diff b1f9bc4..56bc728 symmetric_tensor_categories_fiber_v2.txt
diff --git a/symmetric_tensor_categories_fiber_v2.txt b/symmetric_tensor_categories_fiber_v2.txt
index 63bab3b..77f55fc 100644
--- a/symmetric_tensor_categories_fiber_v2.txt
+++ b/symmetric_tensor_categories_fiber_v2.txt
@@ -57,10 +57,10 @@ 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
+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$.
+category Ver_{p^n}(k). The result covers every prime, including
+p=2, and the height n may depend on mathcal C.calC.
tableofcontents
@@ -72,74 +72,74 @@ tableofcontents
## The basic objects
-Throughout, $k$ is an algebraically closed field of characteristic
-$p>0$.
+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 $mathcal C$calC$
+A finite symmetric tensor category over k is a
+k-linear abelian rigid symmetric monoidal category mathcal CcalC
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
+ 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;
- item the unit object $mathbf 1$bf1$ has $End(mathbf 1)bf1) = k$;
- item there are enough projective 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
+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 $mathcal CcalCto mathcal D$calD$ between finite
-symmetric tensor categories is a $k$-linear exact faithful strong
+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
+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
-categories over $k$ and they form a 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,qquad
- Ver_3(k) = sVec_k,qquad
+ 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
+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 $SL_2$.
+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
@@ -147,9 +147,9 @@ 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
-$mathcal C$calC$ over $k$, there exist an integer $nge 1$ and a fiber
+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).
@@ -157,53 +157,50 @@ functor
This settles the finite case of Conjecture 1.4 of
-Benson--Etingof--Ostrik (2023), including $p=2$. The conjecture in
+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 $Phi$,
+We iteratively apply a semilinear higher Frobenius functor Phi,
each application adding one fresh Verlinde factor while forcing
-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 $mathcal C$,calC$, all first components land in the
+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*]
- item 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
+ 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.
-
- item Higher Frobenius functor.
+ 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 ($Phi(af) = a^{p^r}Phi(f)$).
-
- 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
+ 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.
-
- item Dimension descent. Even when $Phi$ is only
+ 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 $mathcal C_calC_{ex}$.
+ 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
+(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.
@@ -221,72 +218,72 @@ corollaries.
The definition of finite symmetric tensor category was chosen so
that the usual Tannakian toolkit applies. We make this explicit.
-[$mathcal C$calC$ is modules over a finite algebra]
+[mathcal CcalC is modules over a finite algebra]
-Let $mathcal C$calC$ be a finite symmetric tensor category. Then the
+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 $mathcal C$calC$ is equivalent, as a $k$-linear
+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
-$A = End_{mathcal C}calC(P)^{op}$.
+over the finite-dimensional k-algebra
+A = End_{mathcal C}calC(P)^{op}.
Proof.
-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 $Wsimeq V$. The endomorphism ring of the unit has no
-idempotents, so $V=0$.
-
-For the projective generator: pick surjections $Q_ito S_i$ from
+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
+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
+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 $mathcal AcalAboxtimesmathcal B$calB$ 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
-$mathcal AcalAboxtimesmathcal BcalBsimeq (A ⊗ _k B)-mod$,
+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).
+the two factors (here we use k algebraically closed).
[combining exact tensor functors]
-Let $mathcal A_1,...,calA_1,...,mathcal A_s,calA_s,mathcal Z$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
+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,
- 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
+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
-$mathcal Z$calZ$ gives the tensor constraint by rearranging factors, and
+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
@@ -295,34 +292,34 @@ also left exact.
## Frobenius--Perron dimension
-For finite tensor categories the Grothendieck ring $K_0(mathcal C)$calC)$
+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
-$d = FPdim : K_0(mathcal C)calC)tomathbb R$bbR$ with $d(mathbf 1)=1$bf1)=1$
-and $d$ multiplicative on classes. Key facts, all in Etingof--
+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 $mathcal C$calC$ be a finite tensor category.
+Let mathcal CcalC 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 $0to Xto Yto Zto 0$ is exact, then
- $d(Y)=d(X)+d(Z)$.
- item On a Deligne product, $d(Aboxtimes B)=d(A),d(B)$.
- item If $F:mathcal CcalCtomathcal D$calD$ is an exact strong monoidal
+ 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 $X$.
+ 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--
+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.
+mod-p congruence.
% =================================================================
# Step 1: Restricted tilting modules
@@ -331,31 +328,31 @@ mod-$p$ congruence.
