Flatness of integral Kostant toral closures

The integral Kostant toral closure is the subgroup of a general linear group generated by its represented root subgroups and weight torus. The coordinate algebras of these generators are torsion-free over ℤ, so the common-kernel quotient defining the carrier is torsion-free and therefore flat over ℤ.

This controls scalar torsion in the integral carrier before comparison with the pinned simply connected group scheme. It does not assert that its special fibers are smooth, or that formation of the generated subgroup commutes with specialization.

References

• J. E. Humphreys, Linear Algebraic Groups, §26.
• The Stacks Project, Tag 0AUW, for flatness of torsion-free modules over Dedekind domains.

The integral carrier is TauCeti.UniversalEnvelopingAlgebra.kostantToralDefiningIdeal; flatness follows from the common-kernel scalar-torsion criterion.

instance TauCeti.UniversalEnvelopingAlgebra.isTorsionFree_kostantToralCoordinateHopfAlgebra {L : Type u} [LieRing L] [LieAlgebra ℚ L] {I : Type w} {κ : Type} [Finite κ] {V : Type} [AddCommGroup V] [Module ℚ V] (e : I → L) (h : κ → L) (ρ : UniversalEnvelopingAlgebra ℚ L →ₐ[ℚ] Module.End ℚ V) (M : AddSubgroup V) (hM : ∀ u ∈ kostantForm e h, ∀ m ∈ M, (ρ u) m ∈ M) (hnil : ∀ (i : I), IsNilpotent (ρ ((UniversalEnvelopingAlgebra.ι ℚ) (e i)))) {n : ℕ} (b : Module.Basis (Fin n) ℤ ↥M) (wt : Fin n → κ → ℤ) : Module.IsTorsionFree ℤ ↑(CommHopfAlgCat.quotient (GeneralLinear.coordinateHopfAlgebra ℤ n) (kostantToralDefiningIdeal e h ρ M hM hnil b wt))

The coordinate algebra of an integral Kostant toral closure has no scalar torsion. In particular it is flat over ℤ.