Results on symmetric tensor categories in positive characteristic and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Finite symmetric tensor categories as module categories — Lemma 1.3. In a finite symmetric tensor category the unit is simple, there is a projective generator P, and Hom(P, -) is an exact faithful equivalence onto the finite-dimensional modules over End(P)^op.
• Frobenius-Perron dimension under tensor functors — Lemma 3.2. Exact tensor functors between finite symmetric tensor categories, linear or semilinear, preserve Frobenius-Perron dimension; a simple object of dimension one is invertible.
• Fibre functors to the Verlinde tower — Theorem 5.1. Every finite symmetric tensor category over an algebraically closed field of characteristic p > 0 admits a fibre functor to Ver_{p^n} for some n, including p = 2. Proved in the eighth lesson, using the named results N1 to N9.
• The unipotent degree of a simple module — Lemma 2.3. The upper unitriangular group acts on the simple SL_2-module L(h) by matrices whose entries are polynomials of degree at most h, with degree exactly h.
• Tensor products of tilting modules for SL_2 — Proposition 3.5. Tensor products of SL_2-modules with good filtrations have good filtrations, and tensor products of tilting modules are tilting, by the determinant exact sequence.
• Steinberg modules and bounded weights — Lemma 5.1. If all composition factors of M have highest weight below p^(m+1), then St_m (x) M is tilting.
• A bundle on the plane with bounded weights — Proposition 4.2. Every representation of (Z/p)^r embedded in the unipotent group is the fibre at (1,0) of an SL_2-equivariant vector bundle on the plane whose origin fibre has highest weights below p^r.
• Comparison with a tilting origin fibre — Lemma 3.1. For an SL_2-equivariant vector bundle on the plane whose origin fibre T is tilting, the fibre at (1,0) is T restricted to the unipotent group.
• The restricted tilting theorem — Theorem 4.1. For every embedding of (Z/p)^r in the unipotent group of SL_2 and every representation D, the restriction of St_(r-1) tensored with D is the restriction of a tilting module (a conjecture of Coulembier and Flake).
• What the Coulembier-Flake functor kills — Lemma 3.1. The symmetric tensor functor Theta from representations of (Z/p)^r to Ver_(p^r) kills projective modules and modules induced from proper subgroups, and does not kill restrictions of tilting modules that are not projective.
• An exactness criterion — Proposition 5.1. The higher Frobenius functor Phi(X) = Theta(X^(x)q) is exact as soon as Theta(Hom(Q, P^(x)q)) is nonzero for some projective objects P and Q. Uses the construction of Coulembier and Flake (named result N5).
• The norm criterion — Lemma 1.1. A representation of a finite p-group is projective exactly when its norm map from coinvariants to invariants is an isomorphism; the norm map of every projective representation of a finite group is an isomorphism.
• Total symmetrizers — Lemma 2.3. Every nonzero tensor ideal of the symmetric group algebras contains a total symmetrizer. Uses Kleshchev's faithfulness theorem (named result N7).
• A first nonprojective tensor power — Proposition 4.1. For a projective generator P there is a least r with Hom(P, P^(x)p^r) not projective over (Z/p)^r; its restrictions to proper subgroups are projective. Uses Chouinard's theorem (named result N6).
• The Moore-matrix selection — Lemma 4.1. For F_p-independent alpha_1, ..., alpha_r there are F_p-independent lambda_1, ..., lambda_r with sum alpha_i lambda_i^(p^j) = 0 for 0 <= j <= r - 2.
• An exact higher Frobenius functor — Theorem 5.1. For every finite symmetric tensor category there are r and an embedding of (Z/p)^r for which the higher Frobenius functor is exact. Uses rank varieties (named result N6).
• A congruence for Frobenius components — Lemma 2.1. In the Grothendieck ring modulo p, [Y]^p = sum i [F_i(Y)] for the components F_i of the Frobenius functor of Etingof and Ostrik.
• Equality of dimensions — Lemma 3.1. If a simple S has a composition factor A (x) B of Phi(S) with FPdim A = FPdim S, then S is Frobenius exact, B is invertible and A is the only first component.
• Finite incompressible symmetric tensor categories — Corollary 2.2. A finite symmetric tensor category is incompressible exactly when it is a higher Verlinde category or its even part (with Vec and sVec at the bottom). Uses the subcategories and incompressibility of the tower (named result N1).
• Reconstruction over Ver_(p^n) — Corollary 4.1. Every finite symmetric tensor category is the representation category of an affine group scheme internal to a higher Verlinde category. Uses the generalized Tannaka theorem of Coulembier, Etingof and Ostrik (named result N10).

Reading order

Nine lessons, to be read in order. The first sets up finite symmetric tensor categories and the Verlinde tower and states the theorem. The second collects the representation theory of SL_2 in characteristic p. The third and fourth prove the restricted tilting theorem with an equivariant vector bundle on the plane. The fifth records the higher Frobenius functors of Coulembier and Flake, the sixth and seventh choose the rank and the embedding that make them exact, the eighth descends to a fibre functor, and the ninth proves the consequences. The deep inputs (the construction of the Verlinde tower, the Coulembier-Flake functor, the Frobenius functor of Etingof and Ostrik, Chouinard's theorem and rank varieties, generalized Tannaka duality) are used without proof, as the named results N1 to N10.