Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

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
Download PDF
⟨ Previous Next ⟩

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 $n+2$.

Keywords

References

  1. A. Brown, A Structure theorem for a class of grade three perfect ideals, J. Algebra 105 (1987), 308-327. https://doi.org/10.1016/0021-8693(87)90196-7
  2. D. A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions and some structure theorems for ideals for codimension 3, Amer. J. Math. 99 (1977), no. 3 447-485. https://doi.org/10.2307/2373926
  3. E. J. Choi, O.-J. Kang and H. J. Ko, A structure theorem for complete intersections, Bull. Korean. Math. Soc. 46 (2009), no 4, 657-671.
  4. J. A. Eagon and D. G. Hochster, Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. https://doi.org/10.2307/2373744
  5. O.-J. Kang, Y. S. Cho and H. J. Ko, Structure theory for some classes of grade three perfect ideals, J. Algebra. 322 (2009), 2680-2708. https://doi.org/10.1016/j.jalgebra.2009.07.021
  6. O.-J. Kang and H. J. Ko, The structure theorem for complete intersections of grade 4, Algebra Collo. 12 (2005), no. 2, 181-197. https://doi.org/10.1142/S1005386705000179
  7. O.-J. Kang and J. Kim, New construction of the Eagon-Northcott complex, Submitted.
  8. R. Sanchez, A Structure theorem for type 3 grade 3 perfect ideals, J. Algebra. 123 (1989), 263-288. https://doi.org/10.1016/0021-8693(89)90047-1