Behavior of local cohomology modules under polarization

Let $S=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, ..., $x_n$, $\mmmm$ the irrelevant maximal ideal of $S$, $I$ a monomial ideal in $S$ and $I'$ the polarization of $I$ in the polynomial ring $S'$ with $\rho$ variables. We show that each graded piece $H_\mmmm^i(S/I)_\aaa$, $\aaa\in\ZZZ^n$, of the local cohomology module $H_\mmmm^i(S/I)$ is isomorphic to a specific graded piece $H_{\mmmm'}^{i+\rho-n}(S'/I')_\alphaaa$, $\alphaaa\in\ZZZ^\rho$, of the local cohomology module $H_{\mmmm'}^{i+\rho-n}(S'/I')$, where $\mmmm'$ is the irrelevant maximal ideal of $S'$.