Illinois Journal of Mathematics

Powers of complete intersections: graded Betti numbers and applications

Elena Guardo and Adam Van Tuyl

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Let $I = (F_1,\ldots,F_r)$ be a homogeneous ideal of the ring $R = k[x_0,\ldots,x_n]$ generated by a regular sequence of type $(d_1,\ldots,d_r)$. We give an elementary proof for an explicit description of the graded Betti numbers of $I^s$ for any $s \geq 1$. These numbers depend only upon the type and $s$. We then use this description to: (1) write $H_{R/I^s}$, the Hilbert function of $R/I^s$, in terms of $H_{R/I}$; (2) verify that the $k$-algebra $R/I^s$ satisfies a conjecture of Herzog-Huneke-Srinivasan; and (3) obtain information about the numerical invariants associated to sets of fat points in $\mathbb{P}^n$ whose support is a complete intersection or a complete intersection minus a point.

Article information

Illinois J. Math., Volume 49, Number 1 (2005), 265-279.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Secondary: 13D02: Syzygies, resolutions, complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 14A15: Schemes and morphisms


Guardo, Elena; Van Tuyl, Adam. Powers of complete intersections: graded Betti numbers and applications. Illinois J. Math. 49 (2005), no. 1, 265--279. doi:10.1215/ijm/1258138318.

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