Cohen-Macaulayness inherits to quotient (Matsumura, CRT, Exercise 17.4)

Question

A well-known theorem in commutative algebra states the fact that if $R$ is a Cohen-Macaulay ring, and $a_1,...,a_r$ is an $R$-sequence, then $R/I$ is Cohen-Macaulay, where $I=(a_1,...,a_r)$.

Now,

Is it true that for any positive integer $n$ the ring $R/I^n$ is Cohen-Macaulay?

I know, somehow, that we should resort to induction and use the short exact sequence $0\to I^n/I^{n+1} \to R/I^{n+1}\to R/I^n\to 0$ but I could not continue. Is there an elementary proof (or counterexample) for that?

Thanks for cooperation!

Answer 1

Suppose $(R,\mathfrak m)$ is local. We want to show that $\operatorname{depth}R/I^n=\operatorname{depth}R/I$. By Theorem 1.18 from Bruns and Herzog we get $(R/I)[X_1,\dots,X_r]\simeq\operatorname{gr}_I(R)$. It follows that $I^{k}/I^{k+1}\simeq(R/I)^{m_k}$ for all $k\ge0$. Now one can show inductively that $\operatorname{Ext}_R^i(R/\mathfrak m,R/I^n)=0$ for $i<\operatorname{depth}R/I$, and $\operatorname{Ext}_R^i(R/\mathfrak m,R/I^n)\ne0$ for $i=\operatorname{depth}R/I$.

Since $\dim R/I^n=\dim R/I$ we are done.