The splitting algebra of a single object

The argument of Deligne §2.11 tensors together a splitting algebra for every object of the category and a section for every short exact sequence. The index family is then as large as the category itself, and the ring of scalars of the result has no dimension bound.

Only one object need be split. Over a simple algebra the free modules on the unit and on the odd line are simple, so a free mixed module is semisimple of finite length and every subquotient of it is again free mixed; the class of split objects is therefore closed under subobjects and quotients as well as sums, tensor products and duals, and a tensor generator drags the whole category into it.

This module supplies the entry point: for a single object, a splitting algebra that is countably presented, which is what makes its scalars a field of countable dimension over the complex numbers, hence the complex numbers themselves.

theorem RS.exists_generator_algebra {C : Type v} [CategoryTheory.SmallCategory C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.SymmetricCategory C] [CategoryTheory.Abelian C] [CategoryTheory.RigidCategory C] [CategoryTheory.MonoidalPreadditive C] (ψ : ℂ ≃+* CategoryTheory.End (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) (P : SchurPackage) (P₀ : SchurPackage) (L : OddLine (CategoryTheory.Ind C)) (X₀ : C) (lam : YoungDiagram) (hkill : SchurKilled P X₀ lam) (hlen : ∀ (Z : C), ∃ (N : ℕ), LengthLE Z N) : ∃ (p : ℕ) (q : ℕ) (𝔸 : CategoryTheory.Ind C) (x : CategoryTheory.MonObj 𝔸) (_ : CategoryTheory.IsCommMonObj 𝔸), CategoryTheory.MonObj.one ≠ 0 ∧ CountablyPresented 𝔸 ∧ Nonempty (freeMod 𝔸 (indOf.obj X₀) ≅ freeMod 𝔸 (L.mix p q))

The splitting algebra of a single object, countably presented. Proposition 2.9 makes the object locally mixed, and the countable descent replaces the witnessing algebra by a countably presented one.