Algebraic Torus - Isogenies

Isogenies

An isogeny is a surjective morphism of tori whose kernel is a finite flat group scheme. Equivalently, it is an injection of the corresponding weight lattices with finite cokernel. The degree of the isogeny is defined to be the order of the kernel, i.e., the rank of its structure sheaf as a locally free -module, and it is a locally constant function on the base. One can also define the degree to be order of the cokernel of the corresponding linear transformation on weight lattices. Two tori are called isogenous if there exists an isogeny between them. An isogeny is an isomorphism if and only if its degree is one. Note that if S doesn't have a map to Spec Q, then the kernel may not be smooth over S.

Given an isogeny f of degree n, one can prove using linear algebra on weights and faithfully flat descent that there exists a dual isogeny g such that gf is the nth power map on the source torus. Therefore, isogeny is an equivalence relation on the category of tori. T. Ono pointed out that two tori over a field are isogenous if and only if their weight lattices are rationally equivalent as Galois modules, where rational equivalence means we tensor over Z with Q and get equivalent vector spaces with Galois action. This extends naturally from Galois modules to fpqc sheaves, where Z and Q are constant sheaves rather than plain groups.