Generators of reductions of ideals in a local Noetherian ring with finite residue field

TL;DR

This paper explores the existence and properties of reductions of ideals in local Noetherian rings with finite residue fields, providing formulas, examples, and counterexamples across different dimensions.

Contribution

It offers new formulas for minimal generators of reductions in one-dimensional rings and constructs counterexamples in higher dimensions, addressing open questions.

Findings

In one-dimensional rings, every ideal has a principal reduction iff the number of maximal ideals in the normalization is at most |k|.

In higher dimensions, some ideals lack n-generated reductions, even for large n.

Constructs a two-dimensional regular local ring example with an ideal that has no 2-generated reduction.

Abstract

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of $R$ and the number of generators needed for a reduction in the case $k$ is a finite field. When $R$ is one-dimensional, we give a formula for the smallest integer $n$ for which every ideal has an $n$-generated reduction. It follows that in a one-dimensional local Noetherian ring every ideal has a principal reduction if and only if the number of maximal ideals in the normalization of the reduced quotient of $R$ is at most $|k|$. In higher dimensions, we show that for any positive integer, there exists an ideal of $R$ that does not have an $n$-generated reduction and that if $n \geq \dim R$ this ideal can be chosen to be $\mathfrak{m}$-primary. In the case where $R$ is a two-dimensional regular local ring, we construct an example of an integrally closed $\mathfrak{m}$-primary ideal that does not have a $2$-generated reduction and thus answer in the negative a question raised by Heinzer and Shannon.