diff b34550f..b1f9bc4 symmetric_tensor_categories_fiber_v2.txt
diff --git a/symmetric_tensor_categories_fiber_v2.txt b/symmetric_tensor_categories_fiber_v2.txt
index 045ba64..63bab3b 100644
--- a/symmetric_tensor_categories_fiber_v2.txt
+++ b/symmetric_tensor_categories_fiber_v2.txt
@@ -16,7 +16,7 @@
-\usetikzlibrary{arrows.meta,positioning,calc,fit}
+usetikzlibrary{arrows.meta,positioning,calc,fit}
@@ -47,46 +47,29 @@
-# Fiber functors for finite symmetric tensor categories\\
+# Fiber functors for finite symmetric tensor categories
in positive characteristic
-\maketitle
+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 $\mathcal C$.
+category $Ver_{p^n}(k)$. The result covers every prime, including
+$p=2$, and the height $n$ may depend on $mathcal C$.calC$.
-\tableofcontents
+tableofcontents
% =================================================================
# Introduction and main result
% =================================================================
-## What this paper is about, in one paragraph
-
-In characteristic zero, Deligne's theorem says that any
-``sufficiently nice'' symmetric tensor category embeds faithfully
-into supervector spaces. In positive characteristic there is no
-single universal target: to catch every finite symmetric tensor
-category one needs a whole tower of finite abelian categories
-\[
- \Ver_p(k) \subset \Ver_{p^2}(k)\subset
- \Ver_{p^3}(k)\subset...,
-\]
-called the higher Verlinde categories of Benson--Etingof--Ostrik.
-The claim of Benson--Etingof--Ostrik is that this tower does
-work: every finite symmetric tensor category is a category of
-representations internal to some finite level of the tower. The
-present paper proves that claim.
-
## The basic objects
Throughout, $k$ is an algebraically closed field of characteristic
@@ -95,67 +78,67 @@ $p>0$.
[finite symmetric tensor category]
A finite symmetric tensor category over $k$ is a
-$k$-linear abelian rigid symmetric monoidal category $\mathcal C$
+$k$-linear abelian rigid symmetric monoidal category $mathcal C$calC$
such that:
-[label=(\arabic*)]
- \item the tensor product is biexact;
- \item every $\Hom(X,Y)$ is finite dimensional over $k$;
- \item every object has finite length;
- \item there are finitely many isomorphism classes of simple
+[label=(arabic*)]
+ item the tensor product is biexact;
+ item every $Hom(X,Y)$ is finite dimensional over $k$;
+ item every object has finite length;
+ item there are finitely many isomorphism classes of simple
objects;
- \item the unit object $\mathbf 1$ has $\End(\mathbf 1) = k$;
- \item there are enough projective objects.
+ item the unit object $mathbf 1$bf1$ has $End(mathbf 1)bf1) = k$;
+ item 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 $\mathbb Z/2$-graded
+$Vec_k$, 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 $\mathcal C\to \mathcal D$ between finite
+A fiber functor $mathcal CcalCto mathcal D$calD$ between finite
symmetric tensor categories is a $k$-linear exact faithful strong
symmetric monoidal functor. ``Fiber functor'' in the classical
-sense is the special case $\mathcal D = \Vec_k$; the target being a
+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
+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
+The categories $Ver_{p^n}(k)$ are finite symmetric tensor
categories over $k$ and they form a tower
-\[
- \Ver_p(k)\subset \Ver_{p^2}(k)\subset\Ver_{p^3}(k)\subset...,
-\]
+[
+ Ver_p(k)subset Ver_{p^2}(k)subsetVer_{p^3}(k)subset...,
+]
with each inclusion exact and symmetric monoidal. The bottom cases
are
-\[
- \Ver_2(k) = \Vec_k,\qquad
- \Ver_3(k) = \sVec_k,\qquad
- \Ver_3^+(k) = \Vec_k.
-\]
+[
+ Ver_2(k) = Vec_k,qquad
+ Ver_3(k) = sVec_k,qquad
+ Ver_3^+(k) = Vec_k.
+]
For every $p^n > 2$ one has a symmetric tensor subcategory
-$\Ver_{p^n}^+(k)⊆\Ver_{p^n}(k)$; concretely
-$\Ver_4^+(k)$ is the category of representations of the connected
-group scheme $\alpha_2(k)$ with its unique symmetric structure, as
+$Ver_{p^n}^+(k)⊆Ver_{p^n}(k)$; 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
-$\Rep_k(\mathbb Z/p)$: one kills every morphism $f$ with $\tr(gf)=0$
+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
+$Vec_k$; 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
@@ -166,11 +149,11 @@ Here is the result we prove.
For every prime $p$, every algebraically closed field $k$ of
characteristic $p$, and every finite symmetric tensor category
-$\mathcal C$ over $k$, there exist an integer $n\ge 1$ and a fiber
+$mathcal C$calC$ over $k$, there exist an integer $nge 1$ and a fiber
functor
-\[
- \mathcal C → \Ver_{p^n}(k).
-\]
+[
+ mathcal CcalC → Ver_{p^n}(k).
+]
This settles the finite case of Conjecture 1.4 of
@@ -181,41 +164,41 @@ implies the finite part of all the consequences we discuss.
## Proof outline
-We iteratively apply a semilinear higher Frobenius functor $\Phi$,
+We iteratively apply a semilinear higher Frobenius functor $Phi$,
each application adding one fresh Verlinde factor while forcing
-the original $\mathcal C$ factor to shrink into its Frobenius-exact
+the original $mathcal C$calC$ factor to shrink into its Frobenius-exact
subcategory. After at most $N$ iterations, where $N$ is the number
-of simple objects of $\mathcal C$, all first components land in the
+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*]
- \item Restricted tilting modulo a Steinberg module.
+[label=Step arabic*.,ref=step:arabic*]
+ item Restricted tilting modulo a Steinberg module.
For every finite-dimensional module $D$ over the
- elementary abelian group $E_r = (\mathbb Z/p)^r$, the
- tensor product $\St_{r-1} ⊗ D$ of a fixed Steinberg
+ 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.
- \item Higher Frobenius functor.
+ item 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
- $\Phi:\mathcal C\to \mathcal C\boxtimes \Ver_{p^r}(k)$
- that is only semilinear ($\Phi(af) = a^{p^r}\Phi(f)$).
+ $Phi:mathcal CcalCto mathcal CcalCboxtimes Ver_{p^r}(k)$
+ that is only semilinear ($Phi(af) = a^{p^r}Phi(f)$).
- \item Exactness by support. For a well-chosen
- embedding of $E_r$ (with $r$ minimal), $\Phi$ is actually
+ item 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
+ module $M = Hom(P,P^{ ⊗ p^r})$, and the theory of
the projectivised rank variety.
- \item Dimension descent. Even when $\Phi$ is only
+ item 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
+ 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 $\mathcal C_{ex}$.
+ Frobenius-exact tensor subcategory $mathcal C_calC_{ex}$.
After these four steps, a few more tasks finish the argument:
@@ -238,64 +221,64 @@ corollaries.
The definition of finite symmetric tensor category was chosen so
that the usual Tannakian toolkit applies. We make this explicit.
-[$\mathcal C$ is modules over a finite algebra]
+[$mathcal C$calC$ is modules over a finite algebra]
-Let $\mathcal C$ be a finite symmetric tensor category. Then the
+Let $mathcal C$calC$ be a finite symmetric tensor category. Then the
unit object is simple, and there is a projective generator
-$P\in\mathcal C$. The functor $\omega = \Hom_{\mathcal C}(P,-)$ is
-exact and faithful, and $\mathcal C$ is equivalent, as a $k$-linear
+$Pinmathcal C$.calC$. The functor $omega = Hom_{mathcal C}calC(P,-)$ is
+exact and faithful, and $mathcal C$calC$ is equivalent, as a $k$-linear
abelian category, to the category of finite-dimensional modules
over the finite-dimensional $k$-algebra
-$A = \End_{\mathcal C}(P)^{op}$.
