A CLASS OF GRADE THREE DETERMINANTAL IDEALS

Abstract

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