Document Type : Original Manuscript


Faculty of Sciences, Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 56199-11367, Ardabil, Iran.


Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local
cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp.
R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions
of local cohomology modules, and show that if the Lynch's conjecture, in [11], holds in
R[[X]], then it will holds in R.


  1. K. Bahmanpour, Annihilators of local cohomology modules, Comm. Algebra, 43
    (2015), 2509–2515.
  2.  K. Bahmanpour, A note on Lynch’s conjecture, Comm. Algebra, 45 (2017), 2738–2745.
  3.  K. Bahmanpour, J. A’zami and G. Ghasemi, On the annihilators of local cohomology modules, J. Algebra, 363 (2012), 8–13.
  4.  K. Bahmanpour and M. Seidali Samani, On the cohomological dimension of finitely generated modules, Bull. Korean Math. Soc., 55 (2018), 311–317.
  5.  M. P. Brodmann and R. Y. Sharp, Local Cohomology; An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.
  6.  W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge University Press, Cambridge, 1998.
  7.  K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc., 130 (2002), 3537–3544.
  8.  C. Faith, Associated primes in commutative polynomial rings, Comm. Algebra, 28 (2000), 3983–3986.
  9.  A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lecture Notes in Math., Vol. 862, Springer, New York, 1966.
  10.  C. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambridge Philos. Soc., 110 (1991), 421–429.
  11.  L. R. Lynch, Annihilators of top local cohomology, Comm. Algebra, 40 (2012), 542–551.
  12.  H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.
  13.  P. Schenzel, Cohomological annihilators, Math. Proc. Cambridge Philos. Soc., 91 (1982), 345–350.
  14.  P. Schenzel, On the use of local cohomology in algebra and geometry, Six lectures on commutative algebra, Bellaterra 1996, 241–292.