{0\}$, $g\mapsto ge_1$,
is a right $U$-torsor. Indeed its fibres are right $U$-cosets,
and on the opens $x\ne0$ and $y\ne0$ it has the respective sections
\[
\begin{pmatrix}x&0\\y&x^{-1}\end{pmatrix},\qquad
\begin{pmatrix}x&-y^{-1}\\y&0\end{pmatrix}.
\]
Form the associated vector bundle
\[
\mathcal E=G\times^U\widehat D,
\qquad (g,d)\sim(gu,u^{-1}d),
\]
on this punctured plane, with $G$ acting by left multiplication.
The fibre identification $d\mapsto[1,d]$ at $e_1$ gives precisely
the prescribed $U$-action on $\widehat D$.
We next extend the bundle and its equivariance across the origin.
A coherent extension $\mathcal F$ of $\mathcal E$ exists on the
noetherian affine plane. Its double dual
$\mathcal M=\mathcal F^{**}$ is coherent and reflexive, and still
restricts to $\mathcal E$. On a regular surface a reflexive coherent
sheaf is locally free: its stalk has depth two at a closed point,
so the Auslander--Buchsbaum formula gives projective dimension zero;
the stalks of dimensions zero and one are free as well.
These standard extension statements can be found in
\cite[Tags 0G41, 0AY4, and 0B3N]{StacksProject}.
Write $a:G\times\mathbb A^2\to\mathbb A^2$ for the action and
$\pi_2$ for the projection. Over the punctured plane the bundle
action is an isomorphism $\pi_2^*\mathcal M\to a^*\mathcal M$.
Both sheaves are vector bundles on the regular variety
$G\times\mathbb A^2$. Their homomorphism sheaf is locally free,
and its sections extend uniquely across $G\times\{0\}$, which
has codimension two. This is the usual extension property for
reflexive sheaves \cite[Tag 0EBJ]{StacksProject}. Extend the isomorphism
and its inverse; their composites are the identity by uniqueness.
The identity and cocycle conditions hold on the punctured plane
and therefore, by the same uniqueness on $\mathbb A^2$ and
$G\times G\times\mathbb A^2$, hold everywhere. Thus
$\mathcal M$ is a $G$-equivariant vector bundle.
It remains to bound the highest weights of $M_0$. The line
$\ell=ke_1$ is fixed pointwise by $U$. For $a\in k^\times$ put
$h_a=\operatorname{diag}(a,a^{-1})$. The bundle action of $h_a$
identifies the fibres at $e_1$ and $ae_1$, and
\[
h_a^{-1}u(t)h_a=u(a^{-2}t).
\]
Consequently the $U$-action on every nonzero fibre of
$\mathcal M|_\ell$ has degree less than $q$.
To pass this bound to the origin, let
$A=\Gamma(\ell,\mathcal End(\mathcal M|_\ell))$.
Since $U$ acts trivially on the base $\ell$, its bundle action is
an element of $A\otimes_k k[t]$, say
$\sum_{j=0}^{N} b_jt^j$. For $j\geq q$, the section $b_j$
vanishes on every nonzero fibre. The field $k$ is infinite and
$\mathcal End(\mathcal M|_\ell)$ is locally free, so $b_j=0$.
Specialization at zero therefore gives an action on $M_0$ of
degree less than $q$. This bound passes to submodules and
quotients by writing the action in a basis adapted to a submodule.
Finally, if $L(h)$ is the simple $G$-module of highest weight $h$,
its $U$-action has degree exactly $h$. To see this, write
$h=\sum_i h_ip^i$ with $0\leq h_i<p$. Steinberg's tensor product
theorem, in the form recalled in \cite[Lemma 1.1 and the proof of
Theorem 2.1]{DotyHenke2005}, gives
\[
L(h)\cong\bigotimes_i
\bigl(\operatorname{Sym}^{h_i}(k^2)\bigr)^{(i)}.