+$A = End_{mathcal C}calC(P)^{op}$.
Proof.
-Simple unit is the Deligne--Milne argument. If $U\subset\mathbf 1$
-is a proper subobject with cokernel $V\ne0$, the epimorphism
-$U\to V ⊗ U$ coming from $\mathbf 1\twoheadrightarrow V$ has
-zero composite with the subobject $V ⊗ U\twoheadrightarrow V$.
-So $V ⊗ U=0$. Put $W=\ker(\mathbf 1\to U^*)$. Then $W$
+Simple unit is the Deligne--Milne argument. If $Usubsetmathbf 1$bf1$
+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 $W\simeq V$. The endomorphism ring of the unit has no
+$W$ gives $Wsimeq V$. The endomorphism ring of the unit has no
idempotents, so $V=0$.
-For the projective generator: pick surjections $Q_i\to S_i$ from
+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 $P\to$ image. To identify
-$\mathcal C\simeq A-mod$, note that presenting objects by
+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,
-$P\twoheadrightarrow\mathbf 1$, since the unit is one of the
+$Ptwoheadrightarrowmathbf 1$,bf1$, since the unit is one of the
simples.
## Deligne products
-For finite $k$-linear abelian categories $\mathcal A,\mathcal B$, the
-Deligne product $\mathcal A\boxtimes\mathcal B$ is the
+For finite $k$-linear abelian categories $mathcal A,calA,mathcal B$,calB$, the
+Deligne product $mathcal AcalAboxtimesmathcal B$calB$ is the
universal recipient of functors that are $k$-bilinear and right
-exact in each variable. When $\mathcal A\simeq A-mod$ and
-$\mathcal B\simeq B-mod$, we have
-$\mathcal A\boxtimes\mathcal B\simeq (A ⊗ _k B)-mod$,
+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 $\mathcal A_1,...,\mathcal A_s,\mathcal Z$ be finite
-symmetric tensor categories and let $F_i:\mathcal A_i\to\mathcal Z$
+Let $mathcal A_1,...,calA_1,...,mathcal A_s,calA_s,mathcal Z$calZ$ be finite
+symmetric tensor categories and let $F_i:mathcal A_icalA_itomathcal Z$calZ$
be exact $k$-linear strong symmetric monoidal functors. Then
-\[
- H:\mathcal A_1\boxtimes...\boxtimes\mathcal A_s\to\mathcal Z,
- \qquad
- H(X_1\boxtimes...\boxtimes X_s)
+[
+ H:mathcal A_1calA_1boxtimes...boxtimesmathcal A_scalA_stomathcal Z,calZ,
+ qquad
+ H(X_1boxtimes...boxtimes X_s)
= F_1(X_1) ⊗ ... ⊗ F_s(X_s),
-\]
+]
extends uniquely to an exact $k$-linear strong symmetric monoidal
functor.
@@ -303,7 +286,7 @@ functor.
Proof.
The displayed multilinear functor has a right-exact extension by
the universal property of the Deligne product. Symmetry in
-$\mathcal Z$ gives the tensor constraint by rearranging factors, and
+$mathcal Z$calZ$ 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
@@ -312,31 +295,31 @@ also left exact.
## Frobenius--Perron dimension
-For finite tensor categories the Grothendieck ring $K_0(\mathcal C)$
+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
-$d = \FPdim : K_0(\mathcal C)\to\mathbb R$ with $d(\mathbf 1)=1$
+$d = FPdim : K_0(mathcal C)calC)tomathbb R$bbR$ with $d(mathbf 1)=1$bf1)=1$
and $d$ multiplicative on classes. Key facts, all in Etingof--
Gelaki--Nikshych--Ostrik (2015):
[FP dimension toolkit]
-Let $\mathcal C$ be a finite tensor category.
-[label=(\roman*)]
- \item $d(X)\ge 1$ for every nonzero $X$; equality if and only if
+Let $mathcal C$calC$ be a finite tensor category.
+[label=(roman*)]
+ item $d(X)ge 1$ for every nonzero $X$; equality if and only if
$X$ is invertible.
- \item If $0\to X\to Y\to Z\to 0$ is exact, then
+ item If $0to Xto Yto Zto 0$ is exact, then
$d(Y)=d(X)+d(Z)$.
- \item On a Deligne product, $d(A\boxtimes B)=d(A)\,d(B)$.
- \item If $F:\mathcal C\to\mathcal D$ is an exact strong monoidal
+ item On a Deligne product, $d(Aboxtimes B)=d(A),d(B)$.
+ item If $F:mathcal CcalCtomathcal D$calD$ 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}
+$dim_{cat}(B)in k$ 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.
@@ -348,30 +331,30 @@ mod-$p$ congruence.
## The statement
-Let $G = SL_2(k)$, and let $U\simeq\mathbb G_a$ be its upper
+Let $G = SL_2(k)$, and let $Usimeqmathbb G_a$bbG_a$ be its upper
unipotent subgroup, identified with $k$ via
-$u(t) = (1&t\\0&1)$.
-For $r\ge 1$ put $E_r = (\mathbb Z/p)^r$ and let
-$\iota:E_r↪ U(k)$ be any injective homomorphism.
-Write $R = R_\iota$ for the restriction of $G$-modules along $\iota$.
+$u(t) = (1&t0&1)$.
+For $rge 1$ put $E_r = (mathbb Z/p)^r$bbZ/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
-$h\ge0$; 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
+$hge0$; 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
+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 $r\ge 1$, $E_r = (\mathbb Z/p)^r$, and $\iota:E_r\to U(k)$ any
+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
-\[
- R_\iota(\St_{r-1}) ⊗ D \cong R_\iota(T)
-\]
+[
+ R_iota(St_{r-1}) ⊗ D cong R_iota(T)
+]
as $kE_r$-modules.
@@ -381,15 +364,15 @@ Conjecture 1 of Coulembier--Flake (2025), for every prime.
## The plan of the proof
We do three things:
-[label=(\alph*)]
- \item lift the arbitrary $kE_r$-module $D$ to a coherent rational
- $U$-module $\widehat D$ of bounded ``degree'' (a bound on
+[label=(alph*)]
+ item lift the arbitrary $kE_r$-module $D$ to a coherent rational
+ $U$-module $widehatwidehat D$D$ of bounded ``degree'' (a bound on
polynomial degree of the unipotent action matrix);
- \item use $\widehat D$ to construct a $G$-equivariant vector
- bundle $\mathcal M$ on the plane $\mathbb A^2_k$ whose
+ item use $widehatwidehat D$D$ to construct a $G$-equivariant vector
+ bundle $mathcal M$calM$ on the plane $mathbb A^2_k$bbA^2_k$ whose
origin fibre has all simple composition factors below a
height bound;
- \item tensor by $\St_{r-1}$, observe the resulting bundle's
+ item tensor by $St_{r-1}$, observe the resulting bundle's
origin fibre is tilting, and compare origin fibre with
$e_1$-fibre via a formal construction on jets.
@@ -400,89 +383,89 @@ We then recover $D$ from the $e_1$-fibre.
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.
+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 $\widehat D$ of degree $< p^r$
-such that $\widehat D|_{E_r}\cong D$ along $\iota$.