## The statement
-Let $G = SL_2(k)$, and let $Usimeqmathbb G_a$bbG_a$ be its upper
-unipotent subgroup, identified with $k$ via
-$u(t) = (1&t0&1)$.
-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
-$hge0$; 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
+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 $a$.
+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
+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.
+as kE_r-modules.
This proves Conjecture 5.4.2(1) of Coulembier--Flake (2026) and
@@ -365,141 +362,141 @@ Conjecture 1 of Coulembier--Flake (2025), for every prime.
We do three things:
[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
+ 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);
- 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
+ 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;
- item tensor by $St_{r-1}$, observe the resulting bundle's
+ 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.
+ e_1-fibre via a formal construction on jets.
-We then recover $D$ from the $e_1$-fibre.
+We then recover D from the e_1-fibre.
-## Lifting $D$ to a bounded $U$-module
+## 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.
+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 $widehatwidehat D$D$ of degree $< p^r$
-such that $widehatwidehat D|_D|_{E_r}cong D$ along $iota$.
+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
+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
+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,
- qquad (g,d)sim(gu,u^{-1}d),
+ (g,d)sim(gu,u^{-1}d),
]
-on the punctured plane, with $G$ acting by left multiplication.
-The fibre at $e_1$ is $widehatwidehat D$D$ itself.
+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 $mathcal E$calE$ extends to a $G$-equivariant vector bundle
-$mathcal M$calM$ on the whole affine plane, and every simple
+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
-$M_0 = mathcal M|_0$calM|_0$ has highest weight $< q$.
+M_0 = mathcal M|_0calM|_0 has highest weight < q.
Proof.
-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
+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 $mathcal M$calM$ is a vector bundle.
+trivially free. Hence mathcal McalM is a vector bundle.
Equivariance. Over the punctured plane we have a
-$G$-equivariant bundle isomorphism
-$pi_2^*mathcal McalMcong a^*mathcal M$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
+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
+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
+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
-$h<q$.
+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
+We use two standard facts about SL_2-tilting modules, both due to
Donkin and both valid in every positive characteristic including
-$p=2$:
+p=2:
- L(a) = T(a)quad(0le a<p),
- qquad
+ L(a) = T(a)(0le a<p),
+
T(a) ⊗ T(b)^{(1)}simeq T(a+pb)
- quad(p-1le ale 2p-2, bge 0),
+ (p-1le ale 2p-2, bge 0),
-where $(1)$ means pullback along the Frobenius. See Benson--
+where (1) means pullback along the Frobenius. See Benson--
Etingof--Ostrik (2023), Proposition 3.1.
-[Steinberg $ ⊗ $ bounded weight is tilting]
+[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.
+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 $M = L(h)$.
+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 $0le d<p$ 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
@@ -507,94 +504,94 @@ Steinberg's tensor product theorem gives
(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
+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:
+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 $T$
+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$.
+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 $mathcal N$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.
+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
+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.
+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 $j$,
+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 $mathfrak m^j$frakm^j$ to $s_{j-1}$; the obstruction is exactly the
-Ext group just computed. Extending scalars, $s_j$ gives an
-$A/mathfrak m^j$-linearfrakm^j$-linear map
-$(A/mathfrak m^j)frakm^j) ⊗ _k Tto N/mathfrak m^jfrakm^j N$, which is an
+Starting from the identity s_1:Tto N/mathfrak mfrakm N, we lift
+successively to G-maps s_j:Tto N/mathfrak m^jfrakm^j N that reduce
+modulo mathfrak m^jfrakm^j to s_{j-1}; the obstruction is exactly the
+Ext group just computed. Extending scalars, s_j gives an
+A/mathfrak m^j-linearfrakm^j-linear map
+(A/mathfrak m^j)frakm^j) ⊗ _k Tto N/mathfrak m^jfrakm^j N, which is an
isomorphism by Nakayama (both sides free of the same rank and the
map is an isomorphism on the residue field). The inverse limit is a
-$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
+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 $k$.
+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 $mathcal M$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
+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 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.
+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.
% =================================================================
@@ -602,29 +599,29 @@ does $T$. Set $T_{out} := T$; the theorem follows.
% =================================================================
-## The functor $Θ$ from group modules to Verlinde
+## 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
-$St_r = L(p^r-1)$.
+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
-$mathcal T/I_r$calT/I_r$ whenever $age p^r - 1$; this is the standing
+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 $mathcal T/I_rcalT/I_rtomathcal W$calW$ is an equivalence onto a
+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
-$mathcal T/I_rcalT/I_rtoStab(E) :=
-Rep_k(E)/(projectives)$.
