Generalizations of Hilbert's Syzygy theorem Hilbert's Syzygy theorem states that a minimal free resolution of a finitely generated graded module over a (standard graded) polynomial ring in $n$ variables $k[x_1, \ldots, x_n]$ does not have more than $n+1$ terms in it. To what rings other than the polynomial ring has Hilbert's theorem been generalized? Does it hold for polynomial rings which are not standard graded? Please give me a reference if the answers to these are known.

As suggested by Jason I put my comments into an answer. Hilbert's theorem means that all modules over $k[x_1,\ldots ,x_n]$ have projective dimension $\leq n$ -- one says that the _global dimension_ (aka homological dimension) of $k[x_1,\ldots ,x_n]$ is $n$. This is a property of the ring $k[x_1,\ldots ,x_n]$, it does not depend on the grading. It applies to many other rings: by a famous theorem of Serre, a local (commutative) ring has finite global dimension if and only if it is regular.