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Given a compact stratified pseudomanifold with a Thom-Mather stratificationand a class of riemannian metrics over its regular part, we study therelationships between the $L^{2}$ de Rham and Hodge cohomology and theintersection cohomology of $X$ associated to some perversities. More precisely,to a kind of metric which we call \emph{quasi edge with weights}, we associatetwo 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 isomorphicto the maximal $L^{2}$ de Rham cohomology and this is in turn isomorphic to theintersection cohomology associated to the perversity $q_{g}$. Moreover we provethat the relative $L^{2}$ Hodge cohomology is isomorphic to the minimal $L^{2}$de Rham cohomology and this is in turn isomorphic to the intersectioncohomology associated to the perversity $p_{g}$.

General perversities and L^2 de Rham and Hodge theorems for stratified pseudomanifolds