diff 56bc728..f6bf760 symmetric_tensor_categories_fiber_v2.txt
diff --git a/symmetric_tensor_categories_fiber_v2.txt b/symmetric_tensor_categories_fiber_v2.txt
index 77f55fc..2058a0b 100644
--- a/symmetric_tensor_categories_fiber_v2.txt
+++ b/symmetric_tensor_categories_fiber_v2.txt
@@ -60,7 +60,7 @@ every finite symmetric tensor category over an algebraically closed
field k of characteristic p>0 admits a k-linear exact faithful
strong symmetric monoidal functor to a finite higher Verlinde
category Ver_{p^n}(k). The result covers every prime, including
-p=2, and the height n may depend on mathcal C.calC.
+p=2, and the height n may depend on C.
tableofcontents
@@ -78,7 +78,7 @@ p>0.
[finite symmetric tensor category]
A finite symmetric tensor category over k is a
-k-linear abelian rigid symmetric monoidal category mathcal CcalC
+k-linear abelian rigid symmetric monoidal category C
such that:
[label=(arabic*)]
the tensor product is biexact;
@@ -95,14 +95,14 @@ 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
+when p>2 its partner is sVec_k, the Z/2-gradedbbZ/2-graded
version with the Koszul sign in the symmetry.
[fiber functor]
-A fiber functor mathcal CcalCto mathcal DcalD between finite
+A fiber functor C D between finite
symmetric tensor categories is a k-linear exact faithful strong
symmetric monoidal functor. ``Fiber functor'' in the classical
-sense is the special case mathcal DcalD = Vec_k; the target being a
+sense is the special case D = Vec_k; the target being a
non-vector-space category is what forces the tower in positive
characteristic.
@@ -134,7 +134,7 @@ 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
+Rep_k(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
@@ -149,10 +149,10 @@ 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 CcalC over k, there exist an integer nge 1 and a fiber
+C over k, there exist an integer nge 1 and a fiber
functor
[
- mathcal CcalC → Ver_{p^n}(k).
+ C → Ver_{p^n}(k).
]
@@ -166,15 +166,15 @@ implies the finite part of all the consequences we discuss.
We iteratively apply a semilinear higher Frobenius functor Phi,
each application adding one fresh Verlinde factor while forcing
-the original mathcal CcalC factor to shrink into its Frobenius-exact
+the original C factor to shrink into its Frobenius-exact
subcategory. After at most N iterations, where N is the number
-of simple objects of mathcal C,calC, all first components land in the
+of simple objects of C, all first components land in the
Serre closure of that subcategory. The essential ingredients are:
[label=Step arabic*.,ref=step:arabic*]
Restricted tilting modulo a Steinberg module.
For every finite-dimensional module D over the
- elementary abelian group E_r = (mathbb Z/p)^r,bbZ/p)^r, the
+ elementary abelian group E_r = (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
@@ -183,7 +183,7 @@ Serre closure of that subcategory. The essential ingredients are:
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)
+ Phi:C C 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
@@ -195,7 +195,7 @@ Serre closure of that subcategory. The essential ingredients are:
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 C_{ex}.
After these four steps, a few more tasks finish the argument:
@@ -218,15 +218,15 @@ corollaries.
The definition of finite symmetric tensor category was chosen so
that the usual Tannakian toolkit applies. We make this explicit.
-[mathcal CcalC is modules over a finite algebra]
+[C is modules over a finite algebra]
-Let mathcal CcalC be a finite symmetric tensor category. Then the
+Let C be a finite symmetric tensor category. Then the
unit object is simple, and there is a projective generator
-Pinmathcal C.calC. The functor omega = Hom_{mathcal C}calC(P,-) is
-exact and faithful, and mathcal CcalC is equivalent, as a k-linear
+PinC. The functor omega = Hom_{C},-) is
+exact and faithful, and C is equivalent, as a k-linear
abelian category, to the category of finite-dimensional modules
over the finite-dimensional k-algebra
-A = End_{mathcal C}calC(P)^{op}.
+A = End_{C}{op}.
Proof.
@@ -236,7 +236,7 @@ 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
+W gives W~= V. The endomorphism ring of the unit has no
idempotents, so V=0.
For the projective generator: pick surjections Q_ito S_i from
@@ -244,7 +244,7 @@ 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
+C~= 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.
@@ -256,22 +256,22 @@ simples.