\]
Here $(i)$ denotes Frobenius twist. The action on the $i$th factor
has degree $h_ip^i$. On the tensor product, the coefficient from
$\bigotimes_i e_2^{h_i}$ to $\bigotimes_i e_1^{h_i}$ is $t^h$,
so the upper bound $h$ is attained. Hence every simple factor
$L(h)$ of $M_0$ satisfies $h<q$, as required. The argument applies
unchanged when $p=2$.
\end{proof}
\subsection{The origin fibre after tensoring with a Steinberg module}
\label{subsec:bounded-tilting}
We next convert the highest-weight bound on the origin fibre into a tilting
module. We use the theory of tilting modules developed by Donkin
\cite{Donkin1993}. Tilting modules are closed under extensions and tensor products.
For $0\leq a<p$ one has $L(a)=T(a)$, and Donkin's formula is
\begin{equation}\label{eq:donkin-sl2}
T(a)\otimes T(b)^{(1)}\simeq T(a+pb)
\qquad(p-1\leq a\leq2p-2,\quad b\geq0),
\end{equation}
where $(1)$ means pullback along the Frobenius morphism of $G$.
The identity $L(a)=T(a)$ in the stated range and Formula~\eqref{eq:donkin-sl2}
hold in every positive characteristic, including $2$;
see \cite[Proposition~3.1]{BEO2023}.
The following is the extension-closed form of
\cite[Lemma~3.3]{BEO2023}, also used in the proof of
\cite[Lemma~3.1.6]{CoulembierFlakeSteinberg}. We recall its short proof
to make the highest-weight bound explicit.
\begin{lemma}\label{lem:bounded-tilting}
Let $k$ be algebraically closed of characteristic $p>0$, let $m\geq0$,
and let $M$ be a finite-dimensional rational $SL_2(k)$-module.
If every simple composition factor of $M$ is $L(h)$ with
$0\leq h<p^{m+1}$, then $\operatorname{St}_m\otimes M$ is tilting.
Here $\operatorname{St}_0=k$.
\end{lemma}
\begin{proof}
Tensoring a composition series of $M$ by $\operatorname{St}_m$ and splicing
Weyl and costandard filtrations reduces the assertion to $M=L(h)$.
We induct on $m$. When $m=0$, the inequality $h<p$ gives
$\operatorname{St}_0\otimes L(h)=T(h)$.
For $m>0$, write $h=d+ph'$ with $0\leq d<p$ and $h'<p^m$.
Steinberg's tensor product theorem gives
\[
\operatorname{St}_m\otimes L(h)
\simeq
L(d)\otimes\operatorname{St}_1\otimes
\bigl(\operatorname{St}_{m-1}\otimes L(h')\bigr)^{(1)}.
\]
By induction, the module in parentheses is a direct sum of tilting
modules $T(b)$. Formula~\eqref{eq:donkin-sl2}, applied with $a=p-1$,
shows that tensoring its Frobenius twist with $\operatorname{St}_1$
is tilting. Tensoring once more with $L(d)=T(d)$ preserves that property.
\end{proof}
Apply Lemma~\ref{lem:bounded-tilting} to the origin fibre $M_0$ with
$m=r-1$. Its established bound $h<p^r$ shows that
\[
T:=\operatorname{St}_{r-1}\otimes M_0
\]
is tilting. This includes $r=1$, when the Steinberg factor is trivial.
It remains to compare this origin fibre with the given module on the
punctured plane.
\subsection{Comparing the origin and the unipotent fibre}
\label{subsec:formal-comparison}
The bundle $\mathcal N=\operatorname{St}_{r-1}\otimes_k\mathcal M$
has the tilting origin fibre $T$ constructed above. We now recover its
representation on the punctured orbit from that fibre. The following
lemma applies to any equivariant bundle with tilting origin fibre:
we first trivialize it equivariantly on successive infinitesimal
neighbourhoods of $0$, then compare the generic point of the $U$-fixed
line with $e_1$.
\begin{lemma}[Comparison with a tilting origin fibre]
\label{lem:formal-comparison}
Let $k$ be an algebraically closed field of arbitrary positive
characteristic. Let $\mathcal N$ be a $G$-equivariant vector bundle on $V$,
and suppose that its origin fibre $T=\mathcal N_0$ is a tilting
$G$-module. Then
\[
\mathcal N_{e_1}\cong T|_U
\]
as rational $U$-modules over $k$.
