Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal generated by a regular sequence. I want to show that:
$\bigoplus_{n \geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring over $R/I$ in as many variables as generators of $I$.
My attempt: Let $S$ be the polynomial ring $(R/I)[x_{0},...,x_{n}]$. Then we have $S=\bigoplus_{d \geq 0}S_d$, which $S_d$ is the set of linear combinations of monomials with total weight $d$. I tried to show that there is an isomorphism between $I^{n}/I^{n+1}$ and $S_{n}$ for $n \geq 0$. Let $(a_{0},...,a_{m})$ be the regular sequence by which $I$ is generated. Consider $\beta : I^{n}/I^{n+1} \rightarrow S_n$ with $\beta(r.a_{i_1}...a_{i_n}+I^{n+1})= (r+I).x_{i_1}...x_{i_n}. $ I'm not sure if it works since I haven't used the fact that $(a_{0},...,a_{m})$ is a regular sequence.