A short note on cohomological dimension

Let $(R,\mathfrak m)$ be a Noehterian regular local ring and $\mathfrak p$ be a prime ideal of $R$. In this paper it is shown that if the set $S:=\{n\in\mathbb N\colon R/\mathfrak p^{(n)}\ \text{is Cohen--Macaulay}\}$ is infinite, then $\mathrm{cd}(\mathfrak p,R)=\mathrm{height}(\mathfrak p)$.