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

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.