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!
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