Saturations of powers of certain determinantal ideals

TL;DR

This paper investigates the associated primes and saturation of powers of determinantal ideals generated by maximal minors of a matrix over a Noetherian local ring, under certain grade conditions, especially in Cohen-Macaulay rings.

Contribution

It provides new results on the associated primes of powers of determinantal ideals and explicitly computes their saturations in Cohen-Macaulay rings with specific matrix entries.

Findings

Characterization of associated primes of $I^n$ for all $n > 0$

Explicit computation of the saturation of $I^n$ for $1 \

in Cohen-Macaulay rings with entries as powers of elements forming an sop.

Abstract

Let $R$ be a Noetherian local ring and $m$ a positive integer. Let $I$ be the ideal of $R$ generated by the maximal minors of an $m \times (m + 1)$ matrix $M$ with entries in $R$. Assuming that the grade of the ideal generated by the $k$-minors of $M$ is at least $m - k + 2$ for $1 \leq \forall k \leq m$, we will study the associated primes of $I^n$ for $\forall n > 0$. Moreover, we compute the saturation of $I^n$ for $1 \leq \forall n \leq m$ in the case where $R$ is a Cohen-Macaulay ring and the entries of $M$ are powers of elements that form an sop for $R$.