• 제목/요약/키워드: skew-symmetrizable matrix

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PERFECT IDEALS OF GRADE THREE DEFINED BY SKEW-SYMMETRIZABLE MATRICES

  • Cho, Yong-Sung;Kang, Oh-Jin;Ko, Hyoung-June
    • 대한수학회보
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    • 제49권4호
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    • pp.715-736
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    • 2012
  • Brown provided a structure theorem for a class of perfect ideals of grade 3 with type ${\lambda}$ > 0. We introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4 in a Noetherian local ring. We construct a class of perfect ideals I of grade 3 with type 2 defined by a certain skew-symmetrizable matrix. We present the Hilbert function of the standard $k$-algebras R/I, where R is the polynomial ring $R=k[v_0,v_1,{\ldots},v_m]$ over a field $k$ with indeterminates $v_i$ and deg $v_i=1$.

HILBERT FUNCTIONS OF STANDARD k-ALGEBRAS DEFINED BY SKEW-SYMMETRIZABLE MATRICES

  • Kang, Oh-Jin
    • 대한수학회지
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    • 제54권5호
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    • pp.1379-1410
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    • 2017
  • 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 $I_i{\cap}J_i$.

ON THE STRUCTURE OF THE GRADE THREE PERFECT IDEALS OF TYPE THREE

  • Choi, Eun-Jeong;Kang, Oh-Jin;Ko, Hyoung-June
    • 대한수학회논문집
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    • 제23권4호
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    • pp.487-497
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    • 2008
  • Buchsbaum and Eisenbud showed that every Gorenstein ideal of grade 3 is generated by the submaximal order pfaffians of an alternating matrix. In this paper, we describe a method for constructing a class of type 3, grade 3, perfect ideals which are not Gorenstein. We also prove that they are algebraically linked to an even type grade 3 almost complete intersection.