Proof. We recall the unit argument of Deligne–Milne (Deligne and Milne 1982, Proposition 1.17). Let \(i:U\hookrightarrow\mathbf1\) and \(V=\operatorname{coker}i\). The epimorphism \(V\otimes U\leftarrow U\) induced by \(\mathbf1\twoheadrightarrow V\) has zero composite with the monomorphism \(V\otimes i:V\otimes U\to V\). Thus \(V\otimes U=0\). Put \(W=\ker(i^*:\mathbf1\to U^*)\). For a subobject \(j:T\hookrightarrow X\), adjunction and exactness give \[T\otimes U=0 \quad\Longleftrightarrow\quad (i^*\otimes\operatorname{id}_X)j:T\longrightarrow U^*\otimes X \text{ is zero}.\] Consequently \(W\otimes X\) is the largest subobject of \(X\) annihilated by tensoring with \(U\). Taking \(X=\mathbf1,V\) gives \(W\otimes U=0\) and \(W\otimes V\simeq V\). Tensoring \(0\to U\to\mathbf1\to V\to0\) with \(W\) now shows that the natural map \(W\to V\) is an isomorphism. Hence \(\mathbf1=U\oplus W\). Since \(\operatorname{End}(\mathbf1)=k\) has no nontrivial idempotents, \(\mathbf1\) is simple.

Choose representatives \(S_1,\dots,S_t\) of the simple objects and projective objects \(Q_i\) with epimorphisms \(Q_i\twoheadrightarrow S_i\). Set \(P=\bigoplus_iQ_i\). Induction on length, using projectivity to lift an epimorphism onto a simple quotient, shows that every object is a quotient of a finite direct sum of copies of \(P\). Thus \(P\) is a projective generator. Evaluation by \(P\) is exact; it is faithful because a nonzero image of a morphism receives a nonzero map from \(P\), which lifts across the epimorphism onto that image. Writing \(A=\operatorname{End}(P)^{\mathrm{op}}\), finite presentations by copies of \(P\) identify \(\mathcal C\) with \(A\)-modules of finite dimension. Indeed, evaluation is fully faithful on such presentations, and every finite \(A\)-module has a presentation by finite free \(A\)-modules. The algebra \(A\) is finite-dimensional by the Hom-space hypothesis. ◻

Proof. The displayed multilinear functor extends to a right-exact functor by the universal property. Symmetry in \(\mathcal Z\) supplies the tensor constraint by reordering the factors. The constraint, its inverse, and the coherence identities extend from exterior products by that same property. The resulting strong monoidal functor preserves duals. Dualizing a short exact sequence, applying right exactness, and dualizing back proves left exactness. The identical argument, with the componentwise tensor structure, proves the last assertion. ◻

Proof. We first lift \(D\) to a rational \(U\)-module of degree less than \(q\). Let \(H\subset k\) be the image of \(E_r\) under its additive embedding. The vector space \[C_q=k[t]_{<q}=\operatorname{span}_k\{1,t,\ldots,t^{q-1}\}\] is a subcoalgebra of \(k[t]\), with comultiplication \(\Delta f(t,s)=f(t+s)\) and counit \(f\mapsto f(0)\). Evaluation induces a coalgebra map \[\operatorname{ev}_H:C_q\longrightarrow k^H=\mathcal O(E_r).\] It is bijective: a polynomial of degree less than \(q\) vanishing at the \(q\) distinct elements of \(H\) is zero, and both spaces have dimension \(q\). Transporting the \(\mathcal O(E_r)\)-coaction of \(D\) across this isomorphism and then including \(C_q\subset k[t]\) gives a rational \(U\)-module \(\widehat D\) of degree less than \(q\). Its restriction is \(D\). This is a lift of individual representations; we do not require the lifting operation to preserve tensor products.

