Free resolutions of lex-ideals over a Koszul toric ring

Murai, Satoshi

doi:10.1090/S0002-9947-2010-05074-3

Free resolutions of lex-ideals over a Koszul toric ring
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Trans. Amer. Math. Soc. 363 (2011), 857-885 Request permission

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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