  Given a compact stratified pseudomanifold with a Thom-Mather stratification
and a class of riemannian metrics over its regular part, we study the
relationships between the $L^{2}$ de Rham and Hodge cohomology and the
intersection cohomology of $X$ associated to some perversities. More precisely,
to a kind of metric which we call \emph{quasi edge with weights}, we associate
two general perversities in the sense of G. Friedman, $p_{g}$ and its dual
$q_{g}$. We then show that the absolute $L^{2}$ Hodge cohomology is isomorphic
to the maximal $L^{2}$ de Rham cohomology and this is in turn isomorphic to the
intersection cohomology associated to the perversity $q_{g}$. Moreover we prove
that the relative $L^{2}$ Hodge cohomology is isomorphic to the minimal $L^{2}$
de Rham cohomology and this is in turn isomorphic to the intersection
cohomology associated to the perversity $p_{g}$.