The orbit map \(G\to\mathbb A^2\setminus\{0\}\), \(g\mapsto ge_1\), is a right \(U\)-torsor. Indeed its fibres are right \(U\)-cosets, and on the opens \(x\ne0\) and \(y\ne0\) it has the respective sections \[\begin{pmatrix}x&0\\y&x^{-1}\end{pmatrix},\qquad \begin{pmatrix}x&-y^{-1}\\y&0\end{pmatrix}.\] Form the associated vector bundle \[\mathcal E=G\times^U\widehat D, \qquad (g,d)\sim(gu,u^{-1}d),\] on this punctured plane, with \(G\) acting by left multiplication. The fibre identification \(d\mapsto[1,d]\) at \(e_1\) gives precisely the prescribed \(U\)-action on \(\widehat D\).

We next extend the bundle and its equivariance across the origin. A coherent extension \(\mathcal F\) of \(\mathcal E\) exists on the noetherian affine plane. Its double dual \(\mathcal M=\mathcal F^{**}\) is coherent and reflexive, and still restricts to \(\mathcal E\). On a regular surface a reflexive coherent sheaf is locally free: its stalk has depth two at a closed point, so the Auslander–Buchsbaum formula gives projective dimension zero; the stalks of dimensions zero and one are free as well. These standard extension statements can be found in (The Stacks Project Authors 2026, Tags 0G41, 0AY4, and 0B3N).

Write \(a:G\times\mathbb A^2\to\mathbb A^2\) for the action and \(\pi_2\) for the projection. Over the punctured plane the bundle action is an isomorphism \(\pi_2^*\mathcal M\to a^*\mathcal M\). Both sheaves are vector bundles on the regular variety \(G\times\mathbb A^2\). Their homomorphism sheaf is locally free, and its sections extend uniquely across \(G\times\{0\}\), which has codimension two. This is the usual extension property for reflexive sheaves (The Stacks Project Authors 2026, Tag 0EBJ). Extend the isomorphism and its inverse; their composites are the identity by uniqueness. The identity and cocycle conditions hold on the punctured plane and therefore, by the same uniqueness on \(\mathbb A^2\) and \(G\times G\times\mathbb A^2\), hold everywhere. Thus \(\mathcal M\) is a \(G\)-equivariant vector bundle.

It remains to bound the highest weights of \(M_0\). The line \(\ell=ke_1\) is fixed pointwise by \(U\). For \(a\in k^\times\) put \(h_a=\operatorname{diag}(a,a^{-1})\). The bundle action of \(h_a\) identifies the fibres at \(e_1\) and \(ae_1\), and \[h_a^{-1}u(t)h_a=u(a^{-2}t).\] Consequently the \(U\)-action on every nonzero fibre of \(\mathcal M|_\ell\) has degree less than \(q\).

To pass this bound to the origin, let \(A=\Gamma(\ell,\mathcal End(\mathcal M|_\ell))\). Since \(U\) acts trivially on the base \(\ell\), its bundle action is an element of \(A\otimes_k k[t]\), say \(\sum_{j=0}^{N} b_jt^j\). For \(j\geq q\), the section \(b_j\) vanishes on every nonzero fibre. The field \(k\) is infinite and \(\mathcal End(\mathcal M|_\ell)\) is locally free, so \(b_j=0\). Specialization at zero therefore gives an action on \(M_0\) of degree less than \(q\). This bound passes to submodules and quotients by writing the action in a basis adapted to a submodule.

Finally, if \(L(h)\) is the simple \(G\)-module of highest weight \(h\), its \(U\)-action has degree exactly \(h\). To see this, write \(h=\sum_i h_ip^i\) with \(0\leq h_i<p\). Steinberg’s tensor product theorem, in the form recalled in (Doty and Henke 2005, Lemma 1.1 and the proof of Theorem 2.1), gives \[L(h)\cong\bigotimes_i \bigl(\operatorname{Sym}^{h_i}(k^2)\bigr)^{(i)}.\] Here \((i)\) denotes Frobenius twist. The action on the \(i\)th factor has degree \(h_ip^i\). On the tensor product, the coefficient from \(\bigotimes_i e_2^{h_i}\) to \(\bigotimes_i e_1^{h_i}\) is \(t^h\), so the upper bound \(h\) is attained. Hence every simple factor \(L(h)\) of \(M_0\) satisfies \(h<q\), as required. The argument applies unchanged when \(p=2\). ◻

