On the length function of saturations of ideal powers

TL;DR

This paper investigates the polynomial behavior of the length function of local cohomology modules of ideal powers in local rings, providing explicit formulas under certain conditions and counterexamples otherwise.

Contribution

It establishes polynomiality of the length function for specific classes of ideals and computes the coefficients explicitly, advancing understanding of local cohomology in commutative algebra.

Findings

The length function is polynomial for principal ideals and certain almost p-standard systems.

Explicit formulas for polynomial coefficients are derived in special cases.

Counterexamples show polynomiality does not always hold for all ideals.

Abstract

For an ideal $I$ in a local ring $(R, \fm)$, we prove that the integer-valued function $\ell_R(H^0_\fm(R/I^{n+1}))$ is a polynomial for $n$ big enough if either $I$ is a principle ideal or $I$ is generated by part of an almost p-standard system of parameters. 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 (general) system of parameters such that the function $\ell_R(H^0_\fm(R/I^{n+1}))$ is not a polynomial for $n\gg 0$.