editors: nobody
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% Fiber functors for finite symmetric tensor categories in positive
% characteristic. Aim: readable for a mathematically serious high
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% proof of the main theorem. Self-contained preamble; drop into
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usetikzlibrary{arrows.meta,positioning,calc,fit}
# Fiber functors for finite symmetric tensor categories
in positive characteristic
maketitle
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.calC.
tableofcontents
% =================================================================
# Introduction and main result
% =================================================================
## The basic objects
Throughout, k is an algebraically closed field of characteristic
p>0.
[finite symmetric tensor category]
A finite symmetric tensor category over k is a
k-linear abelian rigid symmetric monoidal category mathcal CcalC
such that:
[label=(arabic*)]
the tensor product is biexact;
every Hom(X,Y) is finite dimensional over k;
every object has finite length;
there are finitely many isomorphism classes of simple
objects;
the unit object mathbf 1bf1 has End(mathbf 1)bf1) = k;
there are enough projective objects.
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-gradedbbZ/2-graded
version with the Koszul sign in the symmetry.
[fiber functor]
A fiber functor mathcal CcalCto mathcal DcalD 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 DcalD = Vec_k; the target being a
non-vector-space category is what forces the tower in positive
characteristic.
## The higher Verlinde categories
We will use the following facts about Ver_{p^n}(k), proved in
Benson--Etingof--Ostrik (2023).
[properties of the Verlinde tower]
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)subsetVer_{p^3}(k)subset...,
]
with each inclusion exact and symmetric monoidal. The bottom cases
are
[
Ver_2(k) = Vec_k,
Ver_3(k) = sVec_k,
Ver_3^+(k) = Vec_k.
]
For every p^n > 2 one has a symmetric tensor subcategory
Ver_{p^n}^+(k)⊆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).
Intuition for Ver_p: it is the ``semisimplification'' of
Rep_k(mathbb Z/p):bbZ/p): one kills every morphism f with tr(gf)=0
for all g; the simple objects correspond to Jordan blocks of
lengths 1,...,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.
## The main theorem
Here is the result we prove.
[main theorem]
For every prime p, every algebraically closed field k of
characteristic p, and every finite symmetric tensor category
mathcal CcalC over k, there exist an integer nge 1 and a fiber
functor
[
mathcal CcalC → Ver_{p^n}(k).
]
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 moderate growth; our
theorem is for finite categories, and the finite case already
implies the finite part of all the consequences we discuss.
## Proof outline
We iteratively apply a semilinear higher Frobenius functor Phi,
each application adding one fresh Verlinde factor while forcing
the original mathcal CcalC factor to shrink into its Frobenius-exact
subcategory. After at most N iterations, where N is the number
of simple objects of mathcal C,calC, all first components land in the
Serre closure of that subcategory. The essential ingredients are:
[label=Step arabic*.,ref=step:arabic*]
Restricted tilting modulo a Steinberg module.
For every finite-dimensional module D over the
elementary abelian group E_r = (mathbb Z/p)^r,bbZ/p)^r, the
tensor product St_{r-1} ⊗ 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.
Higher Frobenius functor.
The previous step lets us define, for every embedding
E_r↪ U(k)⊆ SL_2(k), a
k-linear additive strong symmetric monoidal functor
Phi:mathcal CcalCto mathcal CcalCboxtimes Ver_{p^r}(k)
that is only semilinear (Phi(af) = a^{p^r}Phi(f)).
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^{ ⊗ p^r}), and the theory of
the projectivised rank variety.
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_calC_{ex}.
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 sec:groundwork--sec:serre-finish carry out the
four steps. Section sec:consequences records the standard
corollaries.
% =================================================================
# Groundwork
% =================================================================
## 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.
[mathcal CcalC is modules over a finite algebra]
Let mathcal CcalC be a finite symmetric tensor category. Then the
unit object is simple, and there is a projective generator
Pinmathcal C.calC. The functor omega = Hom_{mathcal C}calC(P,-) is
exact and faithful, and mathcal CcalC 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}calC(P)^{op}.
Proof.
Simple unit is the Deligne--Milne argument. If Usubsetmathbf 1bf1
is a proper subobject with cokernel Vne0, the epimorphism
Uto V ⊗ U coming from mathbf 1bf1twoheadrightarrow V has
zero composite with the subobject V ⊗ Utwoheadrightarrow V.
So V ⊗ U=0. Put W=ker(mathbf 1bf1to U^*). Then W
sits inside a short exact sequence with U, V, and tensoring by
W gives Wsimeq V. The endomorphism ring of the unit has no
idempotents, so V=0.
For the projective generator: pick surjections Q_ito S_i from
projectives onto the finitely many simples; set
P=⊕_i Q_i. Projectivity lifts surjections to P, so
every object is a quotient of P^{ ⊕ m} for some m. Exact
faithfulness comes from lifting along Pto image. To identify
mathcal CcalCsimeq A-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.
We fix such a P and such an A once and for all. In particular,
Ptwoheadrightarrowmathbf 1,bf1, since the unit is one of the
simples.
## Deligne products
For finite k-linear abelian categories mathcal A,calA,mathcal B,calB, the
Deligne product mathcal AcalAboxtimesmathcal BcalB is the
universal recipient of functors that are k-bilinear and right
exact in each variable. When mathcal AcalAsimeq A-mod and
mathcal BcalBsimeq B-mod, we have
mathcal AcalAboxtimesmathcal BcalBsimeq (A ⊗ _k B)-mod,
and its simple objects are the exterior products of the simples of
the two factors (here we use k algebraically closed).