Proof. Tensoring a composition series of \(M\) by \(\operatorname{St}_m\) and splicing Weyl and costandard filtrations reduces the assertion to \(M=L(h)\). We induct on \(m\). When \(m=0\), the inequality \(h<p\) gives \(\operatorname{St}_0\otimes L(h)=T(h)\). For \(m>0\), write \(h=d+ph'\) with \(0\leq d<p\) and \(h'<p^m\). Steinberg’s tensor product theorem gives \[\operatorname{St}_m\otimes L(h) \simeq L(d)\otimes\operatorname{St}_1\otimes \bigl(\operatorname{St}_{m-1}\otimes L(h')\bigr)^{(1)}.\] By induction, the module in parentheses is a direct sum of tilting modules \(T(b)\). Formula (1), applied with \(a=p-1\), shows that tensoring its Frobenius twist with \(\operatorname{St}_1\) is tilting. Tensoring once more with \(L(d)=T(d)\) preserves that property. ◻

Proof. We first construct compatible equivariant trivializations on every infinitesimal neighbourhood of \(0\). Let \(A=k[x,y]\), let \(\mathfrak m=(x,y)\), and let \(N=\Gamma(V,\mathcal N)\), a possibly infinite-dimensional rational \(G\)-module. Its quotients \(N/\mathfrak m^jN\) are finite dimensional. Local freeness gives canonical \(G\)-isomorphisms \[ \mathfrak m^jN/\mathfrak m^{j+1}N \cong (\mathfrak m^j/\mathfrak m^{j+1})\otimes_k T \cong \operatorname{Sym}^j(V^*)\otimes_k T \qquad(j\geq 0). \tag{2}\]

Recall that a good filtration has costandard successive quotients, and a Weyl filtration has standard successive quotients. For \(G=\mathrm{SL}_2\), the costandard module of highest weight \(a\) is \(\nabla(a)=\operatorname{Sym}^a(V^*)\). Tensor products of two such modules have good filtrations. Indeed, for \(a,b\geq1\), multiplication of polynomials gives the exact sequence \[ 0\longrightarrow \operatorname{Sym}^{a-1}(V^*)\otimes\operatorname{Sym}^{b-1}(V^*) \xrightarrow{\;d\;} \operatorname{Sym}^{a}(V^*)\otimes\operatorname{Sym}^{b}(V^*) \longrightarrow \operatorname{Sym}^{a+b}(V^*) \longrightarrow0. \tag{3}\] Here \(d=x_1y_2-y_1x_2\) is the determinant in two copies of \(V^*\). Multiplication by \(d\) is injective, its image lies in the kernel, and dimensions prove exactness. Induction on \(\min(a,b)\), starting with a trivial factor, gives the assertion. Refining filtrations then proves the same tensor-product assertion for arbitrary modules with good filtrations. This argument is unchanged in characteristic two.

The module \(T\) has both a Weyl filtration and a good filtration. Consequently, the target of the next Ext group has a good filtration, and the standard–costandard Ext vanishing theorem of Cline–Parshall–Scott–van der Kallen gives \[ \operatorname{Ext}^1_G \bigl(T,\operatorname{Sym}^j(V^*)\otimes T\bigr)=0 \qquad(j\geq0). \tag{4}\] We use Ext in the category of rational \(G\)-modules; the vanishing follows from \(\operatorname{Ext}^1_G(\Delta(a),\nabla(b))=0\) by induction on the two filtrations; see (Riche, n.d., chap. 1, Proposition 2.6, and Chapter 4, Proposition 1.1).

