HILBERT FUNCTIONS OF STANDARD k-ALGEBRAS DEFINED BY SKEW-SYMMETRIZABLE MATRICES
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Kang and Ko introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4. Let $R=k[w_0, w_1, w_2, \ldots, w_m]$ be the polynomial ring over an algebraically closed field $k$ with indetermiantes $w_l$ and $\deg w_l=1,$ and $I_i$ a homogeneous perfect ideal of grade 3 with type $t_i$ defined by a skew-symmetrizable matrix $G_i (1 \leq t_i \leq 4).$ We show that for $m=2$ the Hilbert function of the zero dimensional standard $k$-algebra $R/I_i$ is determined by $CI$-sequences and a Gorenstein sequence. As an application of this result we show that for $i=1,2,3$ and for $m=3$ a Gorenstein sequence $h(R/H_i)=(1,4,h_2,\ldots,h_s)$ is unimodal, where $H_i$ is the sum of homogeneous perfect ideals $I_i$ and $J_i$ which are geometrically linked by a homogeneous regular sequence $z$ in $I_i \cap J_i.$
Abstract
Kang and Ko introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4. Let $R=k[w_0, w_1, w_2, \ldots, w_m]$ be the polynomial ring over an algebraically closed field $k$ with indetermiantes $w_l$ and $\deg w_l=1,$ and $I_i$ a homogeneous perfect ideal of grade 3 with type $t_i$ defined by a skew-symmetrizable matrix $G_i (1 \leq t_i \leq 4).$ We show that for $m=2$ the Hilbert function of the zero dimensional standard $k$-algebra $R/I_i$ is determined by $CI$-sequences and a Gorenstein sequence. As an application of this result we show that for $i=1,2,3$ and for $m=3$ a Gorenstein sequence $h(R/H_i)=(1,4,h_2,\ldots,h_s)$ is unimodal, where $H_i$ is the sum of homogeneous perfect ideals $I_i$ and $J_i$ which are geometrically linked by a homogeneous regular sequence $z$ in $I_i \cap J_i.$
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