+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
+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
@@ -632,55 +629,55 @@ symmetric monoidal functor}
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
+cref{thm:restricted-tilting} produces Θ, once and for all
embeddings.
-$Θ$ is not exact; it is additive and symmetric monoidal.
+Θ is not exact; it is additive and symmetric monoidal.
-[$Θ$ detects nonprojectives]
+[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 $Θ$.
+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
+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,
-$Θ(Ind_H^E D)=0$.
+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 $mathcal C$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 $mathcal W$,calW$, hence an object of
-$mathcal CcalCboxtimesmathcal W$.calW$. This is the internal
-extension $Θ_{mathcal C}calC$ of $Θ$ (Coulembier--Flake
+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).
+additivity of Θ, not exactness).
[evaluation at projectives]
-For $Qinmathcal C$calC$ projective and $Z$ an $E$-object of
-$mathcal C$,calC$,
+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))
@@ -689,16 +686,16 @@ $mathcal C$,calC$,
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
-$Hom_A(Q,-)$.
+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 $Phi$
+## The tensor-power functor Phi
-Give $X^{ ⊗ q}$ the $E$-action coming from regular permutation
+Give X^{ ⊗ q} the E-action coming from regular permutation
of tensor positions. Define
Phi(X) = Θ_{mathcal C}calC(X^{ ⊗ q})
@@ -707,23 +704,23 @@ of tensor positions. Define
[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 $ain k$.
+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
+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
+(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.
@@ -733,24 +730,24 @@ 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 $Q$.
+for every projective Q.
## The nonprojective image criterion
-[$Phi$ exact iff image nonzero]
+[Phi exact iff image nonzero]
-If there are projective objects $P,Qinmathcal C$calC$ with
-$Θ(Hom_{mathcal C}calC(Q,P^{ ⊗ q}))ne0$, then $Phi$ is
+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
+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.
@@ -759,127 +756,127 @@ 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 $M(r) := Hom_{mathcal C}calC(P,P^{ ⊗ p^r})$ (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.
+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
+## 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
-$sum_{hin H} h:Xto X$.
+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]
+[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.
+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 $M'=0$.
+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
+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 $X_iinmathcal C$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
+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
-$(X ⊗ Y)^{Htimes K}cong X^H ⊗ Y^K$
+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.
+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 $r$
+## Existence of a minimal r
-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.
+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.
+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 $Ptwoheadrightarrowmathbf 1$bf1$ ($mathfrak S_d$-equivariant,frakS_d$-equivariant,
-trivial action on the target) and exactness of $omega$ followed by
+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
-$P^{ ⊗ d}$.
+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
+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,
+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.
+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
+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
+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) :
@@ -890,45 +887,45 @@ a finite-dimensional module $D$ is
Two standard facts (Benson 2017; Coulembier--Flake 2025):
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 $[alpha]insupp(M)$ has
-$mathbb F_p$-linearlybbF_p$-linearly independent coordinates.
+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 $E_r/Hcong
-mathbb Z/p$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 $sum_i c_ialpha_i=0$, then $h_alphain(z^2)$, so
-$h_alpha^{p-1}=0$, incompatible with a free module of dimension
-$p$; so $N$ is not free.
+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
+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
+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
@@ -937,112 +934,112 @@ $g_imapsto u(lambda_i)$. Write $R_lambda$ for restriction. Then
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
+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.
+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
-$M_j : sum_ialpha_ilambda_i^{p^j}=0$, 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.
+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 $alpha_1,...,alpha_rin k$ be $mathbb F_p$-linearlybbF_p$-linearly
-independent. Then there exist $mathbb F_p$-linearlybbF_p$-linearly independent
-$lambda_1,...,lambda_rin k$ with
+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,
- qquad 0le jle r-2.
+ 0le jle r-2.
-For $r=1$ the system is empty.
+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
-$cinmathbb F_p^r$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).
+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
+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 $[alpha]insupp(M)$; coordinates are $mathbb F_p$-linearlybbF_p$-linearly
-independent by cref{lem:support-independent}. Choose $lambda$ by
+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
+[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 $mathcal C$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
+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
+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
+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.
+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},
+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.
+So Θ(M)ne0. Cref{prop:phi-exactness} with P=Q shows
+Phi is exact.
% =================================================================
@@ -1050,34 +1047,34 @@ $Phi$ is exact.
