  Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$
be a $\Z_p^d$-extension, unramified outside a finite set of places of $K$, with
$\Gal(L/K)=\Gamma$. Let $\Lambda(\Gamma):=\Z_p[[\Gamma]]$ denote the Iwasawa
algebra. In this paper, we study how the characteristic ideal of the
$\Lambda(\Gamma)$-module $X_L$, the dual $p$-primary Selmer group, varies when
$L/K$ is replaced by a intermediate $\Z_p^e$-extension.