[combining exact tensor functors]
Let mathcal A_1,...,calA_1,...,mathcal A_s,calA_s,mathcal ZcalZ be finite
symmetric tensor categories and let F_i:mathcal A_icalA_itomathcal ZcalZ
be exact k-linear strong symmetric monoidal functors. Then
[
H:mathcal A_1calA_1boxtimes...boxtimesmathcal A_scalA_stomathcal Z,calZ,
H(X_1boxtimes...boxtimes X_s)
= F_1(X_1) ⊗ ... ⊗ F_s(X_s),
]
extends uniquely to an exact k-linear strong symmetric monoidal
functor.
Proof.
The displayed multilinear functor has a right-exact extension by
the universal property of the Deligne product. Symmetry in
mathcal ZcalZ 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.
## Frobenius--Perron dimension
For finite tensor categories the Grothendieck ring K_0(mathcal C)calC)
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)calC)tomathbb RbbR with d(mathbf 1)=1bf1)=1
and d multiplicative on classes. Key facts, all in Etingof--
Gelaki--Nikshych--Ostrik (2015):
[FP dimension toolkit]
Let mathcal CcalC be a finite tensor category.
[label=(roman*)]
d(X)ge 1 for every nonzero X; equality if and only if
X is invertible.
If 0to Xto Yto Zto 0 is exact, then
d(Y)=d(X)+d(Z).
On a Deligne product, d(Aboxtimes B)=d(A),d(B).
If F:mathcal CcalCtomathcal DcalD is an exact strong monoidal
functor of finite tensor categories, then
d(F(X))=d(X) for every X.
Beware: the categorical dimension
dim_{cat}(B)in k that appears in cref{sec:descent}
is a different, k-valued invariant; it is not Frobenius--
Perron dimension. We will only use categorical dimension for one
mod-p congruence.
% =================================================================
# Step 1: Restricted tilting modules
% =================================================================
## The statement
Let G = SL_2(k), and let Usimeqmathbb G_abbG_a be its upper
unipotent subgroup, identified with k via
u(t) = (1&t0&1).
For rge 1 put E_r = (mathbb Z/p)^rbbZ/p)^r and let
iota:E_r↪ 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
hge0; let St_m = L(p^m-1) be the m-th 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
tilting if it has both a costandard filtration and a Weyl
filtration; T(a) is the indecomposable tilting module of highest
weight a.
[restricted-tilting theorem]
Let rge 1, E_r = (mathbb Z/p)^r,bbZ/p)^r, and iota:E_rto 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}) ⊗ D cong R_iota(T)
]
as kE_r-modules.
This proves Conjecture 5.4.2(1) of Coulembier--Flake (2026) and
Conjecture 1 of Coulembier--Flake (2025), for every prime.
## The plan of the proof
We do three things:
[label=(alph*)]
lift the arbitrary kE_r-module D to a coherent rational
U-module widehatwidehat DD of bounded ``degree'' (a bound on
polynomial degree of the unipotent action matrix);
use widehatwidehat DD to construct a G-equivariant vector
bundle mathcal McalM on the plane mathbb A^2_kbbA^2_k whose
origin fibre has all simple composition factors below a
height bound;
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.
We then recover D from the e_1-fibre.
## Lifting D to a bounded U-module
For a finite-dimensional rational U-module D, its
degree is the largest exponent of t appearing in the action
matrix rho_D(u(t)); this does not depend on a chosen basis.
[bounded lift]
For every finite-dimensional kE_r-module D, there is a
finite-dimensional rational U-module widehatwidehat DD of degree < p^r
such that widehatwidehat D|_D|_{E_r}cong D along iota.
Proof.
Let H⊆ k be the image of E_r under iota. The
vector space
C_q = span_k{1,t,...,t^{q-1}}⊆ 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
ev_H : C_q to k^H = mathcal O(E_r),calO(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)-coactioncalO(E_r)-coaction defining D across
ev_H, then including C_q subset k[t], gives a
rational U-module of degree <q. Its restriction is D.
## The G-equivariant bundle on the plane
Now consider the orbit map pi : Gtomathbb A^2∖bbA^2∖{0},
gmapsto 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 xne 0,
yne 0 it has obvious rational sections. Form the associated
vector bundle
[
mathcal EcalE = Gtimes^U widehatwidehat D,D,
(g,d)sim(gu,u^{-1}d),
]
on the punctured plane, with G acting by left multiplication.
The fibre at e_1 is widehatwidehat DD itself.
[extension and weight bound]
The bundle mathcal EcalE extends to a G-equivariant vector bundle
mathcal McalM on the whole affine plane, and every simple
composition factor of the origin fibre
M_0 = mathcal M|_0calM|_0 has highest weight < q.
Proof.
Extension. A coherent extension mathcal FcalF exists on the
noetherian affine plane. Its double dual mathcal McalM = mathcal
F^calF^{**} is a coherent reflexive sheaf restricting to mathcal E.calE.
On the regular surface mathbb A^2bbA^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 McalM is a vector bundle.
