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


Authors: Louiza Fouli and Bruce Olberding
Journal: Proc. Amer. Math. Soc. 146 (2018), 5051-5063
MSC (2010): Primary 13A30, 13B22, 13A15
DOI: https://doi.org/10.1090/proc/14138
Published electronically: September 10, 2018
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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 $ \vert k\vert$. 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.


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Additional Information

Louiza Fouli
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: lfouli@nmsu.edu

Bruce Olberding
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: olberdin@nmsu.edu

DOI: https://doi.org/10.1090/proc/14138
Keywords: Reduction, integral closure, finite field, analytic spread
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
Article copyright: © Copyright 2018 American Mathematical Society

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