Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ALMOST COHEN-MACAULAYNESS OF KOSZUL HOMOLOGY

  • Received : 2018.04.06
  • Accepted : 2019.02.07
  • Published : 2019.03.31
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Abstract

Let (R, m) be a commutative Noetherian ring, I an ideal of R and M a non-zero finitely generated R-module. We show that if M and $H_0(I,M)$ are aCM R-modules and $I=(x_1,{\cdots},x_{n+1})$ such that $x_1,{\cdots},x_n$ is an M-regular sequence, then $H_i(I,M)$ is an aCM R-module for all i. Moreover, we prove that if R and $H_i(I,R)$ are aCM for all i, then R/(0 : I) is aCM. In addition, we prove that if R is aCM and $x_1,{\cdots},x_n$ is an aCM d-sequence, then depth $H_i(x_1,{\cdots},x_n;R){\geq}i-1$ for all i.

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References

  1. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  2. L. Chu, Z. Tang, and H. Tang, A note on almost Cohen-Macaulay modules, J. Algebra Appl. 14 (2015), no. 10, 1550136, 7 pp. https://doi.org/10.1142/S0219498815501364
  3. D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2.
  4. C. Huneke, On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. Algebra 62 (1980), no. 2, 268-275. https://doi.org/10.1016/0021-8693(80)90179-9
  5. C. Huneke, The theory of d-sequences and powers of ideals, Adv. in Math. 46 (1982), no. 3, 249-279. https://doi.org/10.1016/0001-8708(82)90045-7
  6. C. Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043-1062. https://doi.org/10.2307/2374083
  7. C. Huneke, The Koszul homology of an ideal, Adv. in Math. 56 (1985), no. 3, 295-318. https://doi.org/10.1016/0001-8708(85)90037-4
  8. C. Ionescu, More properties of almost Cohen-Macaulay rings, J. Commut. Algebra 7 (2015), no. 3, 363-372. https://doi.org/10.1216/JCA-2015-7-3-363
  9. M.-C. Kang, Almost Cohen-Macaulay modules, Comm. Algebra 29 (2001), no. 2, 781-787. https://doi.org/10.1081/AGB-100001541
  10. M.-C. Kang, Addendum to: "Almost Cohen-Macaulay modules", Comm. Algebra 29 (2001), no. 2, 781-787 https://doi.org/10.1081/AGB-100001541
  11. M.-C. Kang, Addendum to: "Almost Cohen-Macaulay modules", Comm. Algebra, 30(2)(2002), 1049-1052. https://doi.org/10.1081/AGB-120013199
  12. A. Mafi and H. Saremi, A note on the finiteness property related to derived functors, Acta Math. Vietnam. 34 (2009), no. 3, 371-374.
  13. A. Mafi and S. Tabejamaat, Results on almost Cohen-Macaulay modules, J. Algebr. Syst. 3 (2016), no. 2, 147-150, 5 (Persian p.).
  14. S. Tabejamaat and A. Mafi, About a Serre-type condition for modules, J. Algebra Appl. 16 (2017), no. 11, 1750206, 6 pp. https://doi.org/10.1142/S0219498817502061
  15. S. Tabejamaat, A. Mafi, and K. Ahmadi Amoli, Property of almost Cohen-Macaulay over extension modules, Algebra Colloq. 24 (2017), no. 3, 509-518. https://doi.org/10.1142/S1005386717000335
  16. W. Vasconcelos, Integral Closure, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.