In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:



> 
> **Proposition.** Let $R$ be a regular local ring and $I=(x\_1,\ldots,x\_r)\subset R$ an ideal generated by a regular sequence. Let $A:=R/I$. Then, 
>  $$\begin{eqnarray\*}
> \phi: A[T\_1,\ldots,T\_r] &\overset{\_\sim}{\longrightarrow} & \mathrm{gr}\_I(R) = \bigoplus\nolimits\_{d\ge 0} {I^d}/{I^{d+1}} \\
>  T\_i & \longmapsto & x\_i
> \end{eqnarray\*}$$
>  is an isomorphism of graded $A$-algebras.
> 
> 
> 


In Hartshorne, the condition of being regular and local is strengthened to Cohen-Macaulay. However, I only need the above. I tried to look up the proof for the general statement in Matsumuura's book, but it seems rather involved (and honestly, a bit convoluted). I would like to use results about regular local rings and in turn, avoid introducing terminology like Hilbert-Samuel polynomials. 


So, I guess I am asking for an "easy" proof of the above proposition. It seems rather easy for $r=1$, but I am somehow stuck trying to prove it by induction.



A particular Isomorphism of graded algebras over a regular local ring
One reference is Bruns-Herzog, Theorem 1.1.8. You don't really need any assumption on $R$. 


