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HILBERT FUNCTIONS OF STANDARD k-ALGEBRAS DEFINED BY SKEW-SYMMETRIZABLE MATRICES

  • Kang, Oh-Jin (Department of General Studies School of Liberal Arts and Sciences Korea Aerospace University)
  • 투고 : 2015.07.06
  • 심사 : 2016.12.26
  • 발행 : 2017.09.01

초록

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 $I_i{\cap}J_i$.

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참고문헌

  1. A. E. Brown, A Structure theorem for a class of grade three perfect ideals, J. Algebra 105 (1987), no. 2, 308-327. https://doi.org/10.1016/0021-8693(87)90196-7
  2. D. 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. Y. S. Cho, O.-J. Kang, and H. J. Ko, Perfect ideal of grade three defined by skew-symmetrizable matrix, Bull. Korean. Math. Soc. 49 (2012), no. 4, 715-736. https://doi.org/10.4134/BKMS.2012.49.4.715
  4. E. J. Choi, O.-J. Kang, and H. J. Ko, On the structure of grade three perfect ideal of type three, Commun. Korean Math. Soc. 23 (2008), no. 4, 487-497. https://doi.org/10.4134/CKMS.2008.23.4.487
  5. E. J. Choi, O.-J. Kang, and H. J. Ko, The structure for some classes of grade three perfect ideal, Comm. Algebra 39 (2011), no. 9, 3435-3461. https://doi.org/10.1080/00927872.2010.512586
  6. E. Davis, A. Geramita, and F. Orrechia, Hilbert functions of linked varieties, The curves seminar at Queen's, Vol. III (Kingston, Ont., 1983), Exp. No. F, 11 pp., Queen's Papers in Pure and Appl. Math., 67, Queen's Univ., Kingston, ON, 1984.
  7. T. Harima, Some examples of unimodal Gorenstein sequence, J. Pure Appl. Algebra 103 (1995), no. 3, 313-324. https://doi.org/10.1016/0022-4049(95)00109-A
  8. A. Iarrobino and H. Srinivasan, Artinian Gorenstein algebras of embedding dimension four: components of PGOR(H) for H = (1, 4, 7, . . . , 1), J. Pure Appl. Algebra 201 (2005), no. 1-3, 62-96. https://doi.org/10.1016/j.jpaa.2004.12.015
  9. O.-J. Kang, Y. S. Cho, and H. J. Ko, Structure theory for some classes of grade three perfect ideals, J. Algebra. 322 (2009), no. 8, 2680-2708. https://doi.org/10.1016/j.jalgebra.2009.07.021
  10. O.-J. Kang and H. J. Ko, The structure theorem for complete intersections of grade 4, Algebra Colloq. 12 (2005), no. 2, 181-197. https://doi.org/10.1142/S1005386705000179
  11. O.-J. Kang and H. J. Ko, Structure theorem for complete intersetions, Commun. Korean Math. Soc 21 (2006), no. 4, 613-630. https://doi.org/10.4134/CKMS.2006.21.4.613
  12. O.-J. Kang and J. Kim, A class of grade three determinantal ideals, Homan Math. J. 34 (2012), no. 2, 279-287.
  13. H. J. Ko and Y. S. Shin, Unimodal sequences of Gorenstein ideals of codimension 4, Acta. Math. Sinica (N.S.) 14 (1998), no. 4, 563-568. https://doi.org/10.1007/BF02580415
  14. A. Kustin and M. Miller, Structure theory for a class of grade four Gorenstein ideals, Trans. Amer. Math. Soc. 270 (1982), no. 1, 287-307. https://doi.org/10.1090/S0002-9947-1982-0642342-4
  15. S. Seo and H. Srinivasan, On unimodality of Hilbert functions of Gorenstein Artin Algebras of embedding dimension four, Comm. Algebra 40 (2012), no. 8, 2893-2905. https://doi.org/10.1080/00927872.2011.587216
  16. Y. S. Shin, The construction of some Gorenstein ideals of codimension 4, J. Pure Appl. Algebra 127 (1998), no. 3, 289-307. https://doi.org/10.1016/S0022-4049(96)00179-X
  17. R. P. Stanley, Hilbert functions of graded algebras, Adv. Math 28 (1978), no. 1, 57-83. https://doi.org/10.1016/0001-8708(78)90045-2