## Deligne products
-For finite k-linear abelian categories mathcal A,calA,mathcal B,calB, the
-Deligne product mathcal AcalAboxtimesmathcal BcalB is the
+For finite k-linear abelian categories A,B, the
+Deligne product A 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,
+exact in each variable. When A~= A-mod and
+B~= B-mod, we have
+A BcalB~= (A ⊗ _k B)-mod,
and its simple objects are the exterior products of the simples of
the two factors (here we use k algebraically closed).
[combining exact tensor functors]
-Let mathcal A_1,...,calA_1,...,mathcal A_s,calA_s,mathcal ZcalZ be finite
-symmetric tensor categories and let F_i:mathcal A_icalA_itomathcal ZcalZ
+Let A_1,...,calA_1,...,A_s,Z be finite
+symmetric tensor categories and let F_i:A_i ZcalZ
be exact k-linear strong symmetric monoidal functors. Then
[
- H:mathcal A_1calA_1boxtimes...boxtimesmathcal A_scalA_stomathcal Z,calZ,
+ H:A_1 A_scalA_stoZ,
H(X_1boxtimes...boxtimes X_s)
= F_1(X_1) ⊗ ... ⊗ F_s(X_s),
@@ -283,7 +283,7 @@ functor.
Proof.
The displayed multilinear functor has a right-exact extension by
the universal property of the Deligne product. Symmetry in
-mathcal ZcalZ gives the tensor constraint by rearranging factors, and
+Z gives the tensor constraint by rearranging factors, and
coherence extends by the same universal property. The result
preserves duals. A short exact sequence dualises, is sent to a
right-exact sequence, and is dualised back; hence the functor is
@@ -292,24 +292,24 @@ 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(C)
is finitely freely generated by simple classes. Symmetry makes it a
commutative ring, and the Frobenius--Perron theorem identifies a
unique positive normalised real character
-d = FPdim : K_0(mathcal C)calC)tomathbb RbbR with d(mathbf 1)=1bf1)=1
+d = FPdim : K_0(C) RbbR with d(mathbf 1)=1bf1)=1
and d multiplicative on classes. Key facts, all in Etingof--
Gelaki--Nikshych--Ostrik (2015):
[FP dimension toolkit]
-Let mathcal CcalC be a finite tensor category.
+Let C be a finite tensor category.
[label=(roman*)]
d(X)ge 1 for every nonzero X; equality if and only if
X is invertible.
If 0to Xto Yto Zto 0 is exact, then
d(Y)=d(X)+d(Z).
On a Deligne product, d(Aboxtimes B)=d(A),d(B).
- If F:mathcal CcalCtomathcal DcalD is an exact strong monoidal
+ If F:C DcalD is an exact strong monoidal
functor of finite tensor categories, then
d(F(X))=d(X) for every X.
@@ -328,10 +328,10 @@ mod-p congruence.
## The statement
-Let G = SL_2(k), and let Usimeqmathbb G_abbG_a be its upper
+Let G = SL_2(k), and let U~=G_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
+For rge 1 put E_r = (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.
@@ -346,11 +346,11 @@ weight a.
[restricted-tilting theorem]
-Let rge 1, E_r = (mathbb Z/p)^r,bbZ/p)^r, and iota:E_rto U(k) any
+Let rge 1, E_r = (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 ~= R_iota(T)
]
as kE_r-modules.
@@ -366,7 +366,7 @@ We do three things:
U-module widehatwidehat DD of bounded ``degree'' (a bound on
polynomial degree of the unipotent action matrix);
use widehatwidehat DD to construct a G-equivariant vector
- bundle mathcal McalM on the plane mathbb A^2_kbbA^2_k whose
+ bundle M on the plane A^2_k whose
origin fibre has all simple composition factors below a
height bound;
tensor by St_{r-1}, observe the resulting bundle's
@@ -386,7 +386,7 @@ matrix rho_D(u(t)); this does not depend on a chosen basis.
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.
+such that widehatwidehat D|_D|_{E_r}~= D along iota.
Proof.
@@ -395,23 +395,23 @@ 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
+ev_H : C_q to k^H = O(E_r), which is a
bijection: the two sides have dimension q, and a degree-<q
polynomial vanishing at the q distinct points of H is zero.
-Transporting the mathcal O(E_r)-coactioncalO(E_r)-coaction defining D across
+Transporting the 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:
+Now consider the orbit map pi : GtoA^2∖bbA^2∖{0},
+g|-> 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,
+ E = Gtimes^U widehatwidehat D,D,
(g,d)sim(gu,u^{-1}d),
]
on the punctured plane, with G acting by left multiplication.
@@ -419,27 +419,27 @@ The fibre at e_1 is widehatwidehat DD itself.