\end{lemma}
\begin{proof}
We first construct compatible equivariant trivializations on every
infinitesimal neighbourhood of $0$. Let $A=k[x,y]$, let
$\mathfrak m=(x,y)$, and let $N=\Gamma(V,\mathcal N)$, a possibly
infinite-dimensional rational $G$-module. Its quotients
$N/\mathfrak m^jN$ are finite dimensional. Local freeness gives
canonical $G$-isomorphisms
\begin{equation}
\label{eq:jet-kernel}
\mathfrak m^jN/\mathfrak m^{j+1}N
\cong (\mathfrak m^j/\mathfrak m^{j+1})\otimes_k T
\cong \operatorname{Sym}^j(V^*)\otimes_k T
\qquad(j\geq 0).
\end{equation}
Recall that a good filtration has costandard successive quotients, and a
Weyl filtration has standard successive quotients. For $G=\mathrm{SL}_2$,
the costandard module of highest weight $a$ is
$\nabla(a)=\operatorname{Sym}^a(V^*)$. Tensor products of two such modules
have good filtrations. Indeed, for $a,b\geq1$, multiplication of polynomials
gives the exact sequence
\begin{equation}
\label{eq:symmetric-power-filtration}
0\longrightarrow
\operatorname{Sym}^{a-1}(V^*)\otimes\operatorname{Sym}^{b-1}(V^*)
\xrightarrow{\;d\;}
\operatorname{Sym}^{a}(V^*)\otimes\operatorname{Sym}^{b}(V^*)
\longrightarrow \operatorname{Sym}^{a+b}(V^*)
\longrightarrow0.
\end{equation}
Here $d=x_1y_2-y_1x_2$ is the determinant in two copies of $V^*$.
Multiplication by $d$ is injective, its image lies in the kernel, and
dimensions prove exactness. Induction on $\min(a,b)$, starting with a
trivial factor, gives the assertion. Refining filtrations then proves the
same tensor-product assertion for arbitrary modules with good
filtrations. This argument is unchanged in characteristic two.
The module $T$ has both a Weyl filtration and a good filtration.
Consequently, the target of the next Ext group has a good filtration,
and the standard--costandard Ext vanishing theorem of
Cline--Parshall--Scott--van der Kallen gives
\begin{equation}
\label{eq:formal-ext-vanishing}
\operatorname{Ext}^1_G
\bigl(T,\operatorname{Sym}^j(V^*)\otimes T\bigr)=0
\qquad(j\geq0).
\end{equation}
We use Ext in the category of rational $G$-modules; the vanishing follows
from $\operatorname{Ext}^1_G(\Delta(a),\nabla(b))=0$ by induction on the two
filtrations; see \cite[Chapter~1, Proposition~2.6, and Chapter~4,
Proposition~1.1]{Riche}.
Starting with the identity $s_1:T\to N/\mathfrak mN$, lift successively to
$G$-maps
\[
s_j:T\longrightarrow N/\mathfrak m^jN\qquad(j\geq1)
\]
such that $s_{j+1}$ reduces to $s_j$. Pull back the short exact sequence
\[
0\longrightarrow \operatorname{Sym}^j(V^*)\otimes_k T
\longrightarrow N/\mathfrak m^{j+1}N
\longrightarrow N/\mathfrak m^jN\longrightarrow0
\]
along $s_j$. By \eqref{eq:formal-ext-vanishing} the resulting extension
of $T$ splits equivariantly; a splitting gives the lift $s_{j+1}$.
Extending $s_j$ by scalars gives an equivariant
$A/\mathfrak m^j$-linear map
\[
(A/\mathfrak m^j)\otimes_k T\longrightarrow N/\mathfrak m^jN.
