A CLASS OF GRADE THREE DETERMINANTAL IDEALS

• Journal title : Honam Mathematical Journal
• Volume 34, Issue 2,  2012, pp.279-287
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.2.279
Title & Authors
A CLASS OF GRADE THREE DETERMINANTAL IDEALS
Kang, Oh-Jin; Kim, Joo-Hyung;

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