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Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert... In this paper, we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases, the associated graded ring with respect to tight closure filtration turns out to be Cohen–Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

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References (22)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
eISSN
2197-9847
DOI
10.1007/s40687-021-00296-x
Publisher site
See Article on Publisher Site

Abstract

In this paper, we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases, the associated graded ring with respect to tight closure filtration turns out to be Cohen–Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Dec 1, 2021

Keywords: Tight closure; Parameter ideals; Powers of ideals; Tight Hilbert polynomial; Diagonal hypersurface; Primary: 13A35; Secondary: 13D40

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