On the Frobenius functor for symmetric tensor categories in positive characteristic

Proposition A.1.

Let C be a symmetric tensor category with finitely many simple objects. Let X ∈ C and d i = FPdim ⁢ ( S i ⁢ X ) . Then the series f X ⁢ ( z ) = ∑ i ≥ 0 d i ⁢ z i has radius of convergence 1 unless it is a polynomial.

Proof.

Since d i = 0 or d i ≥ 1 for all i, if f X ⁢ ( z ) is not a polynomial, then its radius of convergence R ≤ 1 . So it suffices to show that R ≥ 1 .

Since FPdim ⁢ ( S i ⁢ gr ⁡ X ) ≥ FPdim ⁢ ( gr ⁡ S i ⁢ X ) = FPdim ⁢ ( S i ⁢ X ) for any filtered object X, it suffices to prove the statement for X being semisimple. Thus it suffices to take X to be the direct sum of all simple objects of 𝒞 . Then S 2 ⁢ X has a filtration whose associated graded is contained in mX for some integer m. Hence

Let f X ± be the sum of all the even (resp. odd) terms of f X . Then inequality (A.1) implies

where ≤ is coefficientwise. Since S 2 ⁢ i + 1 ⁢ X is a quotient of X ⊗ S 2 ⁢ i ⁢ X , we also have

Thus

Let R be the radius of convergence of f X . Inequality (A.2) implies that R ≥ R 1 / 2 . Also R ≥ 1 d 1 (as S i ⁢ X is a quotient of X ⊗ i ), so R ≠ 0 . Thus R ≥ 1 , as claimed. ∎

Corollary A.2.

Let A = ⊕ i ≥ 0 A i be a Z + -graded finitely generated commutative algebra in C with A 0 = 1 , and let f ⁢ ( z ) = ∑ i ≥ 0 FPdim ⁢ ( A i ) ⁢ z i . Then f ⁢ ( z ) has radius of convergence 1 unless it is a polynomial.

Proof.

As before, it is clear that if f is not a polynomial, then its radius of convergence R is ≤ 1 . Thus the statement follows from Proposition A.1 and the fact that A is a quotient of SX for some positively graded object X ∈ 𝒞 . ∎

Corollary A.3.

Let A be a commutative ind-algebra in C such that

is a field, and the pairing

is non-degenerate for each Y ∈ C . Then for each Y ∈ C we have

Proof.

We will work in the category 𝒞 K . Let E ⊂ Hom A ⁡ ( A , Y ⊗ A ) be a finite-dimensional K-subspace. Then we have a natural morphism E ⊗ A → Y ⊗ A which is split by non-degeneracy, in particular injective. Let Z ⊂ A be a subobject such that Z ∈ 𝒞 K (as opposed to Z ∈ Ind ⁡ ( 𝒞 K ) ) and the image of E ⊗ 𝟏 in Y ⊗ A is contained in Y ⊗ Z . Let A Z be the subalgebra of A generated by Z. Then we have an injection E ⊗ A Z → Y ⊗ A Z . Put a filtration on A Z by defining the degree of Z to be 1. Then we have an injection E ⊗ F i - 1 ⁢ A Z → Y ⊗ F i ⁢ A Z (as Y sits in degree zero). Thus we have

where g ⁢ ( z ) := ∑ i ≥ 0 FPdim ⁢ ( F i ⁢ A Z ) ⁢ z i . That is, we have

Now, the algebra gr ⁡ ( A Z ) is finitely generated, and we have

where f ( z ) := ∑ i ≥ 0 FPdim ( gr ( A Z ) i ) z i . Thus by Corollary A.2, the left-hand side of (A.3) converges in the open unit disk, hence can be evaluated at z = 1 - ε for any ε > 0 . Thus we have

Sending ε to zero, we then get FPdim ⁢ ( Y ) - dim ⁡ ( E ) ≥ 0 , as claimed. ∎

Conjecture A.4.

f X ⁢ ( z ) is a rational function in z.

Remark A.5.

Conjecture A.4 clearly holds for 𝒞 = Ver p , since the algebra SL i is finite dimensional for any i > 1 (see, e.g., [17]). Thus it holds for any symmetric tensor category that fibers over Ver p , i.e., for Frobenius exact categories. But we do not know if it holds for the categories 𝒞 n from [2] for general n.