Starting with the identity \(s_1:T\to N/\mathfrak mN\), lift successively to \(G\)-maps \[s_j:T\longrightarrow N/\mathfrak m^jN\qquad(j\geq1)\] such that \(s_{j+1}\) reduces to \(s_j\). Pull back the short exact sequence \[0\longrightarrow \operatorname{Sym}^j(V^*)\otimes_k T \longrightarrow N/\mathfrak m^{j+1}N \longrightarrow N/\mathfrak m^jN\longrightarrow0\] along \(s_j\). By (4) the resulting extension of \(T\) splits equivariantly; a splitting gives the lift \(s_{j+1}\). Extending \(s_j\) by scalars gives an equivariant \(A/\mathfrak m^j\)-linear map \[(A/\mathfrak m^j)\otimes_k T\longrightarrow N/\mathfrak m^jN.\] Both modules are free of rank \(\dim_kT\) over the local Artinian ring \(A/\mathfrak m^j\), and the map reduces to the identity on the residue field. Nakayama’s Lemma therefore makes it an isomorphism. Taking the inverse limit produces an isomorphism \[ k[[x,y]]\otimes_kT\xrightarrow{\sim}\widehat N \tag{5}\] compatible with the \(G\)-action at every finite order.

The line \(y=0\) is fixed scheme-theoretically by \(U\). Restrict (5) to this line and then set \(K=k((x))\). This gives a rational \(U_K\)-isomorphism \[\varphi:T_K\xrightarrow{\sim}\mathcal N_{xe_1}.\] To justify the group-scheme assertion explicitly, choose a polynomial frame for \(\mathcal N\) on the line; this is possible because a finite projective \(k[x]\)-module is free. The action is a matrix \(C(x,t)\in\mathrm{GL}_{\dim T}(k[x,t])\), and the action on \(T\) is a matrix \(B(t)\in\mathrm{GL}_{\dim T}(k[t])\). If \(H(x)\) is the matrix of the restricted formal isomorphism, equivariance at every finite order says \[C(x,t)H(x)=H(x)B(t)\pmod{x^j} \qquad\text{for every }j\geq1.\] These matrices have uniformly bounded degree in \(t\), so coefficientwise \(x\)-adic separation gives the equality in \(k[[x]][t]\), and hence over \(K[t]\).

We must compare the two fibres without changing the parameter of \(U\). Put \(h=\operatorname{diag}(x,x^{-1})\in G(K)\), and let \(b_h:\mathcal N_{e_1,K}\to\mathcal N_{xe_1}\) be bundle transport. Write \(\rho_T\) for the \(G\)-action on \(T\) and \(\rho_{e_1},\rho_x\) for the two fibre actions of \(U\). Since \(h u(t)h^{-1}=u(x^2t)\), equivariance gives \[b_h\rho_{e_1}(u(t))=\rho_x(u(x^2t))b_h, \qquad \rho_T(h)\rho_T(u(t))=\rho_T(u(x^2t))\rho_T(h).\] It follows that \[ f=b_h^{-1}\varphi\rho_T(h):T_K\xrightarrow{\sim}\mathcal N_{e_1,K} \tag{6}\] intertwines the original \(U\)-actions. Thus restriction of \(f\) to any fixed finite subgroup of \(U(k)\) uses that subgroup’s original embedding.

Finally, this isomorphism descends to \(k\). In fixed \(k\)-bases, the intertwiner space \[L=\operatorname{Hom}_U(T,\mathcal N_{e_1})\] is the kernel of finitely many linear equations over \(k\): compare the coefficients of \(t\) in the two polynomial action matrices. Hence its scalar extension to \(K\) is the intertwiner space over \(K\). Equation (6) shows that the determinant polynomial on \(L\) is nonzero. Since \(k\) is infinite, this polynomial is nonzero at some point of \(L(k)\), giving the required isomorphism. At no point have we asserted that the formal trivialization algebraizes on the plane. ◻

