On unmixed and equi-dimensional associated graded rings

Abstract

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d\ge 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p>0$ then let $I^{\ast }$ denote the tight closure of $I$. Let $G_I(A)={⨁}_{n\ge 0}{I}^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$. Assume $G_I(A)$ is unmixed and equi-dimensional. We show that either the function $P_{\overline{I}}:n\mapsto \hat{I}»(\overline{I^n}/I^n)$ is a polynomial type of degree $d-1$ or $\overline{I^n}=I^n$ for all $n\ge 1.$ We prove an analogus result for the tight closure filtration if $A$ is of characteristic $p>0$. When $A$ is generalized Cohen-Macaulay and $I$ is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of $I$ and the tight closure filtration of $I$.

Final version