kernel of an isogeny and coker of its induced map on the Tate module

Question

In a proof in Milne's note "Abelian Variety" (top on p.52), I saw an equality: $ \mathrm{Ker}(\beta)(l) = \mathrm{Coker}(T_{l}(\beta))$, here $\beta$ is an (separable) isogeny of an abelian variety $A/k$, $l$ is a prime number different from the characteristic of the base field $k$, $(l)$ means the torsion points of order a power of $l$, $T_{l}(\beta)$ is the induced map of $\beta$ on the Tate module of $A$.

I can't figure out why this equality holds. Do we have a natural map for these two groups?

Answer 1

Let $x$ be an $l^n$-torsion point in the kernel of the isogeny. Take any point $y$ such that $l^ny=x$. $l^n\beta(y)=\beta(l^ny)=\beta(x)=0$, so $\beta(y)$ is an $l^n$-torsion point. This is well-defined up to the image of an $l^n$-torsion point under the isogeny, since a division by $l^n$ is well-defined up to an $l^n$-torsion point. This gives a map from the $l^n$-torsion points of the kernel to the cokernel of the isogeny on $l^n$-torsion points. Take the limit of this chain of maps to get a map on Tate modules.