+finite-dimensional rational $U$-module $widehatwidehat D$D$ 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
+Let $H⊆ k$ be the image of $E_r$ under $iota$. The
vector space
-$C_q = span_k\{1,t,...,t^{q-1}\}⊆ k[t]$
+$C_q = span_k{1,t,...,t^{q-1}}⊆ k[t]$
(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
-$ev_H : C_q \to k^H = \mathcal O(E_r)$, which is a
+$Delta f(t,s) = f(t+s)$. 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 $\mathcal O(E_r)$-coaction defining $D$ across
-$ev_H$, then including $C_q \subset k[t]$, gives a
+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 $\pi : G\to\mathbb A^2∖\{0\}$,
-$g\mapsto g e_1$, with $e_1=(1,0)$. This is a right $U$-torsor:
-the fibres are right $U$-cosets and on the standard opens $x\ne 0$,
-$y\ne 0$ it has obvious rational sections. Form the associated
+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
-\[
- \mathcal E = G\times^U \widehat D,
- \qquad (g,d)\sim(gu,u^{-1}d),
-\]
+[
+ mathcal EcalE = Gtimes^U widehatwidehat D,D,
+ qquad (g,d)sim(gu,u^{-1}d),
+]
on the punctured plane, with $G$ acting by left multiplication.
-The fibre at $e_1$ is $\widehat D$ itself.
+The fibre at $e_1$ is $widehatwidehat D$D$ itself.
[extension and weight bound]
-The bundle $\mathcal E$ extends to a $G$-equivariant vector bundle
-$\mathcal M$ on the whole affine plane, and every simple
+The bundle $mathcal E$calE$ extends to a $G$-equivariant vector bundle
+$mathcal M$calM$ on the whole affine plane, and every simple
composition factor of the origin fibre
-$M_0 = \mathcal M|_0$ has highest weight $< q$.
+$M_0 = mathcal M|_0$calM|_0$ has highest weight $< q$.
Proof.
-Extension. A coherent extension $\mathcal F$ exists on the
-noetherian affine plane. Its double dual $\mathcal M = \mathcal
-F^{**}$ is a coherent reflexive sheaf restricting to $\mathcal E$.
-On the regular surface $\mathbb A^2$ every reflexive coherent sheaf
+Extension. A coherent extension $mathcal F$calF$ 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 $mathbb A^2$bbA^2$ every reflexive coherent sheaf
is locally free: at a closed point the stalk has depth two, so the
Auslander--Buchsbaum formula gives projective dimension zero, and at
the generic point and at the generic point of a curve the stalks are
-trivially free. Hence $\mathcal M$ is a vector bundle.
+trivially free. Hence $mathcal M$calM$ is a vector bundle.
Equivariance. Over the punctured plane we have a
$G$-equivariant bundle isomorphism
-$\pi_2^*\mathcal M\cong a^*\mathcal M$ for the action
-$a:G\times\mathbb A^2\to\mathbb A^2$ and projection
-$\pi_2:G\times\mathbb A^2\to\mathbb A^2$. Both sides are locally
+$pi_2^*mathcal McalMcong a^*mathcal M$calM$ for the action
+$a:Gtimesmathbb A^2bbA^2tomathbb A^2$bbA^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 $G\times\{0\}$; the extensions of
+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 $a\in k^\times$ put
+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
-$A = \Gamma(\ell,\mathcal E nd(\mathcal M|_\ell))$ and write the
+$ell$ 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 $j\ge q$, $b_j$ vanishes on every nonzero
+$sum_j b_j t^j$. 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
-$h=\sum_i h_ip^i$, $0\le h_i<p$. Its $U$-action has degree exactly
+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
@@ -495,10 +478,10 @@ 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)\quad(0\le a<p),
- \qquad
- T(a) ⊗ T(b)^{(1)}\simeq T(a+pb)
- \quad(p-1\le a\le 2p-2,\ b\ge 0),
+ L(a) = T(a)quad(0le a<p),
+ qquad
+ T(a) ⊗ T(b)^{(1)}simeq T(a+pb)
+ quad(p-1le ale 2p-2, bge 0),
where $(1)$ means pullback along the Frobenius. See Benson--
Etingof--Ostrik (2023), Proposition 3.1.
@@ -506,111 +489,111 @@ 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 $0\le h<p^m$ (where
-$m\ge 1$). Then $\St_{m-1} ⊗ M$ is tilting.
+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
+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 $0\le d<p$ and $h'<p^{m-1}$.
+For $m>1$, write $h = d + p h'$ with $0le d<p$ and $h'<p^{m-1}$.
Steinberg's tensor product theorem gives
-\[
- \St_{m-1} ⊗ L(h) \cong
- L(d) ⊗ \St_1 ⊗
- (\St_{m-2} ⊗ L(h'))^{(1)}.
-\]
+[
+ 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
+$St_1$ (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 $M_0 = \mathcal
-M|_0$, we have:
+Applying cref{lem:steinberg-tilting} with $m=r$ to $M_0 = mathcal
+M|_0$,calM|_0$, we have:
- T := \St_{r-1} ⊗ M_0
+ T := St_{r-1} ⊗ M_0
is a tilting $G$-module. The remaining step is to compare $T$
(the origin fibre of
-$\mathcal N = \St_{r-1} ⊗ _k \mathcal M$) with the $e_1$-fibre
-of $\mathcal N$.
+$mathcal NcalN = St_{r-1} ⊗ _k mathcal M$)calM$) with the $e_1$-fibre
+of $mathcal N$.calN$.
## Formal comparison of fibres
[comparison with a tilting origin fibre]
-Let $\mathcal N$ be a $G$-equivariant vector bundle on $\mathbb A^2$
-whose origin fibre $T = \mathcal N|_0$ is tilting. Then
-$\mathcal N|_{e_1}\cong T|_U$ as rational $U$-modules.
+Let $mathcal N$calN$ be a $G$-equivariant vector bundle on $mathbb A^2$bbA^2$
+whose origin fibre $T = mathcal N|_0$calN|_0$ is tilting. Then
+$mathcal N|_calN|_{e_1}cong T|_U$ as rational $U$-modules.
Proof.
-Jet trivialisation. Let $A=k[x,y]$, $\mathfrak m=(x,y)$,
-and $N=\Gamma(\mathbb A^2,\mathcal N)$, a possibly
+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^j N/\mathfrak m^{j+1}N \cong
- \Sym^j(V^*) ⊗ _k T.
+ 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
-$0\to\Sym^{a-1}V^* ⊗ \Sym^{b-1}V^*\to
-\Sym^a V^* ⊗ \Sym^b V^*\to\Sym^{a+b}V^*\to 0$ is exact.
+$0toSym^{a-1}V^* ⊗ Sym^{b-1}V^*to
+Sym^a V^* ⊗ Sym^b V^*toSym^{a+b}V^*to 0$ is exact.
Now $T$ has both a Weyl and a good filtration, so for every $j$,
-\[
- Ext^1_G(T,\Sym^j(V^*) ⊗ T) = 0
-\]
+[
+ Ext^1_G(T,Sym^j(V^*) ⊗ T) = 0
+]
by Cline--Parshall--Scott--van der Kallen.
-Starting from the identity $s_1:T\to N/\mathfrak m N$, we lift
-successively to $G$-maps $s_j:T\to N/\mathfrak m^j N$ that reduce
-modulo $\mathfrak m^j$ to $s_{j-1}$; the obstruction is exactly the
+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 $mathfrak m^j$frakm^j$ to $s_{j-1}$; the obstruction is exactly the
Ext group just computed. Extending scalars, $s_j$ gives an
-$A/\mathfrak m^j$-linear map
-$(A/\mathfrak m^j) ⊗ _k T\to N/\mathfrak m^j N$, which is 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
-$k[[x,y]] ⊗ _k T\xrightarrow{\sim}\widehat N$.
+$k[[x,y]] ⊗ _k Txrightarrow{sim}widehatwidehat N$.N$.
