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ON THE COHOMOLOGICAL DIMENSION OF FINITELY GENERATED MODULES

  • Bahmanpour, Kamal (Faculty of Sciences Department of Mathematics University of Mohaghegh Ardabili) ;
  • Samani, Masoud Seidali (Faculty of Sciences Department of Mathematics University of Mohaghegh Ardabili)
  • Received : 2016.12.26
  • Accepted : 2017.06.08
  • Published : 2018.01.31

Abstract

Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.

Acknowledgement

Supported by : IPM

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