Hilbert polynomials of multigraded filtrations of ideals

Hilbert functions and Hilbert polynomials of ZsZs-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 FF 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 Z2Z2-graded filtration FF to be Cohen–Macaulay.