On the Length Function of Saturations of Ideal Powers

Abstract

For an ideal I in a Noetherian local ring $(R, \mathfrak {m})$ , we prove that the integer-valued function $\ell _{R}(H^{0}_{\mathfrak {m}}(R/I^{n + 1}))$ is a polynomial for n big enough if either I is a principal ideal or I is generated by part of an almost p-standard system of parameters and R is unmixed. Furthermore, we are able to compute the coefficients of this polynomial in terms of length of certain local cohomology modules and usual multiplicity if either the ideal is principal or it is generated by part of a standard system of parameters in a generalized Cohen-Macaulay ring. We also give an example of an ideal generated by part of a system of parameters such that the function $\ell _{R}(H^{0}_{\mathfrak {m}} (R/I^{n + 1}))$ is not a polynomial for n ≫ 0.