[extension and weight bound]
-The bundle mathcal EcalE extends to a G-equivariant vector bundle
-mathcal McalM on the whole affine plane, and every simple
+The bundle E extends to a G-equivariant vector bundle
+M on the whole affine plane, and every simple
composition factor of the origin fibre
-M_0 = mathcal M|_0calM|_0 has highest weight < q.
+M_0 = M|_0calM|_0 has highest weight < q.
Proof.
-Extension. A coherent extension mathcal FcalF exists on the
-noetherian affine plane. Its double dual mathcal McalM = mathcal
-F^calF^{**} is a coherent reflexive sheaf restricting to mathcal E.calE.
-On the regular surface mathbb A^2bbA^2 every reflexive coherent sheaf
+Extension. A coherent extension F exists on the
+noetherian affine plane. Its double dual M = mathcal
+F^calF^{**} is a coherent reflexive sheaf restricting to E.
+On the regular surface A^2 every reflexive coherent sheaf
is locally free: at a closed point the stalk has depth two, so the
Auslander--Buchsbaum formula gives projective dimension zero, and at
the generic point and at the generic point of a curve the stalks are
-trivially free. Hence mathcal McalM is a vector bundle.
+trivially free. Hence M is a vector bundle.
Equivariance. Over the punctured plane we have a
G-equivariant bundle isomorphism
-pi_2^*mathcal McalMcong a^*mathcal McalM for the action
-a:Gtimesmathbb A^2bbA^2tomathbb A^2bbA^2 and projection
-pi_2:Gtimesmathbb A^2bbA^2tomathbb A^2.bbA^2. Both sides are locally
+pi_2^*M a^*M for the action
+a:GtimesA^2 A^2bbA^2 and projection
+pi_2:GtimesA^2 A^2.bbA^2. Both sides are locally
free, so the Hom-sheaf is locally free and its sections extend
across the codimension-two subset Gtimes{0}; the extensions of
the isomorphism and its inverse remain inverse because the two
@@ -453,7 +453,7 @@ 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
+A = Gamma(ell,E nd(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
@@ -477,7 +477,7 @@ p=2:
L(a) = T(a)(0le a<p),
- T(a) ⊗ T(b)^{(1)}simeq T(a+pb)
+ T(a) ⊗ T(b)^{(1)}~= T(a+pb)
(p-1le ale 2p-2, bge 0),
where (1) means pullback along the Frobenius. See Benson--
@@ -499,7 +499,7 @@ tilting by (eq:sl2-tilting-basics).
For m>1, write h = d + p h' with 0le d<p and h'<p^{m-1}.
Steinberg's tensor product theorem gives
[
- St_{m-1} ⊗ L(h) cong
+ St_{m-1} ⊗ L(h) ~=
L(d) ⊗ St_1 ⊗
(St_{m-2} ⊗ L(h'))^{(1)}.
]
@@ -518,25 +518,25 @@ M|_0,calM|_0, we have:
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.
+N = St_{r-1} ⊗ _k M) with the e_1-fibre
+of N.
## Formal comparison of fibres
[comparison with a tilting origin fibre]
-Let mathcal NcalN be a G-equivariant vector bundle on mathbb A^2bbA^2
-whose origin fibre T = mathcal N|_0calN|_0 is tilting. Then
-mathcal N|_calN|_{e_1}cong T|_U as rational U-modules.
+Let N be a G-equivariant vector bundle on A^2
+whose origin fibre T = N|_0calN|_0 is tilting. Then
+N|_calN|_{e_1}~= 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
+and N=Gamma(A^2,N), 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
+ mathfrak m^jfrakm^j N/mathfrak m^frakm^{j+1}N ~=
Sym^j(V^*) ⊗ _k T.
The target has a good filtration. Tensor products of Sym's of
@@ -564,15 +564,15 @@ 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
+phi:T_Kcong N_{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
+f = b_h^{-1}phirho_T(h) intertwines the original
+U-actions (where b_h : N_{e_1}toN_{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
+L = Hom_U(T,N_{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
@@ -580,16 +580,16 @@ required isomorphism over k.
Proof. [Proof of cref{thm:restricted-tilting}]
-Apply cref{prop:extension-bound} to get mathcal McalM with origin
+Apply cref{prop:extension-bound} to get M 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
+N = St_{r-1} ⊗ _kM:calM: its origin fibre is
T, so
[
- mathcal N|_calN|_{e_1} cong T|_U.
+ N|_calN|_{e_1} ~= T|_U.
]
-But by construction mathcal N|_calN|_{e_1} restricts to
+But by construction 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.
@@ -603,20 +603,20 @@ does T. Set T_{out} := T; the theorem follows.
