Let $R$ be a commutative Noetherian $F$-algebra, where $F$ is a
field (perfect, say). Assume that $R \otimes_F \overline F$ is a polynomial ring over the
algebraic closure $\overline F$.
Does it follow that $R$ was already a polynomial ring over $F$?
I doubt it, but haven't had any luck constructing a counterexample.
The question arises because I'm trying to understand Bruhat cells in
flag manifolds for non-split algebraic groups. In split ones,
Bruhat cells are affine spaces.
Best Answer
In his Bourbaki talk of 1994 Kraft tells us that the complex affine plane does not have non-trivial forms and that the corresponding question in higher dimensions is open. 1994 is an age ago, though!
(He also shows that the automorphism groups in high dimension are not amalgamated products of the two obvious subgroups by exhibiting two expressions of an automorphism as a product, contradicting the uniqueness implicit in amalgamation)