Fibre comparison. Restrict to the fixed line $y=0$, pass
to $K = k((x))$: this gives a $U_K$-isomorphism
-$\varphi:T_K\cong \mathcal N_{xe_1}$. To cancel the parameter
+$varphi:T_Kcong mathcal N_calN_{xe_1}$. 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
-$f = b_h^{-1}\varphi\rho_T(h)$ intertwines the original
-$U$-actions (where $b_h : \mathcal N_{e_1}\to\mathcal N_{xe_1}$ is
+$f = b_h^{-1}varphirho_T(h)$ 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
-$L = \Hom_U(T,\mathcal N_{e_1})$ is cut out by finitely many
+$L = Hom_U(T,mathcal N_calN_{e_1})$ 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 $\mathcal M$ with origin
+Proof. [Proof of cref{thm:restricted-tilting}]
+Apply cref{prop:extension-bound} to get $mathcal M$calM$ 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
-$\mathcal N = \St_{r-1} ⊗ _k\mathcal M$: its origin fibre is
+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
-\[
- \mathcal N|_{e_1} \cong T|_U.
-\]
-But by construction $\mathcal N|_{e_1}$ restricts to
-$R(\St_{r-1}) ⊗ D$ along $E_r↪ U(k)$, hence 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.
@@ -621,35 +604,35 @@ does $T$. Set $T_{out} := T$; the theorem follows.
## The functor $Θ$ from group modules to Verlinde
-Fix $r\ge 1$, $q=p^r$, and an embedding
+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 $\mathcal W = \Ver_{p^r}(k)$.
-Let $\mathcal T⊆ \Rep_k(SL_2)$ be the category of tilting
+$SL_2$-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
-$\St_r = L(p^r-1)$.
+$St_r = L(p^r-1)$.
Concretely, $I_r$ is defined by demanding that $T(a)$ (the
indecomposable tilting module of highest weight $a$) vanish in
-$\mathcal T/I_r$ whenever $a\ge p^r - 1$; this is the standing
+$mathcal T/I_r$calT/I_r$ whenever $age p^r - 1$; this is the standing
convention in this circle of papers.
-The functor $\mathcal T/I_r\to\mathcal W$ is an equivalence onto a
+The functor $mathcal T/I_rcalT/I_rtomathcal W$calW$ is an equivalence onto a
full subcategory (Benson--Etingof--Ostrik (2023), Theorem 4.2(i)).
Stable restriction gives a faithful quotient functor
-$\mathcal T/I_r\toStab(E) :=
-\Rep_k(E)/(projectives)$.
+$mathcal T/I_rcalT/I_rtoStab(E) :=
+Rep_k(E)/(projectives)$.
Coulembier--Flake (2026, Theorem 5.4.3) state: if for every
-finite-dimensional $E$-module $D$ the tensor $R(\St_{r-1) ⊗ D$
+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 W
- \quadwith\quad
- Θ(R(T)) \simeq Σ_r(T\bmod I_r).
+ 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
+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.
@@ -657,117 +640,117 @@ $Θ$ 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,
+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 $T\bmod I_r=0$ then $R(T)$ is projective (stable restriction
-result). Otherwise $T\bmod I_r\ne0$, and $Σ_r$ being fully
-faithful implies $Θ(R(T))\ne0$.
+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
+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 $q$
+$St_r$). 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
+$H<E$ 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
+$S ⊗ Ind_H^E D cong Ind_H^E(Res_H^E S ⊗ D)$ therefore
gives a projective $E$-module, on which $Θ$ vanishes. Since
-$Θ(S)\ne0$ and tensoring by a nonzero object is faithful,
-$Θ(\Ind_H^E D)=0$.
+$Θ(S)ne0$ and tensoring by a nonzero object is faithful,
+$Θ(Ind_H^E D)=0$.
## Internal evaluation
-Let $\mathcal C$ be a finite symmetric tensor category with
-projective generator $P_0$. Write $A = \End_{\mathcal C}(P_0)^{op}$,
-so $\mathcal C\simeq A-mod$. An object of
-$\mathcal C\boxtimes\Rep_k(E)$ is an object of $\mathcal C$
+Let $mathcal C$calC$ 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 $mathcal C$calC$
equipped with a commuting $E$-action. Applying $Θ$ to the
underlying $E$-module and to the action maps produces an $A$-module
-object in $\mathcal W$, hence an object of
-$\mathcal C\boxtimes\mathcal W$. This is the internal
-extension $Θ_{\mathcal C}$ of $Θ$ (Coulembier--Flake
+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 $Q\in\mathcal C$ projective and $Z$ an $E$-object of
-$\mathcal C$,
-\[
- (\Hom_{\mathcal C}(Q,-)\boxtimesId_\mathcal W)
- (Θ_{\mathcal C}(Z))
- \cong Θ(\Hom_{\mathcal C}(Q,Z)).
-\]
+For $Qinmathcal C$calC$ 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
+$Hom_A(Q,-)$ 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
-$\Hom_A(Q,-)$.
+$Hom_A(Q,-)$.
-## The tensor-power functor $\Phi$
+## The tensor-power functor $Phi$
Give $X^{ ⊗ q}$ the $E$-action coming from regular permutation
of tensor positions. Define
- \Phi(X) = Θ_{\mathcal C}(X^{ ⊗ q})
- \in \mathcal C\boxtimes \mathcal W.
+ 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 $a\mapsto a^q$, and satisfies
-$\Phi(af) = a^q\Phi(f)$ for scalars $a\in k$.
+$Phi$ 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
-\[
- (X ⊗ Y)^{ ⊗ q}\cong X^{ ⊗ q} ⊗ Y^{ ⊗ q}
-\]
+[
+ (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).
+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
+$Θ$ (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}:
+Combining cref{eq:phi} and cref{lem:projective-evaluation}:
- (\Hom_{\mathcal C}(Q,-)\boxtimesId_\mathcal W)(\Phi(X))
- \cong Θ(\Hom_{\mathcal C}(Q,X^{ ⊗ q}))
+ (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]
+[$Phi$ exact iff image nonzero]
-If there are projective objects $P,Q\in\mathcal C$ with
-$Θ(\Hom_{\mathcal C}(Q,P^{ ⊗ q}))\ne0$, then $\Phi$ is
+If there are projective objects $P,Qinmathcal C$calC$ 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 $\mathcal C$, tensoring by $P$ splits it because tensor
+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
+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,
+$Phi(P)$. Tensoring by the nonzero $Phi(P)$ is exact and faithful,
and reflects exactness; the image sequence is exact.
@@ -776,39 +759,39 @@ and reflects exactness; the image sequence is exact.
% =================================================================
-Now we need to actually find $r$ and $\iota$ for which the criterion
-of \cref{prop:phi-exactness} fires. The strategy: take $r$ minimal
-so that $M(r) := \Hom_{\mathcal C}(P,P^{ ⊗ p^r})$ (as a
+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.
+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,
-$\nu_H:X_H\to X^H$ is the norm map: it factors
-$\sum_{h\in H} h:X\to X$.
+$nu_H:X_Hto X^H$ 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
+$nu_H(M)$ 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 $N_H=\sum_{h\in H}h$;
+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 $ab\mapsto$ coefficient of
+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
-$N_Hm\ne0$, the map $kH\to M$, $a\mapsto am$ is injective and splits
+$N_Hmne0$, 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 $N_H=M'_H=0$
+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
@@ -818,66 +801,66 @@ implies the total norm iso (same argument).
[independent actions multiply]
-Let finite $p$-groups $H_i$ act on objects $X_i\in\mathcal C$
-($i=1,...,s$). If $\omega(X_i)$ is projective over $kH_i$ for
-exact faithful $\omega = \Hom_{\mathcal C}(P,-)$, then
-$\omega(X_1 ⊗ ... ⊗ X_s)$ is projective over
-$k[H_1\times...\times H_s]$ (independent actions), and over every
+Let finite $p$-groups $H_i$ act on objects $X_iinmathcal C$calC$
+($i=1,...,s$). 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, $(X ⊗ Y)_{H\times K}\cong X_H ⊗ Y_K$ and
-$(X ⊗ Y)^{H\times K}\cong X^H ⊗ Y^K$
+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
-$\nu_{H\times K}(X ⊗ Y) = \nu_H(X) ⊗ \nu_K(Y)$. 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 $H\le H_1\times...\times H_s$: $kH_i$ is free over
+$nu_{Htimes K}(X ⊗ Y) = nu_H(X) ⊗ nu_K(Y)$. 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 $E_a=(\mathbb Z/p)^a$ act on $p^a$ positions by its regular
+Let $E_a=(mathbb Z/p)^a$bbZ/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.