Equivariance. Over the punctured plane we have a
G-equivariant bundle isomorphism
pi_2^*mathcal McalMcong a^*mathcal McalM for the action
a:Gtimesmathbb A^2bbA^2tomathbb A^2bbA^2 and projection
pi_2:Gtimesmathbb A^2bbA^2tomathbb A^2.bbA^2. Both sides are locally
free, so the Hom-sheaf is locally free and its sections extend
across the codimension-two subset Gtimes{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.
Weight bound at the origin. The line ell = ke_1 is
fixed pointwise by U. For ain k^times put
h_a = 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 EcalE nd(mathcal M|_calM|_ell)) and write the
U-action as an element of A ⊗ _k k[t],
sum_j b_j t^j. For jge 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.
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, 0le h_i<p. Its U-action has degree exactly
h (the coefficient of t^h from
⊗_i e_2^{h_i} to ⊗_i e_1^{h_i} is nonzero).
Hence every simple composition factor L(h) of M_0 has
h<q.
## 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:
L(a) = T(a)(0le a<p),
T(a) ⊗ T(b)^{(1)}simeq T(a+pb)
(p-1le ale 2p-2, bge 0),
where (1) means pullback along the Frobenius. See Benson--
Etingof--Ostrik (2023), Proposition 3.1.
[Steinberg ⊗ bounded weight is tilting]
Let M be a finite-dimensional rational SL_2(k)-module with every
composition factor of the form L(h) for 0le h<p^m (where
mge 1). Then St_{m-1} ⊗ M is tilting.
Proof.
Reduces by composition series to M = L(h).
For m=1, h<p and St_0=k, so St_0 ⊗ L(h) = T(h) is
tilting by (eq:sl2-tilting-basics).
For m>1, write h = d + p h' with 0le d<p and h'<p^{m-1}.
Steinberg's tensor product theorem gives
[
St_{m-1} ⊗ L(h) cong
L(d) ⊗ St_1 ⊗
(St_{m-2} ⊗ L(h'))^{(1)}.
]
By induction the bracket is a direct sum of tilting modules. Apply
Donkin's formula (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.
Applying cref{lem:steinberg-tilting} with m=r to M_0 = mathcal
M|_0,calM|_0, we have:
T := St_{r-1} ⊗ M_0
is a tilting G-module. The remaining step is to compare T
(the origin fibre of
mathcal NcalN = St_{r-1} ⊗ _k mathcal M)calM) with the e_1-fibre
of mathcal N.calN.
## Formal comparison of fibres
[comparison with a tilting origin fibre]
Let mathcal NcalN be a G-equivariant vector bundle on mathbb A^2bbA^2
whose origin fibre T = mathcal N|_0calN|_0 is tilting. Then
mathcal N|_calN|_{e_1}cong T|_U as rational U-modules.
Proof.
Jet trivialisation. Let A=k[x,y], mathfrak m=(x,y),frakm=(x,y),
and N=Gamma(mathbb A^2,bbA^2,mathcal N),calN), a possibly
infinite-dimensional rational G-module. Local freeness gives
canonical G-isomorphisms
mathfrak m^jfrakm^j N/mathfrak m^frakm^{j+1}N cong
Sym^j(V^*) ⊗ _k T.
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
0toSym^{a-1}V^* ⊗ Sym^{b-1}V^*to
Sym^a V^* ⊗ Sym^b V^*toSym^{a+b}V^*to 0 is exact.
Now T has both a Weyl and a good filtration, so for every j,
[
Ext^1_G(T,Sym^j(V^*) ⊗ T) = 0
]
by Cline--Parshall--Scott--van der Kallen.
Starting from the identity s_1:Tto N/mathfrak mfrakm N, we lift
successively to G-maps s_j:Tto N/mathfrak m^jfrakm^j N that reduce
modulo mathfrak m^jfrakm^j to s_{j-1}; the obstruction is exactly the
Ext group just computed. Extending scalars, s_j gives an
A/mathfrak m^j-linearfrakm^j-linear map
(A/mathfrak m^j)frakm^j) ⊗ _k Tto N/mathfrak m^jfrakm^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]] ⊗ _k Txrightarrow{sim}widehatwidehat N.N.
Fibre comparison. Restrict to the fixed line y=0, pass
to K = k((x)): this gives a U_K-isomorphism
varphi:T_Kcong mathcal N_calN_{xe_1}. To cancel the parameter
mismatch we use the torus element h = diag(x,x^{-1}):
since hu(t)h^{-1} = u(x^2 t), the composite
f = b_h^{-1}varphirho_T(h) intertwines the original
U-actions (where b_h : mathcal N_calN_{e_1}tomathcal N_calN_{xe_1} is
bundle transport).
Descent to k. The space
L = Hom_U(T,mathcal N_calN_{e_1}) is cut out by finitely many
k-linear equations in fixed k-bases, so L_K = L ⊗ _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.
Proof. [Proof of cref{thm:restricted-tilting}]
Apply cref{prop:extension-bound} to get mathcal McalM with origin
fibre M_0 whose simple factors have highest weight <q=p^r.
By cref{lem:steinberg-tilting}, T = St_{r-1} ⊗ M_0 is
tilting. Apply cref{lem:formal-comparison} to
mathcal NcalN = St_{r-1} ⊗ _kmathcal M:calM: its origin fibre is
T, so
[
mathcal N|_calN|_{e_1} cong T|_U.
]
But by construction mathcal N|_calN|_{e_1} restricts to
R(St_{r-1}) ⊗ D along E_r↪ U(k), hence so
does T. Set T_{out} := T; the theorem follows.
