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ON A CLASS OF MONOMIAL IDEALS

Published online by Cambridge University Press:  28 January 2013

KEIVAN BORNA*
Affiliation:
Faculty of Mathematics and Computer Science, Kharazmi University, Tehran, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran
RAHELEH JAFARI
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran email [email protected]
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Abstract

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Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Aramova, A., Crona, K. and De Negri, E., ‘Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions’, J. Pure Appl. Algebra 150 (2000), 215235.CrossRefGoogle Scholar
Bermejo, I., Garcia-Llorente, E. and Gimenez, Ph., ‘monomialideal.lib. A SINGULAR library for computing with monomial ideals’, SINGULAR 3.1.1 2010.Google Scholar
Borna, K., ‘On linear resolution of powers of an ideal’, Osaka J. Math. 46 (4) (2009), 10471058.Google Scholar
Brodmann, M. P. and Sharp, R. Y., Local Cohomology: an Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, 60 (Cambridge University Press, Cambridge, UK, 1998).CrossRefGoogle Scholar
Chardin, M., ‘Powers of ideals and the cohomology of stalks and fibers of morphisms’, arxiv:1009.1271.Google Scholar
CoCoATeam, ‘CoCoA: a system for doing Computations in Commutative Algebra’, available at http://cocoa.dima.unige.it.Google Scholar
Conca, A., ‘Regularity jumps for powers of ideals’, in: Commutative Algebra, Lecture Notes in Pure and Applied Mathematics, 244 (Chapman & Hall/CRC, Boca Raton, FL, 2006), 2132.Google Scholar
Cutkosky, S. D., Herzog, J. and Trung, N. V., ‘Asymptotic behaviour of the Castelnuovo–Mumford regularity’, Compositio Math. 118 (1999), 243261.CrossRefGoogle Scholar
Eisenbud, D., Commutative Algebra With A View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150 (Springer, Berlin, 1995).Google Scholar
Herzog, J. and Hibi, T., ‘The depth of powers of an ideal’, J. Algebra 291 (2005), 534550.CrossRefGoogle Scholar
Herzog, J. and Hibi, T., Monomial Ideals, Graduate Texts in Mathematics, 260 (Springer, Berlin, 2011).CrossRefGoogle Scholar
Herzog, J., Hibi, T. and Zheng, X., ‘Monomial ideals whose powers have a linear resolution’, Math. Scand. 95 (1) (2004), 2332.CrossRefGoogle Scholar
Hoa, L. T. and Trung, N. V., ‘On the Castelnuovo–Mumford regularity and the arithmetic degree of monomial ideals’, Math. Z. 229 (1998), 519537.CrossRefGoogle Scholar
Kodiyalam, V., ‘Asymptotic behaviour of Castelnuovo–Mumford regularity’, Proc. Amer. Math. Soc. 128 (2000), 407411.CrossRefGoogle Scholar
Römer, T., ‘On minimal graded free resolutions’, Illinois J. Math. 45 (2) (2001), 13611376.Google Scholar