+Set $M(a) = omega(P^{ ⊗ p^a})$, an $E_a$-module.
[first nonprojective power]
-There is a least integer $r\ge 1$ for which $M(r)$ is not projective
+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 $H\le\mathfrak S_d$. Every $H$-orbit in
-$\{1,...,d\}$ is regular because $H$ is abelian; orbitwise,
+$p$-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
+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 $k\mathfrak S_d$.
+projective over $kmathfrak S_d$.frakS_d$.
-Now apply $P\twoheadrightarrow\mathbf 1$ ($\mathfrak S_d$-equivariant,
-trivial action on the target) and exactness of $\omega$ followed by
+Now apply $Ptwoheadrightarrowmathbf 1$bf1$ ($mathfrak S_d$-equivariant,frakS_d$-equivariant,
+trivial action on the target) and exactness of $omega$ followed by
coinvariants:
-\[
- \omega(P^{ ⊗ d})_{\mathfrak S_d} \twoheadrightarrow
- \omega(\mathbf 1)\ne0.
-\]
-By \cref{lem:norm-projectivity}, the total symmetrizer
-$b_d = \sum_{\sigma\in\mathfrak S_d}\sigma$ acts nontrivially on
+[
+ 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
-$k\mathfrak S_d\to\End_{\mathcal C}(P^{ ⊗ d})$ cannot be
+$kmathfrak S_dfrakS_dtoEnd_{mathcal C}calC(P^{ ⊗ d})$ cannot be
injective: length grows exponentially in $d$ (bounded by
-$\ell(P)b^{d-1}$ with $b=\max_i\ell(S_i ⊗ P)$) and the
-dimension of $\Hom(X,Y)$ is bounded by $\ell(X)\ell(Y)$, whereas
+$ell(P)b^{d-1}$ 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
@@ -888,125 +871,125 @@ 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
+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
-$\mathfrak m = (g_i-1)_i$. The projectivised rank variety of
+$mathfrak mfrakm = (g_i-1)_i$. The projectivised rank variety of
a finite-dimensional module $D$ is
-\[
- \supp(D) =
- \bigl\{[\alpha]\in\mathbb P^{r-1}(k) :
+[
+ supp(D) =
+ bigl{[alpha]inmathbb P^bbP^{r-1}(k) :
D not free over k[t]/(t^p),
- \ t\mapsto\textstyle\sum_i\alpha_i(g_i-1)
- \bigr\}.
-\]
+ tmapstotextstylesum_ialpha_i(g_i-1)
+ bigr}.
+]
Two standard facts (Benson 2017; Coulembier--Flake 2025):
- \supp(D) = ∅ \iff D projective,
- \qquad
- \supp(D ⊗ D') = \supp(D)\cap\supp(D').
+ supp(D) = ∅ iff D projective,
+ qquad
+ 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 $[\alpha]\in\supp(M)$ has
-$\mathbb F_p$-linearly independent coordinates.
+proper subgroups projective, then every $[alpha]insupp(M)$ has
+$mathbb F_p$-linearlybbF_p$-linearly independent coordinates.
Proof.
-For $c\in(\mathbb F_p^r)^\ast$, let $H=\ker c$ and
-$N = \Ind_H^{E_r}k$, the regular representation of $E_r/H\cong
-\mathbb Z/p$ (identified with $k[z]/(z^p)$, generator acts by
-$1+z$). The element $\sum_i\alpha_i(g_i-1)$ acts on $N$ by
-$h_\alpha(z) = \sum_i\alpha_i((1+z)^{c_i}-1)$, whose linear
-coefficient is $\sum_i c_i\alpha_i$.
-
-If $\sum_i c_i\alpha_i\ne0$, then $t\mapsto h_\alpha(z)$ is an iso
+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 $E_r/Hcong
+mathbb Z/p$bbZ/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 $\sum_i c_i\alpha_i=0$, then $h_\alpha\in(z^2)$, so
-$h_\alpha^{p-1}=0$, incompatible with a free module of dimension
+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
-$\supp(N) = \{[\alpha] : \sum_i c_i\alpha_i=0\}$. The tensor
-identity gives $M ⊗ \Ind_H^{E_r}k\cong\Ind_H^{E_r}(M|_H)$, which
-is projective, so $\supp(M)\cap\supp(N)=∅$. Hence
-$\sum_i c_i\alpha_i\ne0$ for every nonzero $c$. In particular the
-coordinates of every $[\alpha]\in\supp(M)$ are $\mathbb F_p$-linearly
+$supp(N) = {[alpha] : sum_i c_ialpha_i=0}$. 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 $\lambda_1,...,\lambda_r\in k$ be $\mathbb F_p$-linearly
-independent and define $\iota_\lambda : E_r\to SL_2(k)$,
-$g_i\mapsto u(\lambda_i)$. Write $R_\lambda$ for restriction. Then
+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_i\alpha_i\lambda_i^{p^j}=0
- \ \ (0\le j\le r-2)}.
+ 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 $p\times p$
-Jordan block. Then $\sum_i\alpha_i(g_i-1)$ acts by a polynomial in
-$N$ whose linear coefficient is $\sum_i\alpha_i\lambda_i$; by the
-argument of \cref{lem:support-independent}, the resulting
+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
+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
-$M_j : \sum_i\alpha_i\lambda_i^{p^j}=0$, exactly \cref{eq:steinberg-support}.
-For $r=1$ the system is empty and $\St_0=k$ has
-$\supp(\St_0)=\mathbb P^0(k)$, consistent.
+$M_j : sum_ialpha_ilambda_i^{p^j}=0$, 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 $\lambda$
+## Moore's lemma: choosing $lambda$
-[$\lambda$ solving prescribed equations]
+[$lambda$ solving prescribed equations]
-Let $\alpha_1,...,\alpha_r\in k$ be $\mathbb F_p$-linearly
-independent. Then there exist $\mathbb F_p$-linearly independent
-$\lambda_1,...,\lambda_r\in k$ with
+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_i\alpha_i\lambda_i^{p^j} = 0,
- \qquad 0\le j\le r-2.
+ sum_ialpha_ilambda_i^{p^j} = 0,
+ qquad 0le jle r-2.
For $r=1$ the system is empty.
Proof.
-$r=1$: take $\lambda_1\ne0$. For $r\ge2$: step is standard
-graded linear algebra. Put $\beta_i = \alpha_i^{p^{-(r-2)}}$ (using
+$r=1$: 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.
+$B = (beta_i^{p^s})_{s,i}$ be the $(r-1)times r$ matrix.
Rows of $B$ are linearly independent: otherwise a nontrivial
-additive polynomial $f(X)=\sum_{s=0}^{r-2}a_s X^{p^s}$ vanishes on
-the $p^r$-element span of the $\beta_i$, but $\deg f\le p^{r-2}$, so
+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: if
-$c_i = f(\beta_i)\in\mathbb F_p$, then the polynomial $f^p-f$
+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 $\deg f<p^{r-1}$
+$le p^{r-1}$; so $f^p-f=0$, forcing $f=0$ (since $deg f<p^{r-1}$
and $f$ has zero constant term).
-Choose $\lambda\in\ker B$ nonzero. Kernel is one-dimensional. Any
-$c\in\mathbb F_p^r$ with $\sum_i c_i\lambda_i=0$ belongs to the
-annihilator of $\ker B$, i.e., to the row space of $B$, hence is
-$c=0$. So $\lambda_i$ are $\mathbb F_p$-linearly independent.