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
+SL_2-modules along this embedding, and W = Ver_{p^r}(k).
+Let T⊆calT⊆ Rep_k(SL_2) be the category of tilting
modules, and let I_r be the tensor ideal generated by
St_r = L(p^r-1).
Concretely, I_r is defined by demanding that T(a) (the
indecomposable tilting module of highest weight a) vanish in
-mathcal T/I_rcalT/I_r whenever age p^r - 1; this is the standing
+T/I_rcalT/I_r whenever age p^r - 1; this is the standing
convention in this circle of papers.
-The functor mathcal T/I_rcalT/I_rtomathcal WcalW is an equivalence onto a
+The functor T/I_rcalT/I_rtoW 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) :=
+T/I_rcalT/I_rtoStab(E) :=
Rep_k(E)/(projectives).
Coulembier--Flake (2026, Theorem 5.4.3) state: if for every
@@ -624,9 +624,9 @@ 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
+ Θ : Rep_k(E)→ W
quadwithquad
- Θ(R(T)) simeq Σ_r(Tbmod I_r).
+ Θ(R(T)) ~= Σ_r(Tbmod I_r).
The hypothesis is exactly cref{thm:restricted-tilting}. So
cref{thm:restricted-tilting} produces Θ, once and for all
@@ -654,7 +654,7 @@ 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
+S ⊗ Ind_H^E D ~= 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.
@@ -662,26 +662,26 @@ gives a projective E-module, on which Θ vanishes. Since
## Internal evaluation
-Let mathcal CcalC be a finite symmetric tensor category with
-projective generator P_0. Write A = End_{mathcal C}calC(P_0)^{op},
-so mathcal CcalCsimeq A-mod. An object of
-mathcal CcalCboxtimesRep_k(E) is an object of mathcal CcalC
+Let C be a finite symmetric tensor category with
+projective generator P_0. Write A = End_{C}{op},
+so C~= A-mod. An object of
+C is an object of C
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
+object in W, hence an object of
+C W.calW. This is the internal
+extension Θ_{C} of Θ (Coulembier--Flake
2026, Example 3.2.4 and Proposition 3.2.6; requires only
additivity of Θ, not exactness).
[evaluation at projectives]
-For Qinmathcal CcalC projective and Z an E-object of
-mathcal C,calC,
+For QinC projective and Z an E-object of
+C,
[
- (Hom_{mathcal C}calC(Q,-)boxtimesId_mathcal W)calW)
- (Θ_{mathcal C}calC(Z))
- cong Θ(Hom_{mathcal C}calC(Q,Z)).
+ (Hom_{C},-)boxtimesId_W)
+ (Θ_{C}
+ ~= Θ(Hom_{C},Z)).
]
@@ -698,21 +698,21 @@ Hom_A(Q,-).
Give X^{ ⊗ q} the E-action coming from regular permutation
of tensor positions. Define
- Phi(X) = Θ_{mathcal C}calC(X^{ ⊗ q})
- in mathcal CcalCboxtimes mathcal W.calW.
+ Phi(X) = Θ_{C}{ ⊗ q})
+ in C W.
[monoidal structure and semilinearity]
Phi is additive, strong symmetric monoidal, strong k-linear
-after twisting scalars by Frobenius amapsto a^q, and satisfies
+after twisting scalars by Frobenius a|-> 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}~= 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).
@@ -727,8 +727,8 @@ calculation on scalars.
Combining cref{eq:phi} and cref{lem:projective-evaluation}:
- (Hom_{mathcal C}calC(Q,-)boxtimesId_mathcal W)(calW)(Phi(X))
- cong Θ(Hom_{mathcal C}calC(Q,X^{ ⊗ q}))
+ (Hom_{C},-)boxtimesId_W)(
+ ~= Θ(Hom_{C},X^{ ⊗ q}))
for every projective Q.
@@ -736,14 +736,14 @@ for every projective Q.
[Phi exact iff image nonzero]
-If there are projective objects P,Qinmathcal CcalC with
-Θ(Hom_{mathcal C}calC(Q,P^{ ⊗ q}))ne0, then Phi is
+If there are projective objects P,QinC with
+Θ(Hom_{C},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
+sequence in C, tensoring by P splits it because tensor
products of projectives with arbitrary objects are projective.
The additive Phi preserves this split sequence. Monoidality
identifies the result as the original image sequence tensored by
@@ -758,7 +758,7 @@ 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
+so that M(r) := Hom_{C},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.
@@ -782,11 +782,11 @@ 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
+z>]); and a nondegenerate pairing ab|-> 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
+N_Hmne0, the map kHto M, a|-> 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.