Proof of Theorem 4. Given \(D\), choose \(\mathcal M\) as in Proposition 5 and put \(M_0=\mathcal M|_0\). Lemma 6 makes \(\operatorname{St}_{r-1}\otimes M_0\) tilting. Apply Lemma 7 to \(\mathcal N=\operatorname{St}_{r-1}\otimes_k\mathcal M\). Its origin fibre is the tilting module \(T=\operatorname{St}_{r-1}\otimes_kM_0\). By Proposition 5, the fibre of \(\mathcal N\) at \(e_1\) restricts to \(R(\operatorname{St}_{r-1})\otimes D\) along the fixed embedding \(E_r\hookrightarrow U(k)\). The comparison therefore gives \[R(T)\cong R(\operatorname{St}_{r-1})\otimes D,\] as required. This comparison also applies when \(r=1\), with \(\operatorname{St}_0=k\). ◻

Proof. If \(T\bmod I_r=0\), stable restriction shows that \(R(T)\) is projective. Thus nonprojectivity makes \(T\bmod I_r\) nonzero; the fully faithful functor \(\Sigma_r\) and (7) give the first assertion.

Stable restriction also shows that \(R(\operatorname{St}_r)\) is projective. Its dimension is \(q\), so it is the regular module: projectives over the local algebra \(kE\) are free. Equation (7) therefore shows that \(\Theta\) kills all projectives. Set \(S=R(\operatorname{St}_{r-1})\). Its dimension \(p^{r-1}\) is strictly smaller than \(q\), so it is nonprojective and \(\Theta(S)\ne0\).

Let \(H<E\) have rank \(s<r\). For \(s>0\), stable restriction at level \(s\) shows that \(\operatorname{Res}^E_H S\) is projective, because \(\operatorname{St}_{r-1}\in I_s\). The assertion is automatic for \(H=1\). For a \(kH\)-module \(D\), the tensor identity gives \[S\otimes\operatorname{Ind}_H^E D \simeq\operatorname{Ind}_H^E (\operatorname{Res}^E_H S\otimes D),\] which is projective. Applying \(\Theta\) gives \(\Theta(S)\otimes\Theta(\operatorname{Ind}_H^E D)=0\). Tensoring by a nonzero object of a tensor category is faithful, so \(\Theta(\operatorname{Ind}_H^E D)=0\). ◻

Proof. The \(A\)-module corresponding to \(Q\) is a direct summand of \(A^m\) for some \(m\). Thus \(\operatorname{Hom}_A(Q,-)\) is obtained from \(m\) copies of the underlying object by applying a split idempotent. An additive functor preserves finite direct sums and split idempotents. Applying this observation to \(\Theta\) gives (8), including its naturality. ◻

Proof. Equation (10) implies \(\Phi(P)\ne0\). Tensoring a short exact sequence by \(P\) makes it split, since tensor products of projectives with arbitrary objects are projective. The additive functor \(\Phi\) preserves this split sequence. Its monoidal structure identifies the result with the image sequence tensored by \(\Phi(P)\). Tensoring by the nonzero object \(\Phi(P)\) is exact and faithful, and therefore reflects exactness. Thus the image sequence is exact. This argument is the nonzero-projective-image criterion of (Etingof and Ostrik 2021, Proposition 7.6 and Remark 7.7(i)), applied to the higher construction as in (Coulembier and Flake 2026, Proposition 5.2.4). ◻

Proof. For the regular module \(kG\), both coinvariants and invariants are one-dimensional, generated by \([1]\) and \(\sum_{g\in G}g\), respectively; the norm takes the first generator to the second. The assertion therefore holds for free modules and their direct summands.