% =================================================================
-We now have an exact $Phi:mathcal CcalCtomathcal CcalCboxtimesmathcal
-W$,calW$, $q=p^r$, $mathcal W=calW=Ver_{p^r}(k)$, 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
+## 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
+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},
- qquad 1le i<p.
+ 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 $mathcal C$calC$
-itself. $F$ is not assumed exact.
+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 $K_0$]
+[mod-p identity in K_0]
-In $K_0(mathcal C)/pK_0(calC)/pK_0(mathcal C)$,calC)$,
+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 $rge 1$,
+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.
@@ -1085,24 +1082,24 @@ Consequently, for every $rge 1$,
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
+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
+[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 $mathbf 1↪bf1↪ X ⊗ X^*$), so
+objects (from mathbf 1↪bf1↪ X ⊗ X^*), so
composing with FP dimension gives a positive normalised character on
-$K_0(mathcal C)$;calC)$; 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).
@@ -1110,59 +1107,59 @@ dimension of the source. In particular
[first-component inequality]
-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
+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.
+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 $Aboxtimes B$.
+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 $X=S^{ ⊗ q}$:
+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
-$[S^{ ⊗ q}:A]notequiv 0pmod p$.
+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 $d(ell_{i_t})ge1$:
+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 $Sinmathcal C_calC_{ex}$.
+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 $Phi(S)$.
+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
+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 $mathcal C_calC_{ex}$.
+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]
@@ -1182,12 +1179,12 @@ centering
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) {$C$};
+ 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);
@@ -1195,72 +1192,72 @@ centering
draw[edge] (B) -- (C);
draw[edge] (C) -- (A);
node[align=center] at (-2.8,-0.95)
- {$S→ S'$ implies $d(S')leq d(S)$};
+ {S→ S' implies d(S')leq d(S)};
node[align=center] at (1.83,-2.10)
- {$A,B,Cinmathcal C_calC_{ex}$[2pt]
+ {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
+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 $mathcal A$calA$ = $k$-linear
-functors out of $mathcal A^calA^{[sigma]}$, and
-$(mathcal AcalAboxtimesmathcal B)^calB)^{[sigma]}
+Let sigma(a)=a^q. For a k-linear category mathcal A,calA, let
+mathcal A^calA^{[sigma]} have the same objects and additive morphism
+groups but scalar action twisted by sigma^{-1}. Then
+sigma-semilinear functors out of mathcal AcalA = k-linear
+functors out of mathcal A^calA^{[sigma]}, and
+(mathcal AcalAboxtimesmathcal B)^calB)^{[sigma]}
= mathcal A^calA^{[sigma]}boxtimesmathcal
- B^calB^{[sigma]}$, so products of $sigma$-semilinear functors are
-$sigma$-semilinear.
+ 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
-$Ver_{p^s}(k)$.
+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} = (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]
+[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 $Ainmathcal C_calC_{ex}$.
+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 $mathcal C$calC$ (any first
+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
+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 $mathcal D⊆calD⊆mathcal C$calC$ be the full subcategory whose
-simple composition factors lie in $mathcal C_calC_{ex}$. This
-is the Serre closure of $mathcal C_calC_{ex}$.
+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]
-$mathcal D$calD$ is closed under tensor products, duals, extensions, and
+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
+mathcal C.calC. Moreover the functor Psi_N factors through
[
Psi_N :mathcal CcalCto mathcal DcalDboxtimes
mathcal W^calW^{boxtimes N}.
@@ -1269,104 +1266,104 @@ $mathcal C$.calC$. Moreover the functor $Psi_N$ factors through
Proof.
Closed under tensor products: biexactness gives tensor-product
-filtrations, and $mathcal C_calC_{ex}$ is a tensor
+filtrations, and mathcal C_calC_{ex} is a tensor
subcategory closed under subquotients (Etingof--Ostrik 2021,
-Prop 7.4). Duality preserves $mathcal D$calD$ by exactness.
+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 $mathcal D$calD$ is finite with enough
+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}.
+Factoring through mathcal D:calD: cref{lem:first-components}.
% =================================================================
-# Finishing the proof: a fiber functor on $mathcal D$calD$
+# Finishing the proof: a fiber functor on mathcal DcalD
% =================================================================
-## The fiber functor on $mathcal D$calD$
+## The fiber functor on mathcal DcalD
-We need an exact strong symmetric monoidal $E:mathcal
-DcalDtoVer_{p^n}(k)$ for some fixed $n$.
+We need an exact strong symmetric monoidal E:mathcal
+DcalDtoVer_{p^n}(k) for some fixed n.
