Bulletin of the Korean Mathematical Society (대한수학회보)

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STRUCTURE THEOREMS FOR SOME CLASSES OF GRADE FOUR GORENSTEIN IDEALS

  • Cho, Yong Sung (Department of mathematics Education Mokpo National University) ;
  • Kang, Oh-Jin (Department of General Studies School of Liberal Arts and Sciences Korea Aerospace University) ;
  • Ko, Hyoung June (Department of Mathematics Yonsei University)
  • Received : 2015.10.04
  • Published : 2017.01.31
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Abstract

The structure theorems [3, 6, 21] for the classes of perfect ideals of grade 3 have been generalized to the structure theorems for the classes of perfect ideals linked to almost complete intersections of grade 3 by a regular sequence [15]. In this paper we obtain structure theorems for two classes of Gorenstein ideals of grade 4 expressed as the sum of a perfect ideal of grade 3 (except a Gorenstein ideal of grade 3) and an almost complete intersection of grade 3 which are geometrically linked by a regular sequence.

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References

  1. E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957.
  2. H. Bass, On the ubiquity of Gorenstein rings, Math. Z 82 (1963), 8-28. https://doi.org/10.1007/BF01112819
  3. A. 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
  4. L. Burch, On ideals of finite homological dimension in a local rings, Proc. Cambridge Philos. Soc. 64 (1968), 941-948. https://doi.org/10.1017/S0305004100043620
  5. D. A. Buchsbaum and D. Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259-268. https://doi.org/10.1016/0021-8693(73)90044-6
  6. D. A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), no. 3, 447-485. https://doi.org/10.2307/2373926
  7. Y. S. Cho, A structure theorem for a class of Gorenstein ideals of grade four, Honam Math. J. 36 (2014), no. 2, 387-398. https://doi.org/10.5831/HMJ.2014.36.2.387
  8. Y. S. Cho, On a class of Gorenstein ideals of grade four, Honam Math. J. 36 (2014), no. 3, 605-622. https://doi.org/10.5831/HMJ.2014.36.3.605
  9. E. J. Choi, O.-J. Kang, and H. J. Ko, On the structures of the grade three perfect ideals of type 3, Commun. Korean Math. Soc. 23 (2008), no. 4, 487-497. https://doi.org/10.4134/CKMS.2008.23.4.487
  10. 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. https://doi.org/10.4134/BKMS.2009.46.4.657
  11. E. S. Golod, A note on perfect ideals, from the collection "Algebra" (A. I. Kostrikin, Ed), Moscow State Univ. Publishing House, 37-39, 1980.
  12. D. Hilbert, Uber die Theorie von Algebraischen Forman, Math. Ann. 36 (1890), no. 4, 473-534. https://doi.org/10.1007/BF01208503
  13. A. Iarrobino and H. Srinivasan, Artinian Gorenstein Algebras of embedding dimension four: components of ${\mathbb{P}}Gor$(H) for (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
  14. O.-J. Kang, Structure theory for grade three perfect ideals associated with some matrices, Comm. Algebra 43 (2015), no. 7, 2984-3019. https://doi.org/10.1080/00927872.2014.900684
  15. O.-J. Kang, Y. S. Cho, and H. J. Ko, Structure theory for some classes of grade perfect ideals, J. Algebra 322 (2009), no. 8, 2680-2708. https://doi.org/10.1016/j.jalgebra.2009.07.021
  16. 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
  17. S. El Khoury and H. Srinivasan, A class of Gorenstein Artin Algebras of embedding dimension four, Comm. Algebra 37 (2009), no. 9, 3259-3277. https://doi.org/10.1080/00927870802502738
  18. 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
  19. A. Kustin and M. Miller, Tight double linkage of Gorenstein algebras, J. Algebra 95 (1985), no. 2, 384-397. https://doi.org/10.1016/0021-8693(85)90110-3
  20. C. Peskine and L. Szpiro, Liaison des varietes algebriques, Invent. Math. 26 (1974), 271-302. https://doi.org/10.1007/BF01425554
  21. R. Sanchez, A structure theorem for type 3, grade 3 perfect ideals, J. Algebra 123 (1989), no. 2, 263-288. https://doi.org/10.1016/0021-8693(89)90047-1