\]
Both modules are free of rank $\dim_kT$ over the local Artinian ring
$A/\mathfrak m^j$, and the map reduces to the identity on the residue
field. Nakayama's Lemma therefore makes it an isomorphism. Taking the
inverse limit produces an isomorphism
\begin{equation}
\label{eq:formal-trivialization}
k[[x,y]]\otimes_kT\xrightarrow{\sim}\widehat N
\end{equation}
compatible with the $G$-action at every finite order.
The line $y=0$ is fixed scheme-theoretically by $U$. Restrict
\eqref{eq:formal-trivialization} to this line and then set $K=k((x))$.
This gives a rational $U_K$-isomorphism
\[
\varphi:T_K\xrightarrow{\sim}\mathcal N_{xe_1}.
\]
To justify the group-scheme assertion explicitly, choose a polynomial
frame for $\mathcal N$ on the line; this is possible because a finite
projective $k[x]$-module is free. The action is a matrix
$C(x,t)\in\mathrm{GL}_{\dim T}(k[x,t])$, and the action on $T$ is a
matrix $B(t)\in\mathrm{GL}_{\dim T}(k[t])$. If $H(x)$ is the matrix of
the restricted formal isomorphism, equivariance at every finite order
says
\[
C(x,t)H(x)=H(x)B(t)\pmod{x^j}
\qquad\text{for every }j\geq1.
\]
These matrices have uniformly bounded degree in $t$, so coefficientwise
$x$-adic separation gives the equality in $k[[x]][t]$, and hence over
$K[t]$.
We must compare the two fibres without changing the parameter of $U$.
Put $h=\operatorname{diag}(x,x^{-1})\in G(K)$, and let
$b_h:\mathcal N_{e_1,K}\to\mathcal N_{xe_1}$ be bundle transport.
Write $\rho_T$ for the $G$-action on $T$ and $\rho_{e_1},\rho_x$ for
the two fibre actions of $U$. Since
$h u(t)h^{-1}=u(x^2t)$, equivariance gives
\[
b_h\rho_{e_1}(u(t))=\rho_x(u(x^2t))b_h,
\qquad
\rho_T(h)\rho_T(u(t))=\rho_T(u(x^2t))\rho_T(h).
\]
It follows that
\begin{equation}
\label{eq:cancel-torus-conjugation}
f=b_h^{-1}\varphi\rho_T(h):T_K\xrightarrow{\sim}\mathcal N_{e_1,K}
\end{equation}
intertwines the original $U$-actions. Thus restriction of $f$ to any
fixed finite subgroup of $U(k)$ uses that subgroup's original embedding.
Finally, this isomorphism descends to $k$. In fixed $k$-bases, the
intertwiner space
\[
L=\operatorname{Hom}_U(T,\mathcal N_{e_1})
\]
is the kernel of finitely many linear equations over $k$: compare the
coefficients of $t$ in the two polynomial action matrices. Hence its
scalar extension to $K$ is the intertwiner space over $K$.
Equation~\eqref{eq:cancel-torus-conjugation} shows that the determinant
polynomial on $L$ is nonzero. Since $k$ is infinite, this polynomial is
nonzero at some point of $L(k)$, giving the required isomorphism. At no
point have we asserted that the formal trivialization algebraizes on
the plane.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:restricted-tilting}]
Given $D$, choose $\mathcal M$ as in Proposition~\ref{prop:bounded-origin}
and put $M_0=\mathcal M|_0$. Lemma~\ref{lem:bounded-tilting} makes
$\operatorname{St}_{r-1}\otimes M_0$ tilting. Apply Lemma~\ref{lem:formal-comparison} to
$\mathcal N=\operatorname{St}_{r-1}\otimes_k\mathcal M$. Its origin
fibre is the tilting module
$T=\operatorname{St}_{r-1}\otimes_kM_0$. By
Proposition~\ref{prop:bounded-origin}, the fibre of $\mathcal N$ at $e_1$
restricts to $R(\operatorname{St}_{r-1})\otimes D$ along the fixed
embedding $E_r\hookrightarrow U(k)$. The comparison therefore gives
\[
R(T)\cong R(\operatorname{St}_{r-1})\otimes D,
\]
as required. This comparison also applies when $r=1$, with
$\operatorname{St}_0=k$.