% =================================================================
# Step 2: The higher Frobenius functor
% =================================================================
## The functor Θ from group modules to Verlinde
Fix rge 1, q=p^r, and an embedding
E = E_r ↪ U(k). We write R for restriction of
SL_2-modules along this embedding, and mathcal WcalW = Ver_{p^r}(k).
Let mathcal T⊆calT⊆ 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_rcalT/I_r whenever age p^r - 1; this is the standing
convention in this circle of papers.
The functor mathcal T/I_rcalT/I_rtomathcal WcalW 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_rcalT/I_rtoStab(E) :=
Rep_k(E)/(projectives).
Coulembier--Flake (2026, Theorem 5.4.3) state: if for every
finite-dimensional E-module D the tensor R(St_{r-1) ⊗ D
is a restriction of a tilting, then there is a k-linear strong
symmetric monoidal functor}
Θ : Rep_k(E)→ mathcal WcalW
quadwithquad
Θ(R(T)) simeq Σ_r(Tbmod I_r).
The hypothesis is exactly cref{thm:restricted-tilting}. So
cref{thm:restricted-tilting} produces Θ, once and for all
embeddings.
Θ is not exact; it is additive and symmetric monoidal.
[Θ detects nonprojectives]
Let T be a tilting module whose restriction R(T) is not
projective as an E-module. Then Θ(R(T))ne0. Moreover,
if H<E is a proper subgroup, then every module induced from a
kH-module is killed by Θ.
Proof.
If Tbmod I_r=0 then R(T) is projective (stable restriction
result). Otherwise Tbmod I_rne0, and Σ_r being fully
faithful implies Θ(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 Θ 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 ⊗ Ind_H^E D cong Ind_H^E(Res_H^E S ⊗ D) therefore
gives a projective E-module, on which Θ vanishes. Since
Θ(S)ne0 and tensoring by a nonzero object is faithful,
Θ(Ind_H^E D)=0.
## Internal evaluation
Let mathcal CcalC be a finite symmetric tensor category with
projective generator P_0. Write A = End_{mathcal C}calC(P_0)^{op},
so mathcal CcalCsimeq A-mod. An object of
mathcal CcalCboxtimesRep_k(E) is an object of mathcal CcalC
equipped with a commuting E-action. Applying Θ to the
underlying E-module and to the action maps produces an A-module
object in mathcal W,calW, hence an object of
mathcal CcalCboxtimesmathcal W.calW. This is the internal
extension Θ_{mathcal C}calC of Θ (Coulembier--Flake
2026, Example 3.2.4 and Proposition 3.2.6; requires only
additivity of Θ, not exactness).
[evaluation at projectives]
For Qinmathcal CcalC projective and Z an E-object of
mathcal C,calC,
[
(Hom_{mathcal C}calC(Q,-)boxtimesId_mathcal W)calW)
(Θ_{mathcal C}calC(Z))
cong Θ(Hom_{mathcal C}calC(Q,Z)).
]
Proof.
In A-mod, Q is a direct summand of a free module, so
Hom_A(Q,-) is an additive idempotent of the identity functor on
A-mod. An additive functor preserves split idempotents and
direct sums; applying Θ therefore commutes with
Hom_A(Q,-).
## The tensor-power functor Phi
Give X^{ ⊗ q} the E-action coming from regular permutation
of tensor positions. Define
Phi(X) = Θ_{mathcal C}calC(X^{ ⊗ q})
in mathcal CcalCboxtimes mathcal W.calW.
[monoidal structure and semilinearity]
Phi is additive, strong symmetric monoidal, strong k-linear
after twisting scalars by Frobenius amapsto a^q, and satisfies
Phi(af) = a^qPhi(f) for scalars ain k.
Proof.
The monoidal constraint is the E-equivariant reordering
[
(X ⊗ Y)^{ ⊗ q}cong X^{ ⊗ q} ⊗ Y^{ ⊗ 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 ⊕ Y)^{ ⊗ 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
Θ (using cref{lem:theta-detection}); only the two constant
colourings survive. Semilinearity is immediate from the same
calculation on scalars.
Combining cref{eq:phi} and cref{lem:projective-evaluation}:
(Hom_{mathcal C}calC(Q,-)boxtimesId_mathcal W)(calW)(Phi(X))
cong Θ(Hom_{mathcal C}calC(Q,X^{ ⊗ q}))
for every projective Q.
## The nonprojective image criterion
[Phi exact iff image nonzero]
If there are projective objects P,Qinmathcal CcalC with
Θ(Hom_{mathcal C}calC(Q,P^{ ⊗ q}))ne0, then Phi is
exact.
Proof.
Cref{eq:phi-evaluation} gives Phi(P)ne0. For a short exact
sequence in mathcal C,calC, 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.
% =================================================================
# Step 3: Choosing an exact embedding
% =================================================================
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}calC(P,P^{ ⊗ 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}) ⊗ M(r) stays nonprojective.
## Norms and p-group projectivity
For a finite group H acting on a finite-dimensional vector space,
nu_H:X_Hto X^H is the norm map: it factors
sum_{hin H} h:Xto X.
[p-group projectivity via norm]
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.
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_{hin 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/<
z>]); and a nondegenerate pairing abmapsto 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_Hmne0, the map kHto M, amapsto 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).
