TITLE: Affine bundles over varieties
QUESTION [7 upvotes]: Let $\mathbb{A}^n_k$ be the Affine $n$-space over an algebraically closed field $k$.
Let $X$ be a variety over $k$. What would be the right definition of an "Affine bundle" i.e bundle of fiber type $\mathbb{A}^n_k$ over $X$ (I mean local triviality in zarisky topology,or
etale .. )?. When can one get a vector bundle from an "Affine bundle" , more
precisely (I think !) if I assume the structure group of the affine bundle to
be $Aut_{Var_k}(\mathbb{A}^n_k)$, when can one get a reduction of the structure group
to $GL_n(k)$?

REPLY [3 votes]: The very interesting paper  "Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1976/77), no. 3, 279–299. MR 0432626 (55 #5613), by H. Bass, E. H. Connell, and D. L. Wright, gives some answers to this question. It says that if everybody is affine and the bundle is locally trivial in the Zariski-topology, then it is a vector bundle.