# metadata.dc.description.abstract[].value
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 $I^{\text {sat }}$ of $I$ from its Klyachko diagram. We use this description to compute the first local cohomology module $H_B^1(I)$. As an application, we find a formula for the Hilbert function of $I^{\text {sat }}$, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram.

# metadata.dc.relation.ispartof[].value
Algebras And Representation Theory, 2022, vol. 26, p. 1497-1517

# metadata.dc.source[].value
Articles publicats en revistes (Matemàtiques i Informàtica)