Hilbert polynomials of multigraded filtrations of ideals

Abstract

Hilbert functions and Hilbert polynomials of Zs-graded admissible filtrations of ideals {F(n_)}n_∈Zs such that λ(RF(n_)) is finite for all n_∈Zs are studied. Conditions are provided for the Hilbert function HF(n_):=λ(R/F(n_)) and the corresponding Hilbert polynomial PF(n_) to be equal for all n_∈Ns. A formula for the difference HF(n_)−PF(n_) in terms of local cohomology of the extended Rees algebra of F is proved which is used to obtain sufficient linear relations analogous to the ones given by Huneke and Ooishi among coefficients of PF(n_) so that HF(n_)=PF(n_) for all n_∈Ns. A theorem of Rees about joint reductions of the filtration {IrJs‾}r,s∈Z is generalised for admissible filtrations of ideals in two-dimensional Cohen–Macaulay local rings. Necessary and sufficient conditions are provided for the multi-Rees algebra of an admissible Z2-graded filtration F to be Cohen–Macaulay.