-Reading rows of $B$ in reverse order, $(\alpha_i^{p^{-j}})_i$, and
-raising the equations $\sum_i\lambda_i^{p^s}\alpha_i^{p^{-(r-2)}p^s}
+Choose $lambdainker B$ nonzero. Kernel is one-dimensional. Any
+$cinmathbb F_p^r$bbF_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).
@@ -1014,52 +997,52 @@ raising the equations $\sum_i\lambda_i^{p^s}\alpha_i^{p^{-(r-2)}p^s}
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.
+$iota_lambda:E_r↪ U(k)$ with
+$R_lambda(St_{r-1}) ⊗ M$ nonprojective.
Proof.
-Pick $[\alpha]\in\supp(M)$; coordinates are $\mathbb F_p$-linearly
-independent by \cref{lem:support-independent}. Choose $\lambda$ by
-\cref{lem:moore-selection}; \cref{lem:steinberg-support} then gives
-$[\alpha]\in\supp(R_\lambda(\St_{r-1}))$. So the tensor product
+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.
+$R_lambda(St_{r-1}) ⊗ M$ is nonprojective.
## Exactness
[exact higher Frobenius functor]
-Let $\mathcal C$ be a finite symmetric tensor category over $k$ of
-characteristic $p>0$. Then there exist $r\ge1$ and an embedding
+Let $mathcal C$calC$ 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
+cref{sec:higher} gives an exact additive strong symmetric monoidal
functor
-\[
- \Phi:\mathcal C\to\mathcal C\boxtimes\Ver_{p^r}(k)
-\]
-with $\Phi(X)=Θ_{\mathcal C}(X^{ ⊗ p^r})$ and
-$\Phi(af)=a^{p^r}\Phi(f)$. Moreover, for every projective $Q$ the
+[
+ 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
-$M = \Hom_{\mathcal C}(P,P^{ ⊗ p^r})$ is nonprojective with
+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},
-$R_\lambda(\St_{r-1}) ⊗ M\cong R_\lambda(T)$ 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.
+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.
% =================================================================
@@ -1067,200 +1050,200 @@ $\Phi$ is exact.
% =================================================================
-We now have an exact $\Phi:\mathcal C\to\mathcal C\boxtimes\mathcal
-W$, $q=p^r$, $\mathcal W=\Ver_{p^r}(k)$, semilinear.
+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 $Y\in\mathcal C$, let $c$ be the cyclic permutation of the $p$
+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:
-\[
- F_i(Y) = \frac{\ker D\capimD^{i-1}}
- {\ker D\capimD^i},
- \qquad 1\le i<p.
-\]
-If $\ell_i\in\Ver_p(k)$ is the simple object represented by a
+[
+ F_i(Y) = frac{ker DcapimD^{i-1}}
+ {ker DcapimD^i},
+ qquad 1le i<p.
+]
+If $ell_iinVer_p(k)$ is the simple object represented by a
Jordan block of length $i$, then
-$F(Y)=⊕_{i=1}^{p-1}F_i(Y)\boxtimes\ell_i$. In the
-semilinear convention, the $F_i(Y)$ are objects of $\mathcal C$
+$F(Y)=⊕_{i=1}^{p-1}F_i(Y)boxtimesell_i$. In the
+semilinear convention, the $F_i(Y)$ are objects of $mathcal C$calC$
itself. $F$ is not assumed exact.
[mod-$p$ identity in $K_0$]
-In $K_0(\mathcal C)/pK_0(\mathcal C)$,
-\[
- [Y]^p = \sum_{i=1}^{p-1}i[F_i(Y)].
-\]
-Consequently, for every $r\ge 1$,
+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.
+ [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 $K_j = \ker D\capimD^j$
-for $0\le j\le p$. The short exact sequences
-$0\to K_j\toimD^j\toimD^{j+1}\to0$
-give $[Z]=\sum_{j=0}^{p-1}[K_j]$. Summing the successive quotients
+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
+$[Z]=sum_i i[F_i(Y)]+p[K_{p-1}]$. 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
+An exact strong monoidal $sigma$-semilinear functor still preserves
Frobenius--Perron dimension: it sends nonzero objects to nonzero
-objects (from $\mathbf 1↪ X ⊗ X^*$), so
+objects (from $mathbf 1↪bf1↪ X ⊗ X^*$), so
composing with FP dimension gives a positive normalised character on
-$K_0(\mathcal C)$; the FP characterization forces it to be the FP
+$K_0(mathcal C)$;calC)$; the FP characterization forces it to be the FP
dimension of the source. In particular
-\[
- d(\Phi(X)) = d(X).
-\]
+[
+ d(Phi(X)) = d(X).
+]
[first-component inequality]
-Let $S\in\mathcal C$ be simple and $A\boxtimes B$ a composition
-factor of $\Phi(S)$. Then $d(A)\le d(S)$. If equality holds then
-$\Phi(S)\simeq A\boxtimes B$, $B$ is invertible, and
-$S\in\mathcal C_{ex}$ (the subcategory on which the
+Let $Sinmathcal C$calC$ 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 $d\ge 1$ for nonzero objects.
+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 $A\boxtimes B$.
+single term with $d(B)=1$, so $B$ is invertible and $Phi(S)$ is
+simple of the form $Aboxtimes B$.
-$S\in\mathcal C_{ex$.} Let $Q_A$ be the
+$Sinmathcal C_calC_{ex$.} Let $Q_A$ be the
projective cover of $A$. Over $k$ algebraically closed,
-$\dim_k\Hom(Q_A,X) = [X:A]$. Applying (eq:phi-evaluation) to
+$dim_kHom(Q_A,X) = [X:A]$. Applying (eq:phi-evaluation) to
$S$ and $X=S^{ ⊗ q}$:
-\[
- Θ(\Hom_{\mathcal C}(Q_A,S^{ ⊗ q}))
- \cong B.
-\]
+[
+ 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
-$[S^{ ⊗ q}:A]\not\equiv 0\pmod p$.
+$[S^{ ⊗ q}:A]notequiv 0pmod p$.
By (eq:iterated-ord) there are indices
-$i_1,...,i_r\in\{1,...,p-1\}$ with $A$ a composition factor
+$i_1,...,i_rin{1,...,p-1}$ 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 $S\in\mathcal C_{ex}$.
+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
-$\mathcal C$. Form a directed graph on those classes by arrow
-$S\to A$ whenever $A\boxtimes B$ appears in some composition factor
-of $\Phi(S)$.
+$mathcal C$.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
+cref{lem:first-component-inequality}, dimension is nonincreasing
along arrows. On a directed cycle, dimension is constant, all
-vertices belong to $\mathcal C_{ex}$, and each vertex has
+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 $\mathcal C_{ex}$.
-A schematic path is drawn in \cref{fig:dimension-descent}.
+so hits a cycle. Its endpoint is in $mathcal C_calC_{ex}$.
+A schematic path is drawn in cref{fig:dimension-descent}.
[!htbp]
-\centering
+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}
+ 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)
- {$A,B,C\in\mathcal C_{ex}$\\[2pt]
+ 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
+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 $\mathcal C_{ex}$, and a unique outgoing
+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 $\mathcal A$, let
-$\mathcal A^{[\sigma]}$ have the same objects and additive morphism
-groups but scalar action twisted by $\sigma^{-1}$. Then
-$\sigma$-semilinear functors out of $\mathcal A$ = $k$-linear
-functors out of $\mathcal A^{[\sigma]}$, and
-$(\mathcal A\boxtimes\mathcal B)^{[\sigma]}
- = \mathcal A^{[\sigma]}\boxtimes\mathcal
- B^{[\sigma]}$, so products of $\sigma$-semilinear functors are
-$\sigma$-semilinear.
