Hilbert Syzygy Theorem for non-graded modules

Question

The statement of Hilbert Syzygy Theorem is as follows: Let $R = k[x_1 , \ldots , x_n]$ be a polynomial ring over a field $k$ and $M$ be a finitely generated graded $R$-module. Then $\text{pd }M \leq n$, where $\text{pd}$ denotes the projective dimension. Does it also hold for non-graded modules? I mean, can I say that

Let $R = k[x_1 , \ldots , x_n]$ be a polynomial ring over a field $k$ and $M$ be a finitely generated $R$-module. Then $\text{pd }M \leq n$ ?

Answer 1

That is correct.

Actually, the modern statement of this theorem is that any polynomial ring $k[X_1,\dots,X_n]$ over a field $k$ has global homological dimension equal to $n$.