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A CLASS OF GRADE THREE DETERMINANTAL IDEALS

  • Kang, Oh-Jin (Department of Mathematics, University of Incheon) ;
  • Kim, Joo-Hyung (Department of Mathematics Education, Wonkwang University)
  • Received : 2012.05.04
  • Accepted : 2012.05.19
  • Published : 2012.06.25

Abstract

Let $k$ be a field containing the field $\mathbb{Q}$ of rational numbers and let $R=k[x_{ij}{\mid}1{\leq}i{\leq}m,\;1{\leq}j{\leq}n]$ be the polynomial ring over a field $k$ with indeterminates $x_{ij}$. Let $I_t(X)$ be the determinantal ideal generated by the $t$-minors of an $m{\times}n$ matrix $X=(x_{ij})$. Eagon and Hochster proved that $I_t(X)$ is a perfect ideal of grade $(m-t+1)(n-t+1)$. We give a structure theorem for a class of determinantal ideals of grade 3. This gives us a characterization that $I_t(X)$ has grade 3 if and only if $n=m+2$ and $I_t(X)$ has the minimal free resolution $\mathbb{F}$ such that the second dierential map of $\mathbb{F}$ is a matrix defined by complete matrices of grade $n+2$.

Acknowledgement

Supported by : Wonkwang University

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