@@ -798,17 +798,17 @@ implies the total norm iso (same argument).
[independent actions multiply]
-Let finite p-groups H_i act on objects X_iinmathcal CcalC
+Let finite p-groups H_i act on objects X_iinC
(i=1,...,s). If omega(X_i) is projective over kH_i for
-exact faithful omega = Hom_{mathcal C}calC(P,-), then
+exact faithful omega = Hom_{C},-), 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}~= X_H ⊗ Y_K and
+(X ⊗ Y)^{Htimes K}~= 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
@@ -819,7 +819,7 @@ kH for any finite group, so projectivity is preserved.
## Existence of a minimal r
-Let E_a=(mathbb Z/p)^abbZ/p)^a act on p^a positions by its regular
+Let E_a=(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.
@@ -854,7 +854,7 @@ 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
+kmathfrak S_dfrakS_dtoEnd_{C}{ ⊗ 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
@@ -879,9 +879,9 @@ 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) :
+ bigl{[alpha]inP^{r-1}(k) :
D not free over k[t]/(t^p),
- tmapstotextstylesum_ialpha_i(g_i-1)
+ t|->textstylesum_ialpha_i(g_i-1)
bigr}.
]
Two standard facts (Benson 2017; Coulembier--Flake 2025):
@@ -895,18 +895,18 @@ Two standard facts (Benson 2017; Coulembier--Flake 2025):
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.
+F_p-linearlybbF_p-linearly independent coordinates.
Proof.
-For cin(mathbb F_p^r)^bbF_p^r)^ast, let H=ker c and
+For cin(F_p^r)^, 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
+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
+If sum_i c_ialpha_ine0, then t|-> 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
@@ -917,15 +917,15 @@ 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
+coordinates of every [alpha]insupp(M) are F_p-linearlybbF_p-linearly
independent.
[Steinberg support in coordinates]
-Let lambda_1,...,lambda_rin k be mathbb F_p-linearlybbF_p-linearly
+Let lambda_1,...,lambda_rin k be 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
+g_i|-> u(lambda_i). Write R_lambda for restriction. Then
supp(R_lambda(St_{r-1})) =
{[alpha] : sum_ialpha_ilambda_i^{p^j}=0
@@ -942,20 +942,20 @@ 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
+Since St_{r-1}~=⊗_{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.
+supp(St_0)=P^0(k), consistent.
## Moore's lemma: choosing lambda
[lambda solving prescribed equations]
-Let alpha_1,...,alpha_rin k be mathbb F_p-linearlybbF_p-linearly
-independent. Then there exist mathbb F_p-linearlybbF_p-linearly independent
+Let alpha_1,...,alpha_rin k be F_p-linearlybbF_p-linearly
+independent. Then there exist F_p-linearlybbF_p-linearly independent
lambda_1,...,lambda_rin k with
sum_ialpha_ilambda_i^{p^j} = 0,
@@ -975,16 +975,16 @@ 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
+Row space of B contains no nonzero F_p^r-vector:bbF_p^r-vector: if
+c_i = f(beta_i)inF_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
+cinF_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.
+c=0. So lambda_i are 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).
@@ -999,7 +999,7 @@ R_lambda(St_{r-1}) ⊗ M nonprojective.
Proof.
-Pick [alpha]insupp(M); coordinates are mathbb F_p-linearlybbF_p-linearly
+Pick [alpha]insupp(M); coordinates are 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
@@ -1011,15 +1011,15 @@ R_lambda(St_{r-1}) ⊗ M is nonprojective.
[exact higher Frobenius functor]
-Let mathcal CcalC be a finite symmetric tensor category over k of
+Let C 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)
+ Phi:C CcalCboxtimesVer_{p^r}(k)
]
-with Phi(X)=Θ_{mathcal C}calC(X^{ ⊗ p^r}) and
+with Phi(X)=Θ_{C}{ ⊗ p^r}) and
Phi(af)=a^{p^r}Phi(f). Moreover, for every projective Q the
evaluation (eq:phi-evaluation) holds.
@@ -1027,7 +1027,7 @@ evaluation (eq:phi-evaluation) holds.
Proof.
Let P be the projective generator and r the minimal integer of
cref{prop:first-nonprojective}. Then
-M = Hom_{mathcal C}calC(P,P^{ ⊗ p^r}) is nonprojective with
+M = Hom_{C},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.
@@ -1036,7 +1036,7 @@ 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).
+ ~= Θ(R_lambda(St_{r-1})) ⊗ Θ(M).
]
So Θ(M)ne0. Cref{prop:phi-exactness} with P=Q shows
Phi is exact.
