  In this paper, we introduce the notion of a Klyachko diagram for a monomial
ideal $I$ in a certain multi-graded polynomial ring, namely the Cox ring $R$ of
a smooth complete toric variety, with irrelevant maximal ideal $B$. We present
procedures to compute the Klyachko diagram of $I$ from its monomial generators,
and to retrieve the $B-$saturation $I^{\mathrm{sat}}$ of $I$ from its Klyachko
diagram. We use this description to compute the first local cohomology module
$H^{1}_{B}(I)$. As an application, we find a formula for the Hilbert function
of $I^{\mathrm{sat}}$, and a characterization of monomial ideals with constant
Hilbert polynomial, in terms of their Klyachko diagram.
