Central isogenies of diagonalizable groups

The morphism D(N) → D(M) induced by a homomorphism p : M →* N is a central isogeny exactly when p is injective with finite cokernel, over a nonzero commutative base ring. The sufficient direction also holds over the zero ring. Neither character group needs to be finitely generated, and there is no smoothness or characteristic restriction: this criterion includes inseparable isogenies.

The finiteness and faithful-flatness criteria for group-algebra maps supply the isogeny conditions. Cocommutativity of the target coordinate algebra makes its kernel central.

References

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

theorem TauCeti.DiagonalizableGroup.isCentralIsogeny_mapDomainBialgHom_of_injective_of_finite_quotient (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) (hp : Function.Injective ⇑p) [Finite (N ⧸ p.range)] : CommHopfAlgCat.IsCentralIsogeny (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p))

An injective character homomorphism with finite cokernel induces a central isogeny of diagonalizable groups over any commutative base ring.

@[simp]

theorem TauCeti.DiagonalizableGroup.isCentralIsogeny_mapDomainBialgHom_iff (R : Type u) [CommRing R] {M N : Type v} [CommGroup M] [CommGroup N] (p : M →* N) [Nontrivial R] : CommHopfAlgCat.IsCentralIsogeny (CommHopfAlgCat.ofHom (MonoidAlgebra.mapDomainBialgHom R p)) ↔ Function.Injective ⇑p ∧ Finite (N ⧸ p.range)

Over a nonzero commutative ring, a morphism of diagonalizable groups is a central isogeny if and only if its character homomorphism is injective with finite cokernel.