@@ -1047,14 +1047,14 @@ 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:C CcalCboxtimesmathcal
+W,calW, q=p^r, W={p^r}(k), semilinear.
Its image closes up on the first factor to the Serre closure of a
Frobenius-exact subcategory.
## Ordinary Frobenius and a mod-p identity
-For Yinmathcal C,calC, let c be the cyclic permutation of the p
+For YinC, 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:
[
@@ -1065,12 +1065,12 @@ Etingof--Ostrik components:
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
+semilinear convention, the F_i(Y) are objects of C
itself. F is not assumed exact.
[mod-p identity in K_0]
-In K_0(mathcal C)/pK_0(calC)/pK_0(mathcal C),calC),
+In K_0(C)/pK_0(calC)/pK_0(C),
[
[Y]^p = sum_{i=1}^{p-1}i[F_i(Y)].
]
@@ -1099,7 +1099,7 @@ An exact strong monoidal sigma-semilinear functor still preserves
Frobenius--Perron dimension: it sends nonzero objects to nonzero
objects (from mathbf 1↪bf1↪ X ⊗ X^*), so
composing with FP dimension gives a positive normalised character on
-K_0(mathcal C);calC); the FP characterization forces it to be the FP
+K_0(C); the FP characterization forces it to be the FP
dimension of the source. In particular
[
d(Phi(X)) = d(X).
@@ -1107,10 +1107,10 @@ dimension of the source. In particular
[first-component inequality]
-Let Sinmathcal CcalC be simple and Aboxtimes B a composition
+Let SinC 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
+Phi(S)~= Aboxtimes B, B is invertible, and
+SinC_{ex} (the subcategory on which the
ordinary Frobenius functor is exact).
@@ -1122,13 +1122,13 @@ 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
+SinC_{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(Hom_{C},S^{ ⊗ q}))
+ ~= B.
]
Θ preserves categorical dimension, so its argument has
categorical dimension equal to that of B, which is nonzero (its
@@ -1143,23 +1143,23 @@ 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}.
+in particular d(F(S))=d(S), so SinC_{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
+C. 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 C_{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}.
+so hits a cycle. Its endpoint is in C_{ex}.
A schematic path is drawn in cref{fig:dimension-descent}.
[!htbp]
@@ -1192,55 +1192,55 @@ 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')<= d(S)};
node[align=center] at (1.83,-2.10)
- {A,B,Cinmathcal C_calC_{ex}[2pt]
+ {A,B,CinC_{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 C_{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
+Let sigma(a)=a^q. For a k-linear category A, let
+A^{[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
+sigma-semilinear functors out of A = k-linear
+functors out of A^{[sigma]}, and
+(A B)^calB)^{[sigma]}
+ = A^{[sigma]}boxtimesmathcal
B^calB^{[sigma]}, so products of sigma-semilinear functors are
sigma-semilinear.
-Each Ver_{p^s}(k) is defined over mathbb F_p;bbF_p; scalar transport
-along bmapsto b^{p^a} is an exact strong symmetric sigma^a-
+Each Ver_{p^s}(k) is defined over F_p; scalar transport
+along b|-> b^{p^a} is an exact strong symmetric sigma^a-
semilinear autoequivalence J_{sigma^a} of every
Ver_{p^s}(k).
Define Psi_0=Id, Psi_1=Phi, and
Psi_{j+1} = (Phiboxtimes J_sigma^{boxtimes j})circPsi_j
- :mathcal CcalCtomathcal CcalCboxtimesmathcal W^calW^{boxtimes(j+1)}.
+ :C CcalCboxtimesW^{boxtimes(j+1)}.
By induction each Psi_j is exact, strong symmetric monoidal,
and sigma^j-semilinear.
[first components after N iterations]
-Let Psi_N :mathcal CcalCtomathcal CcalCboxtimes
-mathcal W^calW^{boxtimes N} be as above. Then every composition
+Let Psi_N :C CcalCboxtimes
+W^{boxtimes N} be as above. Then every composition
factor Aboxtimes B_1boxtimes...boxtimes B_N of
-Psi_N(X) has Ainmathcal C_calC_{ex}.
+Psi_N(X) has AinC_{ex}.
Proof.
-Exactness of Phi plus the path bound on mathcal CcalC (any first
+Exactness of Phi plus the path bound on C (any first
component of one iteration is a first component of one further
edge); induction on N, starting with simple objects (where the
path argument applies) and finishing with composition series.
@@ -1248,84 +1248,84 @@ Autoequivalences on other factors send simples to simples and
don't touch the first component.
