• Korean Journal of Mathematics
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• 제20권2호
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• pp.161-176
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• 2012

The authors [6] introduced the concept of a complete matrix of grade $g$ > 3 to describe a structure theorem for complete intersections of grade $g$ > 3. We show that a complete matrix can be used to construct the Eagon-Northcott complex [7]. Moreover, we prove that it is the minimal free resolution $\mathbb{F}$ of a class of determinantal ideals of $n{\times}(n+2)$ matrices $X=(x_{ij})$ such that entries of each row of $X=(x_{ij})$ form a regular sequence and the second differential map of $\mathbb{F}$ is a matrix $f$ defined by the complete matrices of grade $n+2$.

• 호남수학학술지
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• 제34권2호
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• pp.279-287
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• 2012

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 $n+2$.

• 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 $I_i{\cap}J_i$.