Hilbert series characterization of regular sequences Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $f_1,\dots,f_n$ be a sequence of $n\le r$ forms of degrees $d_1,\dots,d_n$. If $f_1,\dots,f_n$ is a regular sequence, then it is easy to see that the Hilbert series of the quotient is 
\begin{align}
H_{S/(f_1,\dots,f_n)}(t) = \frac{(1-t^{d_1})\cdots(1-t^{d_n})}{(1-t)^r}, \, \, \, (1).
\end{align} 
Question: Is the converse statement true? I.e., if (1) is true, is it the case that $f_1,\dots,f_n$ is a regular sequence? If yes, how do we see that? (I can see that if $r=n$.)
 A: So, the grade of the ideal generated by $f_1,\dots,f_n$ is $n$. Now the question is whether this entails that $f_1,\dots,f_n$ is a regular sequence. Since $f_i$ are homogeneous of degree $d_i\ge1$ they belong to the maximal irrelevant ideal $\mathfrak m=(x_1,\dots,x_r)$ of $S$, and $(f_1,\dots,f_n)S_{\mathfrak m}$ is an ideal of grade $n$ in $S_{\mathfrak m}$. This shows that $f_1,\dots,f_n$ form a regular sequence in $S_{\mathfrak m}$ (why?), so they form a regular sequence in $S$ (why?).
An alternative approach. Localize $S$ at $\mathfrak m=(X_1,\dots,X_r)$. The dimension equality remains the same since the dimension of a graded $k$-algebra is the height of its irrelevant maximal ideal. (Here one uses that $f_i$'s are homogeneous.) By the dimension equality the images of $f_1,\dots,f_n$ in $S_{\mathfrak m}$ form a system of parameters, and since $S_{\mathfrak m}$ is a CM local ring they form a regular sequence (in any order) (see Eisenbud, Cor. 17.12). Now we have to show the same property for $f_1,\dots,f_n$ in $S$. This follows easily since they are homogeneous.
