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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 $n+2$.
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참고문헌 (8)
참고문헌 신청
A. Brown / 1987 / A Structure theorem for a class of grade three perfect ideals / J. Algebra 105:308~327
D. A. Buchsbaum / 1977 / Algebra structures for finite free resolutions and some structure theorems for ideals for codimension 3 / Amer. J. Math 99(3):447~485
Eun Jeong Choi / 2009 / A structure theorem for complete intersections / 대한수학회보 46(4):657~671
J. A. Eagon / 1971 / Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci / Amer. J. Math 93:1020~1058
O.-J. Kang / 2009 / Structure theory for some classes of grade three perfect ideals / Journal Of Algebra 322(8):2680~2708
D. A. Buchsbaum / 1977 / Algebra structures for finite free resolutions and some structure theorems for ideals for codimension 3 / Amer. J. Math 99(3):447~485
Eun Jeong Choi / 2009 / A structure theorem for complete intersections / 대한수학회보 46(4):657~671
J. A. Eagon / 1971 / Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci / Amer. J. Math 93:1020~1058
O.-J. Kang / 2009 / Structure theory for some classes of grade three perfect ideals / Journal Of Algebra 322(8):2680~2708
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