+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 $mathcal A$calA$ = $k$-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 $\mathbb F_p$; scalar transport
-along $b\mapsto b^{p^a}$ is an exact strong symmetric $\sigma^a$-
-semilinear autoequivalence $J_{\sigma^a}$ of every
-$\Ver_{p^s}(k)$.
+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
+Define $Psi_0=Id$, $Psi_1=Phi$, and
- \Psi_{j+1} = (\Phi\boxtimes J_\sigma^{\boxtimes j})\circ\Psi_j
- :\mathcal C\to\mathcal C\boxtimes\mathcal W^{\boxtimes(j+1)}.
+ 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.
+By induction each $Psi_j$ is exact, strong symmetric monoidal,
+and $sigma^j$-semilinear.
[first components after $N$ iterations]
-Let $\Psi_N :\mathcal C\to\mathcal C\boxtimes
-\mathcal W^{\boxtimes N}$ be as above. Then every composition
-factor $A\boxtimes B_1\boxtimes...\boxtimes B_N$ of
-$\Psi_N(X)$ has $A\in\mathcal C_{ex}$.
+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 $\mathcal C$ (any first
+Exactness of $Phi$ plus the path bound on $mathcal C$calC$ (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.
@@ -1268,123 +1251,123 @@ Autoequivalences on other factors send simples to simples and
don't touch the first component.
-Let $\mathcal D⊆\mathcal C$ be the full subcategory whose
-simple composition factors lie in $\mathcal C_{ex}$. This
-is the Serre closure of $\mathcal C_{ex}$.
+Let $mathcal D⊆calD⊆mathcal C$calC$ 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]
-$\mathcal D$ is closed under tensor products, duals, extensions, and
+$mathcal D$calD$ is closed under tensor products, duals, extensions, and
contains the unit; it is finite and has enough projectives; it
carries a rigid symmetric tensor structure inherited from
-$\mathcal C$. Moreover the functor $\Psi_N$ factors through
-\[
- \Psi_N :\mathcal C\to \mathcal D\boxtimes
- \mathcal W^{\boxtimes N}.
-\]
+$mathcal C$.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 $\mathcal C_{ex}$ is a tensor
+filtrations, and $mathcal C_calC_{ex}$ is a tensor
subcategory closed under subquotients (Etingof--Ostrik 2021,
-Prop 7.4). Duality preserves $\mathcal D$ by exactness.
+Prop 7.4). Duality preserves $mathcal D$calD$ by exactness.
Finiteness, projectives: concretely, write
-$\mathcal C\simeq H-mod$ with $H$ basic finite-dimensional;
+$mathcal CcalCsimeq H-mod$ with $H$ basic finite-dimensional;
let $e$ be the idempotent summing primitive idempotents of the
-non-$\mathcal C_{ex}$ simples; then
-$\mathcal D\simeq (H/HeH)-mod$ by removing 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 $\mathcal D$ is finite with enough
+projective dimension, so $mathcal D$calD$ is finite with enough
projectives.
-Factoring through $\mathcal D$: \cref{lem:first-components}.
+Factoring through $mathcal D$:calD$: cref{lem:first-components}.
% =================================================================
-# Finishing the proof: a fiber functor on $\mathcal D$
+# Finishing the proof: a fiber functor on $mathcal D$calD$
% =================================================================
-## The fiber functor on $\mathcal D$
+## The fiber functor on $mathcal D$calD$
-We need an exact strong symmetric monoidal $E:\mathcal
-D\to\Ver_{p^n}(k)$ for some fixed $n$.
+We need an exact strong symmetric monoidal $E:mathcal
+DcalDtoVer_{p^n}(k)$ for some fixed $n$.
Case $p>2$.:
-$\mathcal C_{ex}$ is Frobenius-exact, so by
+$mathcal C_calC_{ex}$ is Frobenius-exact, so by
Etingof--Ostrik (2021, Theorem 8.1) it has a fiber functor
-$\mathcal C_{ex}\to\Ver_p(k)$. If $p>2$, the Serre closure
-of $\mathcal C_{ex}$ equals $\mathcal C_{ex}$
-(Etingof--Ostrik 2021, Corollary 9.5). So $\mathcal D=\mathcal
-C_{ex}$ and we already have $E:\mathcal D\to\Ver_p(k)$.
+$mathcal C_calC_{ex}toVer_p(k)$. 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 $\mathcal C_{ex}\to\Ver_2(k)=\Vec_k$ is a
-$k$-linear fiber functor. But $\mathcal C_{ex}⊆
-\mathcal D$ contains every simple of $\mathcal D$ by definition,
-and $\mathcal C_{ex}$ is Frobenius-exact. Etingof--Gelaki
+The fiber functor $mathcal C_calC_{ex}toVer_2(k)=Vec_k$ is a
+$k$-linear fiber functor. But $mathcal C_calC_{ex}⊆
+mathcal D$calD$ contains every simple of $mathcal D$calD$ by definition,
+and $mathcal C_calC_{ex}$ is Frobenius-exact. Etingof--Gelaki
(2021, Theorem 2.21(2)) then extends it to
-$E:\mathcal D\to\Ver_4^+(k)⊆\Ver_4(k)$.
+$E:mathcal DcalDtoVer_4^+(k)⊆Ver_4(k)$.
(This is also Etingof--Ostrik 2021, Corollary 9.8.)
So in all cases:
-\[
- E:\mathcal D → \Ver_{p^n}(k), \qquad
- n = 1 & p>2,\\ 2 & p=2.
-\]
+[
+ E:mathcal DcalD → Ver_{p^n}(k), qquad
+ n = 1 & p>2, 2 & p=2.
+]
## Combining factors
-Set $n=\max(2,r)$ and $\mathcal Z = \Ver_{p^n}(k)$. The tower
-inclusions $\Ver_{p^r}(k)⊆\Ver_{p^n}(k)$
+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 $I:\mathcal W\to\mathcal Z$.
-
-The functors $E:\mathcal D\to\mathcal Z$ and $I:\mathcal W\to
-\mathcal Z$ combine, by \cref{lem:combine-functors}, into
-\[
- H:\mathcal D\boxtimes\mathcal W^{\boxtimes N}\to\mathcal Z,
- \qquad
- H(D\boxtimes W_1\boxtimes...\boxtimes W_N)
+strong symmetric monoidal $I:mathcal WcalWtomathcal Z$.calZ$.
+
+The functors $E:mathcal DcalDtomathcal Z$calZ$ and $I:mathcal WcalWto
+mathcal Z$calZ$ combine, by cref{lem:combine-functors}, into
+[
+ H:mathcal DcalDboxtimesmathcal W^calW^{boxtimes N}tomathcal Z,calZ,
+ qquad
+ H(Dboxtimes W_1boxtimes...boxtimes W_N)
= E(D) ⊗ I(W_1) ⊗ ... ⊗ I(W_N).
-\]
+]
$H$ is exact and strong symmetric monoidal.
## Killing the scalar twist
-The composite $H\Psi_N :\mathcal C\to\mathcal Z$ is
-$\sigma^N$-semilinear (each iteration multiplies semilinearity by
-$\sigma$). Let $J_{\sigma^{-N}}$ be scalar transport on $\mathcal Z$
-along $b\mapsto b^{p^{-N}}$. Then
-\[
- \mathcal F := J_{\sigma^{-N}}\circ H\circ\Psi_N :
- \mathcal C→\mathcal Z
-\]
+The composite $HPsi_N :mathcal CcalCtomathcal Z$calZ$ is
+$sigma^N$-semilinear (each iteration multiplies semilinearity by
+$sigma$). Let $J_{sigma^{-N}}$ be scalar transport on $mathcal Z$calZ$
+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 $F:\mathcal C\to\mathcal D$ be an exact strong symmetric
+Let $F:mathcal CcalCtomathcal D$calD$ be an exact strong symmetric
monoidal functor between finite symmetric tensor categories. Then
$F$ is faithful.
Proof.