-Let mathcal D⊆calD⊆mathcal CcalC be the full subcategory whose
-simple composition factors lie in mathcal C_calC_{ex}. This
-is the Serre closure of mathcal C_calC_{ex}.
+Let D⊆calD⊆C be the full subcategory whose
+simple composition factors lie in C_{ex}. This
+is the Serre closure of C_{ex}.
[Serre closure inherits tensor structure]
-mathcal DcalD is closed under tensor products, duals, extensions, and
+D is closed under tensor products, duals, extensions, and
contains the unit; it is finite and has enough projectives; it
carries a rigid symmetric tensor structure inherited from
-mathcal C.calC. Moreover the functor Psi_N factors through
+C. Moreover the functor Psi_N factors through
[
- Psi_N :mathcal CcalCto mathcal DcalDboxtimes
- mathcal W^calW^{boxtimes N}.
+ Psi_N :C D
+ W^{boxtimes N}.
]
Proof.
Closed under tensor products: biexactness gives tensor-product
-filtrations, and mathcal C_calC_{ex} is a tensor
+filtrations, and C_{ex} is a tensor
subcategory closed under subquotients (Etingof--Ostrik 2021,
-Prop 7.4). Duality preserves mathcal DcalD by exactness.
+Prop 7.4). Duality preserves D by exactness.
Finiteness, projectives: concretely, write
-mathcal CcalCsimeq H-mod with H basic finite-dimensional;
+C~= 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
+non-C_{ex} simples; then
+D~= (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
+projective dimension, so D is finite with enough
projectives.
-Factoring through mathcal D:calD: cref{lem:first-components}.
+Factoring through D:calD: cref{lem:first-components}.
% =================================================================
-# Finishing the proof: a fiber functor on mathcal DcalD
+# Finishing the proof: a fiber functor on D
% =================================================================
-## The fiber functor on mathcal DcalD
+## The fiber functor on D
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
+C_{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).
+C_{ex}toVer_p(k). If p>2, the Serre closure
+of C_{ex} equals C_{ex}
+(Etingof--Ostrik 2021, Corollary 9.5). So D=
+C_calC_{ex} and we already have E:D
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
+The fiber functor C_{ex}toVer_2(k)=Vec_k is a
+k-linear fiber functor. But C_{ex}⊆
+D contains every simple of D by definition,
+and C_{ex} is Frobenius-exact. Etingof--Gelaki
(2021, Theorem 2.21(2)) then extends it to
-E:mathcal DcalDtoVer_4^+(k)⊆Ver_4(k).
+E:D+(k)⊆Ver_4(k).
(This is also Etingof--Ostrik 2021, Corollary 9.8.)
So in all cases:
[
- E:mathcal DcalD → Ver_{p^n}(k),
+ E:D → 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
+Set n=max(2,r) and Z = 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.
+strong symmetric monoidal I:W Z.calZ.
-The functors E:mathcal DcalDtomathcal ZcalZ and I:mathcal WcalWto
-mathcal ZcalZ combine, by cref{lem:combine-functors}, into
+The functors E:D ZcalZ and I:W
+Z combine, by cref{lem:combine-functors}, into
[
- H:mathcal DcalDboxtimesmathcal W^calW^{boxtimes N}tomathcal Z,calZ,
+ H:D W^calW^{boxtimes N}toZ,
H(Dboxtimes W_1boxtimes...boxtimes W_N)
= E(D) ⊗ I(W_1) ⊗ ... ⊗ I(W_N).
@@ -1334,13 +1334,13 @@ H is exact and strong symmetric monoidal.
## Killing the scalar twist
-The composite HPsi_N :mathcal CcalCtomathcal ZcalZ is
+The composite HPsi_N :C 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
+sigma). Let J_{sigma^{-N}} be scalar transport on Z
+along b|-> b^{p^{-N}}. Then
[
- mathcal FcalF := J_{sigma^{-N}}circ HcircPsi_N :
- mathcal C→calC→mathcal ZcalZ
+ F := J_{sigma^{-N}}circ HcircPsi_N :
+ C→calC→Z
]
is k-linear, exact, strong symmetric monoidal.
@@ -1348,7 +1348,7 @@ is k-linear, exact, strong symmetric monoidal.
[exact strong monoidal with simple unit is faithful]
-Let F:mathcal CcalCtomathcal DcalD be an exact strong symmetric
+Let F:C DcalD be an exact strong symmetric
monoidal functor between finite symmetric tensor categories. Then
F is faithful.
@@ -1358,12 +1358,12 @@ 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
+imF(f)~= F(imf), and
imfne0, so F(f)ne0.
