The character criterion for diagonalizable group-scheme isogenies

For finitely generated commutative character groups, groupSchemeMap is a central isogeny exactly when the character homomorphism is injective with finite cokernel. This transports the coordinate-algebra criterion to the existing diagonalizable group-scheme functor. In particular, the surjectivity assertion is scheme-theoretic, and the kernel is central on points valued in every test scheme.

The sufficient direction holds over any commutative base ring; the converse needs a nonzero base. No smoothness assumption or restriction on the characteristic is imposed.

References

• J. S. Milne, Algebraic Groups (2017), §12, especially Theorem 12.9.

theorem TauCeti.DiagonalizableGroup.isCentralIsogeny_groupSchemeMap_of_injective_of_finite_quotient (R : Type u) [CommRing R] {G H : FGCommGrpCat} (φ : G ⟶ H) (hφ : Function.Injective ⇑(FGCommGrpCat.toMonoidHom φ)) [Finite (↑H ⧸ (FGCommGrpCat.toMonoidHom φ).range)] : GroupScheme.IsCentralIsogeny (groupSchemeMap R φ)

An injective homomorphism of finitely generated character groups with finite cokernel induces a central isogeny of diagonalizable group schemes over any commutative ring.

@[simp]

theorem TauCeti.DiagonalizableGroup.isCentralIsogeny_groupSchemeMap_iff (R : Type u) [CommRing R] {G H : FGCommGrpCat} (φ : G ⟶ H) [Nontrivial R] : GroupScheme.IsCentralIsogeny (groupSchemeMap R φ) ↔ Function.Injective ⇑(FGCommGrpCat.toMonoidHom φ) ∧ Finite (↑H ⧸ (FGCommGrpCat.toMonoidHom φ).range)

A morphism of diagonalizable group schemes over a nonzero commutative ring is a central isogeny exactly when its character map is injective with finite cokernel.