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

Abstract

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