-Applying cref{lem:faithful} to mathcal FcalF shows mathcal FcalF is
-faithful. Hence mathcal F:calF:mathcal CcalCtoVer_{p^n}(k) is the
+Applying cref{lem:faithful} to F shows F is
+faithful. Hence F:calF:C{p^n}(k) is the
required fiber functor, and cref{thm:main} follows.
% =================================================================
@@ -1376,14 +1376,14 @@ motivation for proving cref{thm:main}.
## Incompressible categories
-Recall that mathcal CcalC is incompressible if every
-k-linear exact strong symmetric monoidal functor from mathcal CcalC
+Recall that C is incompressible if every
+k-linear exact strong symmetric monoidal functor from C
to a symmetric tensor category over k is fully faithful.
[classification of finite incompressible]
Let k be algebraically closed of characteristic p>0. A finite
-symmetric tensor category mathcal CcalC over k is incompressible if
+symmetric tensor category C over k is incompressible if
and only if it is k-linearly equivalent as a symmetric tensor
category to one of the following, according to p:
[
@@ -1407,16 +1407,16 @@ For the Verlinde full and plus categories, see Benson--Etingof--
Ostrik (2023), Theorems 4.71 and 4.72(iii).
Direction ⇒: take a fiber functor
-F:mathcal CcalCtoVer_{p^n}(k) from cref{thm:main}, let
-mathcal IcalI be the image's full subcategory of subquotients.
+F:C{p^n}(k) from cref{thm:main}, let
+I 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,
+functor onto I (Etingof--Gelaki--Nikshych--Ostrik 2015,
Definition 1.8.18, Prop 1.8.19, Section 6.3). Incompressibility
-of mathcal CcalC forces this corestriction to be fully faithful.
+of C forces this corestriction to be fully faithful.
The two FP-dimension inequalities of finiteness (inclusion vs
surjection) then force equality, hence an equivalence.
Benson--Etingof--Ostrik 2023, Corollary 4.61 then restricts
-mathcal IcalI to a full or plus sub-Verlinde at height le n.
+I to a full or plus sub-Verlinde at height le n.
Collecting cases gives the stated list.
@@ -1432,12 +1432,12 @@ finite case of Conjecture~B of
[uniqueness in characteristic two]
Let k be algebraically closed of characteristic two and
-mathcal CcalC a finite symmetric tensor category. Then
-mathcal CcalCtoVer_{2^∞}(k) :=∪_nVer_{2^n}(k) has
+C a finite symmetric tensor category. Then
+C{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
+functor C{2^n}(k) up to monoidal natural
isomorphism.
@@ -1453,33 +1453,33 @@ inclusion; naturality and tensor compatibility hold by faithfulness.
## Tannaka reconstruction over a Verlinde level
-Let mathcal VcalV be a finite symmetric tensor category. The
-fundamental group pi(mathcal V)calV) :=
-Aut^{ ⊗ }(Id_mathcal V)calV) acts
-canonically on Ind(mathcal V).calV). A
-mathcal V-groupcalV-group is an internal affine group scheme G with
-a homomorphism phi:pi(mathcal V)calV)to G whose induced coaction
+Let V be a finite symmetric tensor category. The
+fundamental group pi(V) :=
+Aut^{ ⊗ }(Id_V) acts
+canonically on Ind(V). A
+V-groupcalV-group is an internal affine group scheme G with
+a homomorphism phi:pi(V) G whose induced coaction
matches the canonical one.
[reconstruction over a Verlinde level]
Let k be algebraically closed of characteristic p>0, mathcal
-CcalC finite symmetric tensor, and F:mathcal CcalCto
-mathcal V=calV=Ver_{p^n}(k) a fiber functor (from cref{thm:main}).
-Then there is a mathcal V-groupcalV-group (G,phi) with
-mathcal CcalCsimeqRep(G,phi) over mathcal V,calV, equivalent to
+CcalC finite symmetric tensor, and F:C
+V={p^n}(k) a fiber functor (from cref{thm:main}).
+Then there is a V-groupcalV-group (G,phi) with
+C~=Rep(G,phi) over V, equivalent to
F under the forgetful functor. One may take
G=Aut^{ ⊗ }(F) and
-phi:pi(mathcal V)calV)toAut^{ ⊗ }(F) the
+phi:pi(V){ ⊗ }(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
+Cref{lem:modules-finite-alg} makes C, together with
+V={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
+V-group;calV-group; conversely, forgetting G-structure gives the
inverse functor. Composition is the equivalence.
@@ -1489,7 +1489,7 @@ 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
+n depending on C; 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.