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On Hilbert quasi-polynomial functions of finitely generated bigraded modules over non standard bigraded rings

Université Blaise Pascal, Labo de Maths, Compl. Sci.
Clermont Ferrand II Cézeaux 63177 Aubière France

Labo de Mathématiques et Informatique, Université Nangui Abrogoua,
22 BP 1709 Abidjan 22 Abidjan, Côte d'Ivoire

Mathematics Subject Classification: [2010] : 13A02, 13A15, 13A30, 13B25, 13D40, 13F20, 13F25, 13H15
Keywords : non standard bigraded module, quasi-polynomial function in two variables, Hilbert two variables functions, Poincaré series, multiplicities


In this paper we prove, as main results, that the Hilbert function $H(M,-,-)$ of a finitely generated bigraded module MATH over a non standard bigraded ring MATH is a quasi- polynomial function in two variables if the ring $A_{0,0}$ is artinian and that if this function $H(M,-,-)$ has nonnegative degree $d(M)$, then MATH where $dim_{Kr}\ M$ is the Krull dimension of the module $M$. We show also that, if $A_{0,0}$ is artinian and local, then the cumulative function $H^{\ast }(M,-,-)$ associated with the Hilbert function $H(M,-,-)$ is a quasi- polynomial function whose degree $d^{\ast }(M)$ satisfies the equality MATH

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