Miró-Roig, Rosa M. (Rosa Maria)Salat Moltó, Martí2024-07-112024-07-112022-06-211386-923Xhttps://hdl.handle.net/2445/214509In 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.21 p.application/pdfengcc-by (c) Rosa M. Miró-Roig, 2022http://creativecommons.org/licenses/by/3.0/es/Varietats tòriquesAnells commutatiusPolinomisToric varietiesCommutative ringsPolynomialsinfo:eu-repo/semantics/article7231732024-07-11info:eu-repo/semantics/openAccess