For the converse, \(kH\) is a local self-injective algebra. Its unique simple module is the trivial module, and its socle is the line generated by \(N_H=\sum_{h\in H}h\). The augmentation ideal is nilpotent by induction on \(|H|\): choose a central element \(z\) of order \(p\), use \((z-1)^p=0\), and pass to \(k[H/\langle z\rangle]\). This proves locality. The nondegenerate associative pairing that takes \((a,b)\) to the coefficient of \(1\) in \(ab\) identifies \(kH\) with its vector-space dual as a module, proving self-injectivity. The socle is the space of left \(H\)-invariants in the regular module, hence the stated line. Every nonzero left ideal meets the socle. Thus, if \(N_Hm\ne0\), the map \(kH\to M\), \(a\mapsto am\), is injective and splits by self-injectivity. Removing free summands repeatedly gives \(M\simeq(kH)^a\oplus M'\) with \(N_HM'=0\). If \(M'\ne0\), then \(M'_H\ne0\) by Nakayama’s Lemma, so its zero norm map cannot be an isomorphism. An isomorphic norm therefore forces \(M'=0\). ◻

Proof. By Lemma 11, each evaluated norm \(\omega(\nu_{H_i}(X_i))\) is an isomorphism. Exact faithfulness of \(\omega\) implies that each internal norm is an isomorphism. Exactness of tensor products gives the natural identifications \[(X\otimes Y)_{H\times K}\simeq X_H\otimes Y_K, \qquad (X\otimes Y)^{H\times K}\simeq X^H\otimes Y^K\] when \(H\) acts on \(X\) and \(K\) acts on \(Y\). One obtains these identities by taking kernels or cokernels for one group at a time. Under them, \(\nu_{H\times K}(X\otimes Y)=\nu_H(X)\otimes\nu_K(Y)\). Iteration proves that the product-group norm is an isomorphism. Applying \(\omega\) and Lemma 11 proves the first claim. Restriction preserves projectivity because a finite group algebra is free over the group algebra of any subgroup. ◻

Proof. Suppose first that every \(M(a)\) is projective. Fix \(d\ge1\) and an elementary abelian \(p\)-subgroup \(H\subseteq\mathfrak S_d\). On each \(H\)-orbit, the action is regular for a quotient \(H/K_i\), because \(H\) is abelian. Choose identifications \(H/K_i\simeq E_{a_i}\), allowing \(a_i=0\) for singleton orbits. Regrouping tensor factors identifies \(P^{\otimes d}\) with the tensor product of the corresponding orbit powers. Each evaluated orbit power is projective for its quotient group, by the assumption when \(a_i>0\), and trivially when \(a_i=0\). Lemma 12 applies to the independent product of the quotient groups. The homomorphism \(H\to\prod_iH/K_i\) is injective, since the permutation action of \(H\) is faithful. Restriction along it shows that \(\omega(P^{\otimes d})\) is projective over \(kH\). Chouinard’s projectivity theorem now implies that \(\omega(P^{\otimes d})\) is projective over \(k\mathfrak S_d\) (Chouinard 1976); see also (Benson 2015, 20).

The tensor power of \(P\twoheadrightarrow\mathbf1\) is an \(\mathfrak S_d\)-equivariant epimorphism to the unit with its trivial action. Exact evaluation, followed by coinvariants, therefore gives an epimorphism \[\omega(P^{\otimes d})_{\mathfrak S_d} \twoheadrightarrow\omega(\mathbf1)\ne0.\] Lemma 11 shows that the total symmetrizer \(b_d=\sum_{\sigma\in\mathfrak S_d}\sigma\) acts nontrivially on \(\omega(P^{\otimes d})\), and hence on \(P^{\otimes d}\).

On the other hand, the action maps \[k\mathfrak S_d\longrightarrow \operatorname{End}_{\mathcal C}(P^{\otimes d})\] cannot all be injective. To see this without any dimension theory, write \(\ell(X)\) for the length of \(X\) and put \(b=\max_i\ell(S_i\otimes P)\). Exactness gives \(\ell(P^{\otimes d})\le\ell(P)b^{d-1}\). Since the simples have endomorphism field \(k\), induction on two composition series gives \(\dim_k\operatorname{Hom}(X,Y)\le\ell(X)\ell(Y)\). Thus \(\dim_k\operatorname{End}(P^{\otimes d})\) grows at most exponentially, whereas \(\dim_k k\mathfrak S_d=d!\).