[independent actions multiply]
Let finite p-groups H_i act on objects X_iinmathcal CcalC
(i=1,...,s). If omega(X_i) is projective over kH_i for
exact faithful omega = Hom_{mathcal C}calC(P,-), then
omega(X_1 ⊗ ... ⊗ X_s) is projective over
k[H_1times...times H_s] (independent actions), and over every
subgroup thereof.
Proof.
Categorically, (X ⊗ Y)_{Htimes K}cong X_H ⊗ Y_K and
(X ⊗ Y)^{Htimes K}cong X^H ⊗ Y^K
(take kernels/cokernels one group at a time). So
nu_{Htimes K}(X ⊗ Y) = nu_H(X) ⊗ 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 Hle H_1times...times H_s: kH_i is free over
kH for any finite group, so projectivity is preserved.
## Existence of a minimal r
Let E_a=(mathbb Z/p)^abbZ/p)^a act on p^a positions by its regular
representation; this induces an E_a-action on P^{ ⊗ p^a}.
Set M(a) = omega(P^{ ⊗ p^a}), an E_a-module.
[first nonprojective power]
There is a least integer rge 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.
Proof.
Suppose all M(a) projective. Fix d and an elementary abelian
p-subgroup Hlemathfrak S_d.frakS_d. Every H-orbit in
{1,...,d} is regular because H is abelian; orbitwise,
P^{ ⊗ d} is a product of P^{ ⊗ p^{a_i}}, each
projective over k E_{a_i} by assumption. By
cref{lem:product-actions}, omega(P^{ ⊗ 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 kmathfrak S_d.frakS_d.
Now apply Ptwoheadrightarrowmathbf 1bf1 (mathfrak S_d-equivariant,frakS_d-equivariant,
trivial action on the target) and exactness of omega followed by
coinvariants:
[
omega(P^{ ⊗ d})_{mathfrak S_d}frakS_d twoheadrightarrow
omega(mathbf 1)bf1)ne0.
]
By cref{lem:norm-projectivity}, the total symmetrizer
b_d = sum_{sigmainmathfrak S_d}frakS_dsigma acts nontrivially on
P^{ ⊗ d}.
On the other hand the action map
kmathfrak S_dfrakS_dtoEnd_{mathcal C}calC(P^{ ⊗ d}) cannot be
injective: length grows exponentially in d (bounded by
ell(P)b^{d-1} with b=max_iell(S_i ⊗ 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 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.
## Rank varieties and support
Let kE_r-modules be graded by the augmentation ideal
mathfrak mfrakm = (g_i-1)_i. The projectivised rank variety of
a finite-dimensional module D is
[
supp(D) =
bigl{[alpha]inmathbb P^bbP^{r-1}(k) :
D not free over k[t]/(t^p),
tmapstotextstylesum_ialpha_i(g_i-1)
bigr}.
]
Two standard facts (Benson 2017; Coulembier--Flake 2025):
supp(D) = ∅ iff D projective,
supp(D ⊗ D') = supp(D)capsupp(D').
[support point independence]
If M is a nonprojective kE_r-module with all restrictions to
proper subgroups projective, then every [alpha]insupp(M) has
mathbb F_p-linearlybbF_p-linearly independent coordinates.
Proof.
For cin(mathbb F_p^r)^bbF_p^r)^ast, let H=ker c and
N = Ind_H^{E_r}k, the regular representation of E_r/Hcong
mathbb Z/pbbZ/p (identified with k[z]/(z^p), generator acts by
1+z). The element sum_ialpha_i(g_i-1) acts on N by
h_alpha(z) = sum_ialpha_i((1+z)^{c_i}-1), whose linear
coefficient is sum_i c_ialpha_i.
If sum_i c_ialpha_ine0, then tmapsto h_alpha(z) is an iso
of truncated polynomial algebras, so N is free as k[t]/(t^p);
if sum_i c_ialpha_i=0, then h_alphain(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_ialpha_i=0}. The tensor
identity gives M ⊗ Ind_H^{E_r}kcongInd_H^{E_r}(M|_H), which
is projective, so supp(M)capsupp(N)=∅. Hence
sum_i c_ialpha_ine0 for every nonzero c. In particular the
coordinates of every [alpha]insupp(M) are mathbb F_p-linearlybbF_p-linearly
independent.
[Steinberg support in coordinates]
Let lambda_1,...,lambda_rin k be mathbb F_p-linearlybbF_p-linearly
independent and define iota_lambda : E_rto SL_2(k),
g_imapsto u(lambda_i). Write R_lambda for restriction. Then
supp(R_lambda(St_{r-1})) =
{[alpha] : sum_ialpha_ilambda_i^{p^j}=0
(0le jle r-2)}.
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 ptimes p
Jordan block. Then sum_ialpha_i(g_i-1) acts by a polynomial in
N whose linear coefficient is sum_ialpha_ilambda_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⊗_{j=0}^{r-2}St_1^{(j)}, the tensor
product formula (eq:support-properties) gives an intersection
of hyperplanes
M_j : sum_ialpha_ilambda_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),bbP^0(k), consistent.
## Moore's lemma: choosing lambda
[lambda solving prescribed equations]
Let alpha_1,...,alpha_rin k be mathbb F_p-linearlybbF_p-linearly
independent. Then there exist mathbb F_p-linearlybbF_p-linearly independent
lambda_1,...,lambda_rin k with
sum_ialpha_ilambda_i^{p^j} = 0,
0le jle r-2.