-Suppose $X\ne0$. The coevaluation
-$\mathbf 1\to X ⊗ X^*$ is nonzero (triangle identity) and by
-simplicity of $\mathbf 1$ it is injective; applying exact $F$ shows
-$F(X)\ne0$. For a nonzero $f:X\to Y$: exactness gives
-$imF(f)\cong F(imf)$, and
-$imf\ne0$, so $F(f)\ne0$.
+Suppose $Xne0$. The coevaluation
+$mathbf 1bf1to X ⊗ X^*$ is nonzero (triangle identity) and by
+simplicity of $mathbf 1$bf1$ 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 $\mathcal F$ shows $\mathcal F$ is
-faithful. Hence $\mathcal F:\mathcal C\to\Ver_{p^n}(k)$ is the
-required fiber functor, and \cref{thm:main} follows.
+Applying cref{lem:faithful} to $mathcal F$calF$ shows $mathcal F$calF$ is
+faithful. Hence $mathcal F:calF:mathcal CcalCtoVer_{p^n}(k)$ is the
+required fiber functor, and cref{thm:main} follows.
% =================================================================
# Consequences
@@ -1392,34 +1375,34 @@ required fiber functor, and \cref{thm:main} follows.
% =================================================================
We now record the three structural corollaries that were the original
-motivation for proving \cref{thm:main}.
+motivation for proving cref{thm:main}.
## Incompressible categories
-Recall that $\mathcal C$ is incompressible if every
-$k$-linear exact strong symmetric monoidal functor from $\mathcal C$
+Recall that $mathcal C$calC$ is incompressible if every
+$k$-linear exact strong symmetric monoidal functor from $mathcal C$calC$
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 $\mathcal C$ over $k$ is incompressible if
+symmetric tensor category $mathcal C$calC$ 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 $p$:
-\[
+[
{c|l}
-p=2 & \Ver_{2^m}(k)\ (m\ge1),\quad
- \Ver_{2^m}^+(k)\ (m\ge2),\\
-p=3 & \Ver_{3^m}(k),\ \Ver_{3^m}^+(k)\quad(m\ge1),\\
-p\ge 5 & \Vec_k,\ \sVec_k,\quad
- \Ver_{p^m}(k),\ \Ver_{p^m}^+(k)\quad(m\ge1).
+p=2 & Ver_{2^m}(k) (mge1),quad
+ Ver_{2^m}^+(k) (mge2),
+p=3 & Ver_{3^m}(k), Ver_{3^m}^+(k)quad(mge1),
+pge 5 & Vec_k, sVec_k,quad
+ Ver_{p^m}(k), Ver_{p^m}^+(k)quad(mge1).
-\]
+]
Proof.
-Direction $⇐$: $\Vec_k$ has trivial
-$\End(\mathbf 1)=k$, so any strong symmetric tensor functor out of
+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).
@@ -1427,23 +1410,23 @@ For the Verlinde full and plus categories, see Benson--Etingof--
Ostrik (2023), Theorems 4.71 and 4.72(iii).
Direction $⇒$: take a fiber functor
-$F:\mathcal C\to\Ver_{p^n}(k)$ from \cref{thm:main}, let
-$\mathcal I$ be the image's full subcategory of subquotients.
+$F:mathcal CcalCtoVer_{p^n}(k)$ from cref{thm:main}, let
+$mathcal I$calI$ be the image's full subcategory of subquotients.
Then $F$ corestricts to a surjective-in-the-tensor-category-sense
-functor onto $\mathcal I$ (Etingof--Gelaki--Nikshych--Ostrik 2015,
+functor onto $mathcal I$calI$ (Etingof--Gelaki--Nikshych--Ostrik 2015,
Definition 1.8.18, Prop 1.8.19, Section 6.3). Incompressibility
-of $\mathcal C$ forces this corestriction to be fully faithful.
+of $mathcal C$calC$ forces this corestriction to be fully faithful.
The two FP-dimension inequalities of finiteness (inclusion vs
surjection) then force equality, hence an equivalence.
Benson--Etingof--Ostrik 2023, Corollary 4.61 then restricts
-$\mathcal I$ to a full or plus sub-Verlinde at height $\le n$.
+$mathcal I$calI$ 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
+powers exponentially. So cref{cor:incompressible} is precisely the
finite case of Conjecture~B of
[cite].
@@ -1452,54 +1435,54 @@ finite case of Conjecture~B of
[uniqueness in characteristic two]
Let $k$ be algebraically closed of characteristic two and
-$\mathcal C$ a finite symmetric tensor category. Then
-$\mathcal C\to\Ver_{2^∞}(k) :=∪_n\Ver_{2^n}(k)$ has
+$mathcal C$calC$ 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 $\mathcal C\to\Ver_{2^n}(k)$ up to monoidal natural
+$Ver_{2^n}(k)$; 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
+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.
+$Ver_{2^∞}(k)$ 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 $\mathcal V$ be a finite symmetric tensor category. The
-fundamental group $\pi(\mathcal V) :=
-\Aut^{ ⊗ }(Id_\mathcal V)$ acts
-canonically on $Ind(\mathcal V)$. A
-$\mathcal V$-group is an internal affine group scheme $G$ with
-a homomorphism $\phi:\pi(\mathcal V)\to G$ whose induced coaction
+Let $mathcal V$calV$ 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 $p>0$, $\mathcal
-C$ finite symmetric tensor, and $F:\mathcal C\to
-\mathcal V=\Ver_{p^n}(k)$ a fiber functor (from \cref{thm:main}).
-Then there is a $\mathcal V$-group $(G,\phi)$ with
-$\mathcal C\simeq\Rep(G,\phi)$ over $\mathcal V$, equivalent to
+Let $k$ be algebraically closed of characteristic $p>0$, $mathcal
+C$calC$ 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
-$\phi:\pi(\mathcal V)\to\Aut^{ ⊗ }(F)$ the
+$G=Aut^{ ⊗ }(F)$ and
+$phi:pi(mathcal V)calV)toAut^{ ⊗ }(F)$ the
canonical homomorphism.
Proof.
-\Cref{lem:modules-finite-alg} makes $\mathcal C$, together with
-$\mathcal V=\Ver_{p^n}(k)$, a ``pretannakian category'' in the
+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; conversely, forgetting $G$-structure gives the
+$mathcal V$-group;calV$-group; conversely, forgetting $G$-structure gives the
inverse functor. Composition is the equivalence.
@@ -1508,8 +1491,8 @@ inverse functor. Composition is the equivalence.
All classification and existence assertions above are confined to
finite categories. Uniqueness is asserted only in characteristic
two, both in the tower union and at each fixed finite stage where a
-fiber functor exists. \Cref{thm:main} supplies some finite height
-$n$ depending on $\mathcal C$; it makes no claim of existence
+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.
@@ -1517,105 +1500,105 @@ claimed.
{99}
-\bibitem[BEO2023]{BEO2023}
+bibitem[BEO2023]{BEO2023}
D.~Benson, P.~Etingof, V.~Ostrik,
New incompressible symmetric tensor categories in positive
-characteristic, Duke Math.\ J.\ 172 (2023), 105--200.
+characteristic, Duke Math. J. 172 (2023), 105--200.
-\bibitem[BE2019]{BensonEtingof2019}
+bibitem[BE2019]{BensonEtingof2019}
D.~Benson, P.~Etingof,
Symmetric tensor categories in characteristic 2,
-Adv.\ Math.\ 351 (2019), 967--999.
+Adv. Math. 351 (2019), 967--999.
-\bibitem[EO2021]{EtingofOstrikFrobenius}
+bibitem[EO2021]{EtingofOstrikFrobenius}
P.~Etingof, V.~Ostrik,
On the Frobenius functor for symmetric tensor categories in
-positive characteristic, J.\ Reine Angew.\ Math.\ 773 (2021),
+positive characteristic, J. Reine Angew. Math. 773 (2021),
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