\end{proof}
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//restricted.tex ====
==== BEGIN Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//support.tex ====
\section{Selecting an embedding by rank support}
\label{sec:support}
The preceding section supplies a nonprojective representation of an
elementary abelian group whose restrictions to all proper subgroups are
projective. We now choose an embedding of that group in the unipotent
subgroup of $\mathrm{SL}_2$ so that its tensor product with the restricted
Steinberg module remains nonprojective. This choice will make the higher
Frobenius functor exact.
Let $k$ be algebraically closed of characteristic $p>0$, let
$E_r=(\mathbb Z/p\mathbb Z)^r$, and fix generators $g_1,\ldots,g_r$.
For a finite-dimensional $kE_r$-module $D$, define
\[
\operatorname{Supp}(D)=
\left\{[\alpha_1:\cdots:\alpha_r]\in\mathbb P^{r-1}(k):
D\text{ is not free over }k[t]/(t^p),\quad
t\mapsto\sum_{i=1}^r\alpha_i(g_i-1)\right\}.
\]
This is the projectivized rank variety. The rank-variety projectivity
and tensor-product theorems state that
\begin{equation}\label{eq:support-properties}
\operatorname{Supp}(D)=\varnothing\iff D\text{ is projective},
\qquad
\operatorname{Supp}(D\otimes D')
=\operatorname{Supp}(D)\cap\operatorname{Supp}(D').
\end{equation}
These statements hold with the indicated coordinates for every prime;
see \cite[Sections~1.9 and~1.12]{Benson2017} and
\cite[Section~2.3]{CoulembierFlakeSteinberg}.
\begin{lemma}\label{lem:support-independent}
Suppose that $M$ is a nonprojective finite-dimensional $kE_r$-module and
that $M|_H$ is projective for every proper subgroup $H<E_r$.
Every point $[\alpha_1:\cdots:\alpha_r]$ of $\operatorname{Supp}(M)$ has
$\mathbb F_p$-linearly independent coordinates.
\end{lemma}
\begin{proof}
For a nonzero linear form $c:E_r\to\mathbb F_p$, write $c(g_i)=c_i$ and
put $H=\ker c$. The tensor identity gives
\[
M\otimes\operatorname{Ind}_H^{E_r}k
\cong\operatorname{Ind}_H^{E_r}(M|_H),
\]
which is projective. We compute the support of
$N=\operatorname{Ind}_H^{E_r}k$ explicitly. This is the regular
representation of $E_r/H$, identified with $k[z]/(z^p)$, where a
generator of the quotient acts by $1+z$. Thus
$\sum_i\alpha_i(g_i-1)$ acts by multiplication by
\[
h_\alpha(z)=\sum_i\alpha_i\big((1+z)^{c_i}-1\big),
\]
with $c_i$ represented by an integer in $\{0,\ldots,p-1\}$.
Its linear coefficient is $\sum_i c_i\alpha_i$.
If this coefficient is nonzero, substitution $t\mapsto h_\alpha(z)$
is an isomorphism of truncated polynomial algebras, so $N$ is free
over $k[t]/(t^p)$. If it is zero, then $h_\alpha\in(z^2)$ and
$h_\alpha^{p-1}=0$, which is impossible on a free module of dimension
$p$. Consequently
\[
\operatorname{Supp}(N)
=\{[\alpha]:\textstyle\sum_i c_i\alpha_i=0\}.
\]
Equation~\eqref{eq:support-properties} shows that this hyperplane misses
$\operatorname{Supp}(M)$. Varying the nonzero vector
$(c_1,\ldots,c_r)\in\mathbb F_p^r$ proves the assertion.
\end{proof}
Every support point of the minimal-rank module therefore has
$\mathbb F_p$-linearly independent coordinates.