For r=1 the system is empty.
Proof.
r=1: take lambda_1ne0. For rge2: 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 fle p^{r-2}, so
f=0.
Row space of B contains no nonzero mathbb F_p^r-vector:bbF_p^r-vector: if
c_i = f(beta_i)inmathbb F_p,bbF_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 lambdainker B nonzero. Kernel is one-dimensional. Any
cinmathbb F_p^rbbF_p^r with sum_i c_ilambda_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-linearlybbF_p-linearly independent.
Reading rows of B in reverse order, (alpha_i^{p^{-j}})_i, and
raising the equations sum_ilambda_i^{p^s}alpha_i^{p^{-(r-2)}p^s}
=0 to the p^{r-2}-th power gives (eq:moore).
[embedding kills no support]
Let M be a nonprojective kE_r-module with projective restriction
to every proper subgroup. Then there is an embedding
iota_lambda:E_r↪ U(k) with
R_lambda(St_{r-1}) ⊗ M nonprojective.
Proof.
Pick [alpha]insupp(M); coordinates are mathbb F_p-linearlybbF_p-linearly
independent by cref{lem:support-independent}. Choose lambda by
cref{lem:moore-selection}; cref{lem:steinberg-support} then gives
[alpha]insupp(R_lambda(St_{r-1})). So the tensor product
support (eq:support-properties) is nonempty. Hence
R_lambda(St_{r-1}) ⊗ M is nonprojective.
## Exactness
[exact higher Frobenius functor]
Let mathcal CcalC be a finite symmetric tensor category over k of
characteristic p>0. Then there exist rge1 and an embedding
E_r↪ U(k) such that the construction of
cref{sec:higher} gives an exact additive strong symmetric monoidal
functor
[
Phi:mathcal CcalCtomathcal CcalCboxtimesVer_{p^r}(k)
]
with Phi(X)=Θ_{mathcal C}calC(X^{ ⊗ p^r}) and
Phi(af)=a^{p^r}Phi(f). Moreover, for every projective Q the
evaluation (eq:phi-evaluation) holds.
Proof.
Let P be the projective generator and r the minimal integer of
cref{prop:first-nonprojective}. Then
M = Hom_{mathcal C}calC(P,P^{ ⊗ 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}) ⊗ M is nonprojective.
By cref{thm:restricted-tilting},
R_lambda(St_{r-1}) ⊗ Mcong R_lambda(T) for a tilting
module T. By cref{lem:theta-detection},
[
0 ne Θ(R_lambda(T))
cong Θ(R_lambda(St_{r-1})) ⊗ Θ(M).
]
So Θ(M)ne0. Cref{prop:phi-exactness} with P=Q shows
Phi is exact.
% =================================================================
# Step 4: Dimension descent
% =================================================================
We now have an exact Phi:mathcal CcalCtomathcal CcalCboxtimesmathcal
W,calW, q=p^r, mathcal W=calW=Ver_{p^r}(k), semilinear.
Its image closes up on the first factor to the Serre closure of a
Frobenius-exact subcategory.
## Ordinary Frobenius and a mod-p identity
For Yinmathcal C,calC, let c be the cyclic permutation of the p
factors of Y^{ ⊗ p} and D = 1-c (so D^p=0). Define the
Etingof--Ostrik components:
[
F_i(Y) = frac{ker DcapimD^{i-1}}
{ker DcapimD^i},
1le i<p.
]
If ell_iinVer_p(k) is the simple object represented by a
Jordan block of length i, then
F(Y)=⊕_{i=1}^{p-1}F_i(Y)boxtimesell_i. In the
semilinear convention, the F_i(Y) are objects of mathcal CcalC
itself. F is not assumed exact.
[mod-p identity in K_0]
In K_0(mathcal C)/pK_0(calC)/pK_0(mathcal C),calC),
[
[Y]^p = sum_{i=1}^{p-1}i[F_i(Y)].
]
Consequently, for every rge 1,
[Y]^{p^r} = sum_{i_1,...,i_r=1}^{p-1}(i_1... i_r)
[F_{i_r}... F_{i_1}(Y)] pmod p.
Proof.
Put Z=Y^{ ⊗ p} and K_j = ker DcapimD^j
for 0le jle p. The short exact sequences
0to K_jtoimD^jtoimD^{j+1}to0
give [Z]=sum_{j=0}^{p-1}[K_j]. Summing the successive quotients
in the filtration K_0⊇...⊇ 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.
## 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↪bf1↪ X ⊗ X^*), so
composing with FP dimension gives a positive normalised character on
K_0(mathcal C);calC); the FP characterization forces it to be the FP
dimension of the source. In particular
[
d(Phi(X)) = d(X).
]
[first-component inequality]
Let Sinmathcal CcalC be simple and Aboxtimes B a composition
factor of Phi(S). Then d(A)le d(S). If equality holds then
Phi(S)simeq Aboxtimes B, B is invertible, and
Sinmathcal C_calC_{ex} (the subcategory on which the
ordinary Frobenius functor is exact).
Proof.
Inequality. d(S)=d(Phi(S))=sum d(A_k)d(B_k)ge d(A_1)
for any factor, using dge 1 for nonzero objects.
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 Aboxtimes B.
