Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE LOCAL COHOMOLOGY OF MINIMAX MODULES

  • Mafi, Amir (University of Kurdistan, School of Mathematics Institute for Research in Fundamental Science (IPM))
  • Received : 2009.05.15
  • Published : 2011.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a commutative Noetherian ring, a an ideal of R, and M a minimax R-module. We prove that the local cohomology modules $H^j_a(M)$ are a-cominimax; that is, $Ext^i_R$(R/a, $H^j_a(M)$) is minimax for all i and j in the following cases: (a) dim R/a = 1; (b) cd(a) = 1, where cd is the cohomological dimension of a in R; (c) dim $R{\leq}2$. In these cases we also prove that the Bass numbers and the Betti numbers of $H^j_a(M)$ are finite.

Keywords

References

  1. J. Azami, R. Naghipour, and B. Vakili, Finiteness properties of local cohomology modules for $\alpha$-minimax modules, Proc. Amer. Math. Soc. 137 (2009), no. 2, 439-448.
  2. K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2359-2363. https://doi.org/10.1090/S0002-9939-08-09260-5
  3. K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), no. 7, 1997-2011. https://doi.org/10.1016/j.jalgebra.2008.12.020
  4. R. Belshoff, S. P. Slattery, and C. Wickham, The local cohomology modules of Matlis reflexive modules are almost cofinite, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2649-2654. https://doi.org/10.1090/S0002-9939-96-03326-6
  5. R. Belshoff, S. P. Slattery, and C. Wickham, Finiteness properties for Matlis reflexive modules, Comm. Algebra 24 (1996), no. 4, 1371-1376. https://doi.org/10.1080/00927879608825640
  6. M. P. Brodmann and R. Y. Sharp, Local Cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, 60. Cambridge University Press, Cambridge, 1998.
  7. D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997), no. 1, 45-52. https://doi.org/10.1016/S0022-4049(96)00044-8
  8. E. Enochs, Flat covers and at cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179-184. https://doi.org/10.1090/S0002-9939-1984-0754698-X
  9. A. Grothendieck, Local Cohomology, Notes by R. Harthsorne, Lecture Notes in Math. 862, Springer-Verlag, 1966.
  10. R. Hartshorne, Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164. https://doi.org/10.1007/BF01404554
  11. K. I. Kawasaki, Cofiniteness of local cohomology modules for principal ideals, Bull. London Math. Soc. 30 (1998), no. 3, 241-246. https://doi.org/10.1112/S0024609397004347
  12. K. Khashyarmanesh and F. Khosh-Ahang, On the local cohomology of Matlis reflexive modules, Comm. Algebra 36 (2008), no. 2, 665-669. https://doi.org/10.1080/00927870701724102
  13. L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), no. 2, 649-668. https://doi.org/10.1016/j.jalgebra.2004.08.037
  14. K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179-191. https://doi.org/10.1017/S0027763000006371
  15. T. Zink, Endlichkeritsbedingungen fur moduln uber einem Noetherschen ring, Math. Nachr. 64 (1974), 239-252. https://doi.org/10.1002/mana.19740640114
  16. H. Zoschinger, Minimax modules, J. Algebra 102 (1986), no. 1, 1-32. https://doi.org/10.1016/0021-8693(86)90125-0
  17. H. Zoschinger, Uber die Maximalbedingung fur radikalvolle Untermoduh, Hokkaido Math. J. 17 (1988), no. 1, 101-116. https://doi.org/10.14492/hokmj/1381517790

Cited by

  1. On the generalized local cohomology of minimax modules vol.15, pp.08, 2016, https://doi.org/10.1142/S0219498816501474
  2. Extension functors of cominimax modules vol.45, pp.2, 2017, https://doi.org/10.1080/00927872.2016.1172613