Let $R = k[x\_1, \ldots, x\_n]$ for $k$ a field of characteristic zero and let $S \subset R$ be a graded sub-$k$-algebra (for the standard grading: $\deg x\_i = 1$) such that $R$ is a free $S$-module of finite rank. Does this imply $S \cong k[y\_1,\ldots,y\_n]$?



If a polynomial ring is finite free over a subring, is the subring polynomial?
Note that $R$ and $S$ each have only one graded maximal ideal, so they are local in the graded sense, so most of the standard results for ungraded local rings are applicable. There is an obvious finite resolution of $k$ by modules that are finitely generated and free over $R$ and thus also over $S$. This implies that $S$ has finite global dimension, and so is a regular local ring by a theorem of Serre (Theorem 19.2 in Matsumura). Together with the grading assumptions this forces $S$ to be polynomial.