-Case $p>2$.:
-$mathcal C_calC_{ex}$ is Frobenius-exact, so by
+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 $mathcal C_calC_{ex}⊆
-mathcal D$calD$ contains every simple of $mathcal D$calD$ by definition,
-and $mathcal C_calC_{ex}$ is Frobenius-exact. Etingof--Gelaki
+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
-$E:mathcal DcalDtoVer_4^+(k)⊆Ver_4(k)$.
+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), qquad
+ 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$.
+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 $E:mathcal DcalDtomathcal Z$calZ$ and $I:mathcal WcalWto
-mathcal Z$calZ$ combine, by cref{lem:combine-functors}, into
+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,
- qquad
+
H(Dboxtimes W_1boxtimes...boxtimes W_N)
= E(D) ⊗ I(W_1) ⊗ ... ⊗ I(W_N).
]
-$H$ is exact and strong symmetric monoidal.
+H is exact and strong symmetric monoidal.
## Killing the scalar twist
-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
+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.
+is k-linear, exact, strong symmetric monoidal.
## Faithfulness
[exact strong monoidal with simple unit is faithful]
-Let $F:mathcal CcalCtomathcal D$calD$ be an exact strong symmetric
+Let F:mathcal CcalCtomathcal DcalD be an exact strong symmetric
monoidal functor between finite symmetric tensor categories. Then
-$F$ is faithful.
+F is faithful.
Proof.
-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 $F(f)ne0$.
+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 $mathcal F$calF$ shows $mathcal F$calF$ is
-faithful. Hence $mathcal F:calF:mathcal CcalCtoVer_{p^n}(k)$ is the
+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.
% =================================================================
@@ -1379,47 +1376,47 @@ motivation for proving cref{thm:main}.
## Incompressible categories
-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.
+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 $mathcal C$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 $p$:
+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),quad
+p=2 & Ver_{2^m}(k) (mge1),
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).
+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
+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
-$mathcal I$calI$ be the image's full subcategory of subquotients.
-Then $F$ corestricts to a surjective-in-the-tensor-category-sense
-functor onto $mathcal I$calI$ (Etingof--Gelaki--Nikshych--Ostrik 2015,
+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 $mathcal C$calC$ forces this corestriction to be fully faithful.
+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
-$mathcal I$calI$ to a full or plus sub-Verlinde at height $le n$.
+mathcal IcalI to a full or plus sub-Verlinde at height le n.
Collecting cases gives the stated list.
@@ -1430,59 +1427,59 @@ powers exponentially. So cref{cor:incompressible} is precisely the
finite case of Conjecture~B of
[cite].
-## Uniqueness in characteristic $2$
+## Uniqueness in characteristic 2
[uniqueness in characteristic two]
-Let $k$ be algebraically closed of characteristic two and
-$mathcal C$calC$ a finite symmetric tensor category. Then
-$mathcal CcalCtoVer_{2^∞}(k) :=∪_nVer_{2^n}(k)$ has
+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
+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.
+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 $mathcal V$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
+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 $p>0$, $mathcal
-C$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
-$phi:pi(mathcal V)calV)toAut^{ ⊗ }(F)$ the
+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
+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
+mathcal V-group;calV-group; conversely, forgetting G-structure gives the
inverse functor. Composition is the equivalence.
@@ -1492,8 +1489,8 @@ 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
+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.
@@ -1538,7 +1535,7 @@ categories, Selecta Math. 32 (2026), Paper 98.
bibitem[CF2025]{CoulembierFlakeSteinberg}
K.~Coulembier, J.~Flake,
-A conjecture on the tensor ideal for an elementary $p$-group
+A conjecture on the tensor ideal for an elementary p-group
generated by the restriction of a Steinberg module,
arXiv:2507.02722v1.
@@ -1576,13 +1573,13 @@ MSRI Publ. 68, Cambridge, 2015, 19--42.
bibitem[B2017]{Benson2017}
D.~Benson,
-Representations of Elementary Abelian $p$-Groups and Vector
+Representations of Elementary Abelian p-Groups and Vector
Bundles, Cambridge Tracts in Math. 208, Cambridge, 2017.
bibitem[DH2005]{DotyHenke2005}
S.~Doty, A.~Henke,
Decomposition of tensor products of modular irreducibles for
-$SL_2$, Quart. J. Math. 56 (2005), 189--207.
+SL_2, Quart. J. Math. 56 (2005), 189--207.
bibitem[D1993]{Donkin1993}
S.~Donkin,