Central isogenies from groups with trivial center

A central isogeny from an affine group with trivial scheme-theoretic center is an isomorphism. In particular, adjoint semisimple groups admit no nontrivial central-isogeny quotients. The result itself needs neither semisimplicity, smoothness, nor finite type of the ambient groups.

For a coordinate morphism f : H ⟶ K, the source group is Spec K: it is the center of K that must be trivial. Centrality puts the kernel inside that center, and the trivial-kernel criterion for isogenies then applies. Both the center and kernel are scheme-theoretic, so the statement also rules out infinitesimal central kernels in positive characteristic.

References

• J. S. Milne, Algebraic Groups (2017), §21.

theorem TauCeti.CommHopfAlgCat.IsCentralIsogeny.isIso_of_centerDefiningIdeal_eq_augmentation {k : Type u} [Field k] {H K : CommHopfAlgCat k} {f : H ⟶ K} (hf : IsCentralIsogeny f) (hK : centerDefiningIdeal K = HopfIdeal.augmentation k ↑K) : CategoryTheory.IsIso f

A central isogeny from an affine group with trivial scheme-theoretic center is an isomorphism. Coordinate arrows reverse, so the trivial-center hypothesis is on K.