Proof of Tate’s conjecture over finite fields

We start by showing that (1) is injective. Take an u ∈ Hom (A,B)⊗ Zl that maps to zero in HomΓ (Tl(A), Tl(B)). Write u = ∑∞ j=0 l uj , uj ∈ Hom(A,B), and [u]n for ∑n j=0 l uj . Note that since Hom (A,B) is a Z-module [u]n is in Hom (A,B). Now (since u maps to zero) [u]n is the zero morphism A[l ] → B[l], so it kills the ltorsion. As it is well known, this implies the existence of a certain ψn ∈ Hom(A,B) such that [u]n = [l ] ◦ ψn. It follows that [u]n belongs to l Hom (A,B) ⊆ l (Hom (A,B)⊗ Zl), and since clearly u−un is in l n (Hom (A,B)⊗ Zl) too we find that u lies in l n (Hom (A,B)⊗ Zl) for every n. As Hom (A,B)⊗ Zl is of finite type, this implies that u = 0 as claimed.