First Coefficient ideals and $R_1$ property of Rees algebras

TL;DR

This paper characterizes when the Rees algebra and its extensions are $R_1$ by relating it to the equality of integral closures and first coefficient ideals of powers of an $rak{m}$-primary ideal in an excellent normal local ring.

Contribution

It establishes equivalences between the $R_1$ property of Rees algebras and the equality of integral closures and first coefficient ideals of ideal powers.

Findings

Rees algebra $A[It]$ is $R_1$ iff $(I^n)^* = (I^n)_1$ for all $n$

Extended Rees algebra $A[It, t^{-1}]$ is $R_1$ iff the same condition holds

The $R_1$ property of $Proj(A[It])$ is also equivalent to this ideal equality

Abstract

Let $(A,\mathfrak{m})$ be an excellent normal local ring of dimension $d \geq 2$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal. Then the following assertions are equivalent: (i) The extended Rees algebra $A[It, t^{-1}]$ is $R_1$. (ii) The Rees algebra $A[It]$ is $R_1$. (iii) $Proj(A[It])$ is $R_1$. (iv) $(I^n)^* = (I^n)_1$ for all $n \geq 1$. Here $(I^n)^*$ is the integral closure of $I^n$ and $(I^n)_1$ is the first coefficient ideal of $I^n$.