Sinmathcal C_calC_{ex.} Let Q_A be the
projective cover of A. Over k algebraically closed,
dim_kHom(Q_A,X) = [X:A]. Applying (eq:phi-evaluation) to
S and X=S^{ ⊗ q}:
[
Θ(Hom_{mathcal C}calC(Q_A,S^{ ⊗ q}))
cong B.
]
Θ 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^{ ⊗ q}:A]notequiv 0pmod p.
By (eq:iterated-ord) there are indices
i_1,...,i_rin{1,...,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 Sinmathcal C_calC_{ex}.
## The finite graph
Let N be the number of simple isomorphism classes in
mathcal C.calC. Form a directed graph on those classes by arrow
Sto A whenever Aboxtimes 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_calC_{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_calC_{ex}.
A schematic path is drawn in cref{fig:dimension-descent}.
[!htbp]
centering
[
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→ S' implies d(S')leq d(S)};
node[align=center] at (1.83,-2.10)
{A,B,Cinmathcal C_calC_{ex}[2pt]
unique outgoing neighbor at each cycle vertex};
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_calC_{ex}, and a unique outgoing
neighbor at every vertex, so the path cannot leave the cycle.}
## Iteration and Serre closure
Let sigma(a)=a^q. For a k-linear category mathcal A,calA, let
mathcal A^calA^{[sigma]} have the same objects and additive morphism
groups but scalar action twisted by sigma^{-1}. Then
sigma-semilinear functors out of mathcal AcalA = k-linear
functors out of mathcal A^calA^{[sigma]}, and
(mathcal AcalAboxtimesmathcal B)^calB)^{[sigma]}
= mathcal A^calA^{[sigma]}boxtimesmathcal
B^calB^{[sigma]}, so products of sigma-semilinear functors are
sigma-semilinear.
Each Ver_{p^s}(k) is defined over mathbb F_p;bbF_p; scalar transport
along bmapsto b^{p^a} is an exact strong symmetric sigma^a-
semilinear autoequivalence J_{sigma^a} of every
Ver_{p^s}(k).
Define Psi_0=Id, Psi_1=Phi, and
Psi_{j+1} = (Phiboxtimes J_sigma^{boxtimes j})circPsi_j
:mathcal CcalCtomathcal CcalCboxtimesmathcal W^calW^{boxtimes(j+1)}.
By induction each Psi_j is exact, strong symmetric monoidal,
and sigma^j-semilinear.
[first components after N iterations]
Let Psi_N :mathcal CcalCtomathcal CcalCboxtimes
mathcal W^calW^{boxtimes N} be as above. Then every composition
factor Aboxtimes B_1boxtimes...boxtimes B_N of
Psi_N(X) has Ainmathcal C_calC_{ex}.
Proof.
Exactness of Phi plus the path bound on mathcal CcalC (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.
Let mathcal D⊆calD⊆mathcal CcalC be the full subcategory whose
simple composition factors lie in mathcal C_calC_{ex}. This
is the Serre closure of mathcal C_calC_{ex}.
[Serre closure inherits tensor structure]
mathcal DcalD 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.calC. Moreover the functor Psi_N factors through
[
Psi_N :mathcal CcalCto mathcal DcalDboxtimes
mathcal W^calW^{boxtimes N}.
]
Proof.
Closed under tensor products: biexactness gives tensor-product
filtrations, and mathcal C_calC_{ex} is a tensor
subcategory closed under subquotients (Etingof--Ostrik 2021,
Prop 7.4). Duality preserves mathcal DcalD by exactness.
Finiteness, projectives: concretely, write
mathcal CcalCsimeq H-mod with H basic finite-dimensional;
let e be the idempotent summing primitive idempotents of the
non-mathcal C_calC_{ex} simples; then
mathcal DcalDsimeq (H/HeH)-mod by removing the
e-isotypic parts, and H/HeH is finite-dimensional and finite
projective dimension, so mathcal DcalD is finite with enough
projectives.
Factoring through mathcal D:calD: cref{lem:first-components}.
% =================================================================
# Finishing the proof: a fiber functor on mathcal DcalD
% =================================================================
## The fiber functor on mathcal DcalD
We need an exact strong symmetric monoidal E:mathcal
DcalDtoVer_{p^n}(k) for some fixed n.
Case p>2.:
mathcal C_calC_{ex} is Frobenius-exact, so by
Etingof--Ostrik (2021, Theorem 8.1) it has a fiber functor
mathcal C_calC_{ex}toVer_p(k). If p>2, the Serre closure
of mathcal C_calC_{ex} equals mathcal C_calC_{ex}
(Etingof--Ostrik 2021, Corollary 9.5). So mathcal D=calD=mathcal
C_calC_{ex} and we already have E:mathcal DcalDtoVer_p(k).
Case p=2.:
The fiber functor mathcal C_calC_{ex}toVer_2(k)=Vec_k is a
k-linear fiber functor. But mathcal C_calC_{ex}⊆
mathcal DcalD contains every simple of mathcal DcalD by definition,
and mathcal C_calC_{ex} is Frobenius-exact. Etingof--Gelaki
(2021, Theorem 2.21(2)) then extends it to
E:mathcal DcalDtoVer_4^+(k)⊆Ver_4(k).
(This is also Etingof--Ostrik 2021, Corollary 9.8.)
So in all cases:
[
E:mathcal DcalD → Ver_{p^n}(k),
n = 1 & p>2, 2 & p=2.
