On the sequential polynomial type of modules

Abstract

Let $M$ be a finitely generated module over a Noetherian local ring $R$. The sequential polynomial type $\mathrm{sp}(M)$ of $M$ was recently introduced by Nhan, Dung and Chau, which measures how far the module $M$ is from the class of sequentially Cohen–Macaulay modules. The present paper purposes to give a parametric characterization for $M$ to have $\mathrm{sp}(M)\le s$, where $s\ge -1$ is an integer. We also study the sequential polynomial type of certain specific rings and modules. As an application, we give an inequality between $\mathrm{sp}(S)$ and $\mathrm{sp}(S^G) $, where $S$ is a Noetherian local ring and $G$ is a finite subgroup of $\mathrm{Aut}S$ such that the order of $G$ is invertible in $S$.