A note on cohomological dimension over Cohen-Macaulay rings
Bull. Korean Math. Soc. 2020 Vol. 57, No. 2, 275-280
https://doi.org/10.4134/BKMS.b190165
Published online August 20, 2019
Printed March 31, 2020
Iraj Bagheriyeh, Kamal Bahmanpour, Ghader Ghasemi
University of Mohaghegh Ardabili; University of Mohaghegh Ardabili; University of Mohaghegh Ardabili
Abstract : Let $(R,\mathfrak m)$ be a Noetherian local Cohen-Macaulay ring and $I$ be a proper ideal of $R$. Assume that $\beta_R(I,R)$ denotes the constant value of ${\rm depth}_R(R/I^{n})$ for $n\gg0$. In this paper we introduce the new notion $\gamma_R(I,R)$ and then we prove the following inequalities: $$\beta_R(I,R)\leq\gamma_R(I,R)\leq \dim R - cd(I,R)\leq \dim R/I.$$ Also, some applications of these inequalities will be included.
Keywords : Canonical module, Cohen-Macaulay ring, cohomological dimension, local cohomology, Noetherian ring
MSC numbers : Primary 13D45, 13E05
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