Generators of reductions of ideals in a local Noetherian ring with finite residue field
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- by Louiza Fouli and Bruce Olberding PDF
- Proc. Amer. Math. Soc. 146 (2018), 5051-5063 Request permission
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.References
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Additional Information
- Louiza Fouli
- Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
- MR Author ID: 835733
- Email: lfouli@nmsu.edu
- Bruce Olberding
- Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
- MR Author ID: 333074
- Email: olberdin@nmsu.edu
- Received by editor(s): August 21, 2017
- Received by editor(s) in revised form: February 21, 2018
- Published electronically: September 10, 2018
- Additional Notes: The first author was partially supported by a grant from the Simons Foundation, grant #244930.
- Communicated by: Irena Peeva
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5051-5063
- MSC (2010): Primary 13A30, 13B22, 13A15
- DOI: https://doi.org/10.1090/proc/14138
- MathSciNet review: 3866845