For clarity, the kernels form a tensor ideal in the permutation category \(\mathsf{Sym}\): its objects are the nonnegative integers, its endomorphism algebra at \(d\) is \(k\mathfrak S_d\), other morphism spaces are zero, and tensor product is addition with the usual block inclusions of symmetric groups. The symmetry of \(\mathcal C\) gives a monoidal functor \(\mathsf{Sym}\to\mathcal C\), \(d\mapsto P^{\otimes d}\), and the kernels form its nonzero kernel ideal. By (Coulembier et al. 2023, Lemma 4.10), every nonzero tensor ideal in \(\mathsf{Sym}\) contains some total symmetrizer \(b_d\) in positive characteristic. Its proof uses Kleshchev’s faithfulness result in the appendix to that paper (Coulembier et al. 2023, Appendix A, Proposition A.1). This contradicts the preceding nonvanishing. The least \(r\) therefore exists.

Finally, let \(H<E_r\) be proper, of rank \(a<r\). Its action on the regular \(E_r\)-set is a disjoint union of regular \(H\)-orbits. Each orbit power has projective evaluation by minimality when \(a>0\), and trivially when \(a=0\). Apply Lemma 12 to the independent orbit groups, then restrict to their diagonal copy of \(H\). This proves the last assertion, including \(r=1\). ◻

Proof. Put \(Z=Y^{\otimes p}\) and \(K_j=\ker D\cap\operatorname{im}D^j\), for \(0\leq j\leq p\). The exact sequences \[0\longrightarrow K_j\longrightarrow\operatorname{im}D^j \xrightarrow{D}\operatorname{im}D^{j+1}\longrightarrow0 \qquad(0\leq j<p)\] give \([Z]=\sum_{j=0}^{p-1}[K_j]\). Since \(K_0\supseteq\cdots\supseteq K_p=0\), summing the successive quotients in these filtrations gives the integral identity \[[Z]=\sum_{i=1}^{p-1}i[F_i(Y)]+p[K_{p-1}].\] Reduction modulo \(p\) proves the first assertion. For the second, raise this identity to successive \(p\)-th powers in the commutative ring \(K_0(\mathcal C)/p\), and apply the first assertion to each concrete object \(F_{i_s}\cdots F_{i_1}(Y)\). This argument does not apply a nonexact functor to a Grothendieck-group relation. ◻

Proof. Every nonzero object has dimension at least one. Thus \(d(S)=d(\Phi(S))\geq d(A)d(B)\geq d(A)\). If the two ends are equal, positivity of the dimensions of all composition factors shows that \(\Phi(S)\) has length one and \(d(B)=1\). A simple object of dimension one is invertible, since \(B\otimes B^*\) has dimension one and contains the unit.

Let \(Q_A\) be the projective cover of \(A\). Over the algebraically closed field \(k\), \[\dim_k\operatorname{Hom}_{\mathcal C}(Q_A,X)=[X:A]\] for every \(X\): the functor is exact and has dimension one on \(A\) and zero on every other simple object. Applying (14) to \(S\) gives \[\Theta\bigl(\operatorname{Hom}_{\mathcal C}(Q_A,S^{\otimes q})\bigr) \simeq B.\] A symmetric monoidal functor preserves categorical dimension. This dimension of the representation on the left is its vector-space dimension viewed in \(k\), whereas the categorical dimension of the invertible object \(B\) is nonzero: its product with the dimension of \(B^*\) is one. Consequently, \[ [S^{\otimes q}:A]\not\equiv0\pmod p. \tag{17}\] This uses categorical dimension only to establish the congruence; the real-valued dimension \(d\) remains Frobenius–Perron dimension.