To make one of them survive after tensoring with a restricted Steinberg
module, we next compute the Steinberg support in terms of the embedding
parameters.
\begin{lemma}\label{lem:steinberg-support}
Let $\lambda_1,\ldots,\lambda_r\in k$ be linearly independent over
$\mathbb F_p$, and restrict $\mathrm{SL}_2$-representations along
\[
\iota_\lambda:E_r\longrightarrow\mathrm{SL}_2(k),
\qquad
g_i\longmapsto
\begin{pmatrix}1&\lambda_i\\0&1\end{pmatrix}.
\]
Write $R_\lambda$ for this restriction. Then
\begin{equation}\label{eq:steinberg-support}
\operatorname{Supp}\big(R_\lambda(\operatorname{St}_{r-1})\big)
=\left\{[\alpha]:
\sum_i\alpha_i\lambda_i^{p^j}=0\quad(0\le j\le r-2)\right\}.
\end{equation}
\end{lemma}
\begin{proof}
This is the Steinberg case of
\cite[Proposition~4.2.5 and Corollary~4.2.7]{CoulembierFlakeSteinberg}.
Here is a direct calculation in the chosen coordinates.
On $\operatorname{St}_1=\operatorname{Sym}^{p-1}(k^2)$, the upper
unipotent element with parameter $a$ acts by
$\exp(aN)=\sum_{b=0}^{p-1}a^bN^b/b!$, where $N$ has one Jordan
block of length $p$. Therefore $\sum_i\alpha_i(g_i-1)$ acts by a
polynomial in $N$ whose linear coefficient is
$\sum_i\alpha_i\lambda_i$. As in the proof of
Lemma~\ref{lem:support-independent}, the resulting module of dimension
$p$ is free exactly when this coefficient is nonzero.
On the $j$-th Frobenius twist, the same calculation replaces
$\lambda_i$ by $\lambda_i^{p^j}$.
Now use
\[
\operatorname{St}_{r-1}
\cong\bigotimes_{j=0}^{r-2}\operatorname{St}_1^{(j)}
\]
and the tensor-product formula in \eqref{eq:support-properties}.
When $r=1$, $\operatorname{St}_0=k$ has support
$\mathbb P^0(k)$, agreeing with the empty system in
\eqref{eq:steinberg-support}.
\end{proof}
Thus a point $[\alpha]$ lies in the Steinberg support precisely when
the embedding parameters satisfy the equations in
\eqref{eq:steinberg-support}. The next lemma solves these equations
while keeping the parameters linearly independent over $\mathbb F_p$,
as required for an embedding.
\begin{lemma}\label{lem:moore-selection}
Let $\alpha_1,\ldots,\alpha_r\in k$ be linearly independent over
$\mathbb F_p$. There are $\mathbb F_p$-linearly independent
$\lambda_1,\ldots,\lambda_r\in k$ such that
\begin{equation}\label{eq:selected-embedding}
\sum_{i=1}^r\alpha_i\lambda_i^{p^j}=0
\qquad(0\le j\le r-2).
\end{equation}
For $r=1$ the list of equations is empty.
\end{lemma}
\begin{proof}
When $r=1$, take any $\lambda_1\ne0$. Suppose $r\ge2$.
Because $k$ is perfect, we may put
$\beta_i=\alpha_i^{p^{-(r-2)}}$. Form the $(r-1)\times r$ matrix
\[
B=(\beta_i^{p^s})_{\substack{0\le s\le r-2\\1\le i\le r}}.
\]
Its rows are linearly independent. Indeed, a nontrivial row relation
would give a nonzero additive polynomial
$f(X)=\sum_{s=0}^{r-2}a_sX^{p^s}$ vanishing on the
$\mathbb F_p$-span of the $\beta_i$. That span has $p^r$ elements,
whereas $\deg f\le p^{r-2}$.
The row space of $B$ contains no nonzero vector in $\mathbb F_p^r$.
To see this, suppose $c_i=f(\beta_i)\in\mathbb F_p$ for such a
polynomial $f$. The additive polynomial $f^p-f$ then vanishes on
the same $p^r$ elements and has degree at most $p^{r-1}$.