]
## Combining factors
Set n=max(2,r) and mathcal ZcalZ = Ver_{p^n}(k). The tower
inclusions Ver_{p^r}(k)⊆Ver_{p^n}(k)
(Benson--Etingof--Ostrik 2023, Theorem 4.34) give a k-linear exact
strong symmetric monoidal I:mathcal WcalWtomathcal Z.calZ.
The functors E:mathcal DcalDtomathcal ZcalZ and I:mathcal WcalWto
mathcal ZcalZ combine, by cref{lem:combine-functors}, into
[
H:mathcal DcalDboxtimesmathcal W^calW^{boxtimes N}tomathcal Z,calZ,
H(Dboxtimes W_1boxtimes...boxtimes W_N)
= E(D) ⊗ I(W_1) ⊗ ... ⊗ I(W_N).
]
H is exact and strong symmetric monoidal.
## Killing the scalar twist
The composite HPsi_N :mathcal CcalCtomathcal ZcalZ is
sigma^N-semilinear (each iteration multiplies semilinearity by
sigma). Let J_{sigma^{-N}} be scalar transport on mathcal ZcalZ
along bmapsto b^{p^{-N}}. Then
[
mathcal FcalF := J_{sigma^{-N}}circ HcircPsi_N :
mathcal C→calC→mathcal ZcalZ
]
is k-linear, exact, strong symmetric monoidal.
## Faithfulness
[exact strong monoidal with simple unit is faithful]
Let F:mathcal CcalCtomathcal DcalD be an exact strong symmetric
monoidal functor between finite symmetric tensor categories. Then
F is faithful.
Proof.
Suppose Xne0. The coevaluation
mathbf 1bf1to X ⊗ X^* is nonzero (triangle identity) and by
simplicity of mathbf 1bf1 it is injective; applying exact F shows
F(X)ne0. For a nonzero f:Xto Y: exactness gives
imF(f)cong F(imf), and
imfne0, so F(f)ne0.
Applying cref{lem:faithful} to mathcal FcalF shows mathcal FcalF is
faithful. Hence mathcal F:calF:mathcal CcalCtoVer_{p^n}(k) is the
required fiber functor, and cref{thm:main} follows.
% =================================================================
# Consequences
% =================================================================
We now record the three structural corollaries that were the original
motivation for proving cref{thm:main}.
## Incompressible categories
Recall that mathcal CcalC is incompressible if every
k-linear exact strong symmetric monoidal functor from mathcal CcalC
to a symmetric tensor category over k is fully faithful.
[classification of finite incompressible]
Let k be algebraically closed of characteristic p>0. A finite
symmetric tensor category mathcal CcalC 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:
[
{c|l}
p=2 & Ver_{2^m}(k) (mge1),
Ver_{2^m}^+(k) (mge2),
p=3 & Ver_{3^m}(k), Ver_{3^m}^+(k)(mge1),
pge 5 & Vec_k, sVec_k,
Ver_{p^m}(k), Ver_{p^m}^+(k)(mge1).
]
Proof.
Direction ⇐: Vec_k has trivial
End(mathbf 1)=k,bf1)=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 ⇒: take a fiber functor
F:mathcal CcalCtoVer_{p^n}(k) from cref{thm:main}, let
mathcal IcalI be the image's full subcategory of subquotients.
Then F corestricts to a surjective-in-the-tensor-category-sense
functor onto mathcal IcalI (Etingof--Gelaki--Nikshych--Ostrik 2015,
Definition 1.8.18, Prop 1.8.19, Section 6.3). Incompressibility
of mathcal CcalC 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 IcalI to a full or plus sub-Verlinde at height le n.
Collecting cases gives the stated list.
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].
## Uniqueness in characteristic 2
[uniqueness in characteristic two]
Let k be algebraically closed of characteristic two and
mathcal CcalC a finite symmetric tensor category. Then
mathcal CcalCtoVer_{2^∞}(k) :=∪_nVer_{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 CcalCtoVer_{2^n}(k) up to monoidal natural
isomorphism.
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^∞}(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.
## Tannaka reconstruction over a Verlinde level
Let mathcal VcalV be a finite symmetric tensor category. The
fundamental group pi(mathcal V)calV) :=
Aut^{ ⊗ }(Id_mathcal V)calV) acts
canonically on Ind(mathcal V).calV). A
mathcal V-groupcalV-group is an internal affine group scheme G with
a homomorphism phi:pi(mathcal V)calV)to G whose induced coaction
matches the canonical one.
[reconstruction over a Verlinde level]
Let k be algebraically closed of characteristic p>0, mathcal
CcalC finite symmetric tensor, and F:mathcal CcalCto
mathcal V=calV=Ver_{p^n}(k) a fiber functor (from cref{thm:main}).
Then there is a mathcal V-groupcalV-group (G,phi) with
mathcal CcalCsimeqRep(G,phi) over mathcal V,calV, equivalent to
F under the forgetful functor. One may take
G=Aut^{ ⊗ }(F) and
phi:pi(mathcal V)calV)toAut^{ ⊗ }(F) the
canonical homomorphism.
Proof.
Cref{lem:modules-finite-alg} makes mathcal C,calC, together with
mathcal V=calV=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;calV-group; conversely, forgetting G-structure gives the
inverse functor. Composition is the equivalence.
## 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;calC; it makes 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.
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