Isogeny

An isogeny between connected algebraic group varieties over a field is a surjective group homomorphism that is a morphism of algebraic varieties and has a finite scheme-theoretic kernel (Milne 2017, p. 46). The kernel is understood as a group scheme, which need not be reduced. Equality of dimensions of source and target alone does not suffice. For example, the constant endomorphism of a positive-dimensional algebraic group has kernel equal to its entire source.

A broader convention allows algebraic groups that are not connected and requires a finite scheme-theoretic kernel and an image of the same dimension as the target. Two algebraic groups are called isogenous when they can be joined by a finite chain of isogenies, with arrows allowed in either direction (Milne 2017, p. 126).

For example, on the multiplicative group is an isogeny for every positive integer . Its scheme-theoretic kernel is . In characteristic , the kernel of is a nonreduced group scheme, even though it has only one point over an algebraic closure. Surjectivity here is a property of the morphism of schemes, not a requirement that every -valued point have a -valued preimage.