I'm trying to understand the proof of the theorem at page 163 of Mumford, Abelian Varieties.
At some point we have the following situation: $X$ is an abelian variety, $D$ is an effective Weil divisor with no multiple components (we can write $D=\sum E_i$), $F$ is a finite subgroup of $X$ and so we have an étale morphism $\pi:X\rightarrow X/F$ (see construction of $X/F$ and $\pi$ in Mumford from page 66 to page 73).
Now, Mumford says:
$D_1=:\pi(Supp D)$ is a closed subset pure of codimension one in $X/F$, which we may consider as a divisor with all components of multiplicity one. Since $\pi$ is étale, $\pi^*(D_1)$ is again a divisor with all components of multiplicity one and has the same support as $D$, so that $D=\pi^*(D_1)$.
I understand the first part: $D_1$ is closed since $\pi$ is a closed morphism and $SuppD$ is closed, and it's pure of codimension one (i.e. every irreducible component has codimension 1) since every étale morphism is codimension preserving. Also, the pullback of an étale morphism is again an étale morphism so we can apply the same argument to $\pi^*$ and say that $\pi^*(D_1)$ is again a divisor with all components of multiplicity one. So everything is clear until this point.
But why does it have the same support as $D$?
Thanks!