Thus $f^p-f=0$, forcing $f=0$ because $f$ has zero constant term.
Choose a nonzero vector $\lambda\in\ker B$. This kernel is a line.
If $c\in\mathbb F_p^r$ satisfied $\sum_i c_i\lambda_i=0$, it would
belong to the annihilator of $\ker B$, which is the row space of $B$.
Hence $c=0$, proving the required independence of the $\lambda_i$.
The rows of $B$, in reverse order, are
$(\alpha_i^{p^{-j}})_i$ for $0\le j\le r-2$.
Raising their nullvector equations to the $p^j$-th power gives
\eqref{eq:selected-embedding}.
\end{proof}
\begin{proposition}\label{prop:support-selection}
Let $M$ be a nonprojective finite-dimensional $kE_r$-module whose
restriction to every proper subgroup is projective. There is an
embedding $\iota_\lambda:E_r\hookrightarrow\mathrm{SL}_2(k)$ into
the upper unipotent subgroup such that
\[
R_\lambda(\operatorname{St}_{r-1})\otimes M
\quad\text{is nonprojective}.
\]
\end{proposition}
\begin{proof}
By \eqref{eq:support-properties}, choose
$[\alpha]\in\operatorname{Supp}(M)$.
Lemma~\ref{lem:support-independent} makes its coordinates
$\mathbb F_p$-linearly independent.
Choose $\lambda$ by Lemma~\ref{lem:moore-selection}.
Its independent coordinates make $\iota_\lambda$ injective, and
Lemma~\ref{lem:steinberg-support} places $[\alpha]$ in the support of
$R_\lambda(\operatorname{St}_{r-1})$.
The tensor-product theorem now gives the conclusion.
\end{proof}
\begin{theorem}[An exact higher Frobenius functor]
\label{thm:exact-higher-frobenius}
Let $\mathcal C$ be a finite symmetric tensor category over an
algebraically closed field $k$ of characteristic $p>0$.
There are an integer $r\ge1$ and an embedding
$E_r\hookrightarrow U(k)$ for which the construction of
Section~\ref{sec:higher} gives an exact additive strong symmetric
monoidal functor
\[
\Phi:\mathcal C\longrightarrow
\mathcal C\boxtimes\operatorname{Ver}_{p^r}(k).
\]
It is given by $\Phi(X)=\Theta_{\mathcal C}(X^{\otimes p^r})$, where
$\Theta$ is the functor in \eqref{eq:theta} for the chosen embedding.
The functor $\Phi$ satisfies
$\Phi(af)=a^{p^r}\Phi(f)$, and its evaluation at every projective
$Q\in\mathcal C$ is given by the natural isomorphism
\eqref{eq:phi-evaluation}.
\end{theorem}
\begin{proof}
Choose a projective generator $P$ and the least rank $r$ of
Proposition~\ref{prop:first-nonprojective}. Then
\[
M=\operatorname{Hom}_{\mathcal C}(P,P^{\otimes p^r})
\]
is nonprojective and has projective restriction to every proper subgroup
of $E_r$. Proposition~\ref{prop:support-selection} supplies an embedding
$\iota_\lambda$ for which
$R_\lambda(\operatorname{St}_{r-1})\otimes M$ remains nonprojective.
Theorem~\ref{thm:restricted-tilting} gives
$R_\lambda(\operatorname{St}_{r-1})\otimes M\cong R_\lambda(T)$
for a tilting module $T$. Lemma~\ref{lem:theta-detection} therefore gives
\[
0\ne\Theta\big(R_\lambda(T)\big)
\cong\Theta\big(R_\lambda(\operatorname{St}_{r-1})\big)
\otimes\Theta(M).
\]
In particular $\Theta(M)\ne0$. Proposition~\ref{prop:higher-exactness},
with both projectives equal to $P$, proves that $\Phi$ is exact.
Its remaining properties and the projective-evaluation identity were
established in Section~\ref{sec:higher}. All steps include $p=2$ and $r=1$.
\end{proof}
==== END Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic//support.tex ====
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