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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Free resolutions of lex-ideals over a Koszul toric ring


Author: Satoshi Murai
Journal: Trans. Amer. Math. Soc. 363 (2011), 857-885
MSC (2010): Primary 13D02; Secondary 05D05, 05E40, 13D40, 13F45, 16S37
Published electronically: September 16, 2010
MathSciNet review: 2728587
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Abstract: In this paper, we study the minimal free resolution of lex-ideals over a Koszul toric ring. In particular, we study in which toric ring $ R$ all lex-ideals are componentwise linear. We give a certain necessity and sufficiency condition for this property, and show that lex-ideals in a strongly Koszul toric ring are componentwise linear. In addition, it is shown that, in the toric ring arising from the Segre product $ \mathbb{P}^1 \times \cdots \times\mathbb{P}^1$, every Hilbert function of a graded ideal is attained by a lex-ideal and that lex-ideals have the greatest graded Betti numbers among all ideals having the same Hilbert function.


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Additional Information

Satoshi Murai
Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Sakyou-ku, Kyoto 606-8502, Japan
Address at time of publication: Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan
Email: murai@math.kyoto-u.ac.jp, murai@yamaguchi-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05074-3
PII: S 0002-9947(2010)05074-3
Keywords: Lex-ideals, Hilbert functions, free resolutions, toric rings, Koszul algebras, componentwise linear ideals
Received by editor(s): December 8, 2008
Received by editor(s) in revised form: April 2, 2009
Published electronically: September 16, 2010
Additional Notes: The author was supported by JSPS Research Fellowships for Young Scientists.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.