By (16) and (17), there are indices \(i_1,\ldots,i_r\in\{1,\ldots,p-1\}\) such that \(A\) is a composition factor of \(Y_r\), where \(Y_0=S\) and \(Y_t=F_{i_t}(Y_{t-1})\). For every \(t\), Equation (15) and \(d(\ell_{i_t})\geq1\) give \[d(Y_{t-1})\geq d(F(Y_{t-1}))\geq d(Y_t).\] Since \(d(Y_r)\geq d(A)=d(S)\), all inequalities are equalities. In particular \(d(F(S))=d(S)\), so \(S\in\mathcal C_{\mathrm{ex}}\). ◻

Proof. Let \(\mathcal C\) be incompressible and choose a fiber functor \(F:\mathcal C\to\operatorname{Ver}_{p^n}(k)\) by Theorem 1. Let \(\mathcal I\) be the full subcategory of subquotients of objects \(F(X)\), \(X\in\mathcal C\). Because \(\mathcal C\) is finite, \(\mathcal I\) is a finite tensor subcategory and the corestriction \(F:\mathcal C\to\mathcal I\) is surjective in the tensor-category sense (Etingof et al. 2015, Definition 1.8.18, Proposition 1.8.19, and Section 6.3). Incompressibility makes this corestriction fully faithful. The injective and surjective Frobenius–Perron-dimension inequalities for finite tensor categories give, respectively, \[\operatorname{FPdim}(\mathcal C)\leq\operatorname{FPdim}(\mathcal I), \qquad \operatorname{FPdim}(\mathcal C)\geq\operatorname{FPdim}(\mathcal I).\] Equality in either inequality makes the corestriction an equivalence (Etingof et al. 2015, Propositions 6.3.3 and 6.3.4).

Corollary 4.61 of (Benson et al. 2023) now restricts \(\mathcal I\) to the full and plus higher Verlinde subcategories at heights \(m\le n\). For odd \(p\), the bottom cases are included using the explicit convention \(\operatorname{Ver}_1=\operatorname{sVec}_k\) and \(\operatorname{Ver}_1^+=\operatorname{Vec}_k\) of (Coulembier et al. 2024, sec. 1.3). With the low-level identifications above, these are exactly the displayed possibilities.

Conversely, the full and plus categories in the list are incompressible for symmetric tensor functors by (Benson et al. 2023, Theorem 4.71 and Remark 4.72(iii)). For \(\operatorname{Vec}_k\), fullness follows from \(\operatorname{End}(\mathbf1)=k\). For \(\operatorname{sVec}_k\) in odd characteristic, a symmetric tensor functor sends the odd invertible generator to an odd invertible object. It cannot be isomorphic to the unit, since their self-symmetries are respectively \(-1\) and \(1\). The functor is therefore fully faithful on the two simple objects, and hence on all their finite direct sums. ◻

Proof. Theorem 1 supplies a fiber functor to a finite stage, and composition with the tower inclusion supplies a fiber functor to the union. By (Coulembier et al. 2024, Theorem 9.4.1), an arbitrary tensor category in characteristic two has at most one \(k\)-linear exact symmetric tensor functor to \(\operatorname{Ver}_{2^\infty}(k)\) up to monoidal natural isomorphism. Applying this to \(\mathcal C\) proves uniqueness in the union. For two fiber functors to the same finite stage, this isomorphism lifts through the fully faithful tower inclusion; its naturality and tensor compatibility hold at that stage by faithfulness. This proves the fixed-stage assertion as well. ◻

Proof. The finite-category conventions, including the simple unit supplied by Lemma 2, make \(\mathcal C\) and \(\mathcal V\) pretannakian categories in the terminology of (Coulembier et al. 2024). The chosen \(F\) is a tensor functor in that paper’s sense. Its Theorem 4.2.1 assigns to \(F\) the \(\mathcal V\)-group displayed above, and the inverse assignment is the forgetful functor from \(\operatorname{Rep}(G,\phi)\). The resulting equivalence over \(\mathcal V\) is precisely the one stated here. ◻