Theorem B.1.

Let C be a finite tensor category and let A ∈ C be an indecomposable algebra. Then A is exact if and only if the following conditions hold:

1. A is simple (i.e., it has no nontrivial two-sided ideals),

2. there exists X ∈ 𝒞 and an embedding of left A - modules

   A * ↪ A ⊗ X .

Proof.

Assume that A is exact. Then A regarded as an A-bimodule is the unit object of the dual category to 𝒞 and hence is simple (see [15, Lemma 3.24]); thus (a) holds. Also injective objects in the category 𝒞 A are projective (see [15, Corollary 3.6]) and any projective is a direct summand of A ⊗ P for some projective object P ∈ 𝒞 ; this implies (b).

Conversely, let A be an algebra satisfying (a) and (b). We would like to show that for any M ∈ 𝒞 A and projective P ∈ 𝒞 the object P ⊗ M ∈ 𝒞 A is projective.

Let Γ ⁢ ( 𝒜 ) be the split Grothendieck group of an additive category 𝒜 ; we also consider Γ ℝ ⁢ ( 𝒜 ) := Γ ⁢ ( 𝒜 ) ⊗ ℤ ℝ . Then Γ ⁢ ( 𝒞 ) is a ring acting on the group Γ ⁢ ( 𝒞 A ) .

Let { X i } i ∈ I be the set of isomorphism classes of simple objects in 𝒞 . For any i let P i ∈ 𝒞 be the projective cover of X i . The following result is standard.

Now let M ∈ 𝒞 A and consider R ⁢ [ M ] ∈ Γ ℝ ⁢ ( 𝒞 A ) . We can write

where 𝒩 is a sum of the classes of indecomposable non-projective objects and 𝒫 is a sum of the classes of indecomposable projective objects. Assume that 𝒩 ≠ 0 . Let P be as in Lemma B.2 (ii). Then [ P ] ⁢ 𝒩 will contain some projective summands; also [ P ] ⁢ 𝒫 is projective. Thus the contribution of projective summands into FPdim ( [ P ] ⁢ R ⁢ [ M ] ) is strictly more than

But on the other hand by Lemma B.3 we have

so the contribution of projective summands is precisely FPdim ( P ) ⁢ FPdim ( 𝒫 ) . This is a contradiction. Thus 𝒩 = 0 , so P i ⊗ M is projective for any i ∈ I and the theorem is proved. ∎

Corollary B.4.

Let C be a tensor subcategory of a finite tensor category D . If A ∈ C is exact, then A regarded as an algebra in D is also exact.

Proof.

Both conditions (a) and (b) of Theorem B.1 are clearly preserved by the inclusion functor 𝒞 ↪ 𝒟 . ∎

Remark B.5.

We recall that an injective (i.e., fully faithful) tensor functor 𝒞 ↪ 𝒟 is an equivalence with a tensor (in particular, abelian) subcategory of 𝒟 , see [14, Proposition 6.3.1]. Thus a fully faithful tensor functor between finite tensor categories sends an exact algebra to an exact one.

Conjecture B.6.

An indecomposable algebra A ∈ 𝒞 is exact if and only if it is simple.

Proposition B.7.

Conjecture B.6 holds if C = Rep H for a finite-dimensional Hopf algebra H.

Proof.

Let A ∈ 𝒞 be a simple algebra, and let us show that the right 𝒞 -module category A ⁢ - ⁢ mod 𝒞 is exact. Since 𝒞 = Rep H , we may view A as an H-module algebra. We have A ⁢ - ⁢ mod 𝒞 = A ⋊ H -mod.

Now let X ∈ A ⁢ - ⁢ mod 𝒞 = A ⋊ H -mod. By [28, Theorem 3.5], the restriction functor

lands in the subcategory of projective A-modules. Thus X is a projective A-module. Hence X ⊗ H = X ⊗ A ( A ⋊ H ) is a projective A ⋊ H -module. Thus tensoring with H takes any object X ∈ A ⁢ - ⁢ mod 𝒞 to a projective object. Hence, A ⁢ - ⁢ mod 𝒞 is an exact right 𝒞 -module category, i.e., A is an exact algebra, as claimed. ∎

Remark B.8.

By using [19, Lemma 3.6] one generalizes Proposition B.7 to the case of quasi-Hopf algebras.