A note on cohomological dimension over Cohen-Macaulay rings

Bull. Korean Math. Soc. Published online August 20, 2019

Iraj Bagheriyeh, Kamal Bahmanpour, and Ghader Ghasemi
University of Mohaghegh Ardabili

Abstract : Let $(R,\m)$ 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 ${\rm depth}_R(R/I^{n}R )$ for $n\gg0$.
In this paper we introduce the new notion $\gamma_R(I,R)$ and the 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