In Eisenbud's Commutative Algebra, Exercise 18.18 is the following fact from a paper of Hartshorne's:

Suppose $(R,P)$ is a local ring containing a field $k$, and let $x_1,...,x_r\in P$ be a sequence of elements. If $x_1,...,x_r$ is a regular sequence, then $R$ is flat as a module over the polynomial ring $k[x_1,...,x_r]$.

My question is: Does this also hold for different situations, i.e. if $R$ is not assumed to be local? A comment in Exercise 6.7 of [Eisenbud] suggests that it does, but unfortunately no reference is provided. I would be particularly interested in the case where $R$ is itself a polynomial ring.

Any help would be appreciated!