Question

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 $GL_n(k)$?

Answer 1

I think the term "affine bundle" is used for at least two things: (1) A map $p:Y\to X$ such that for some open cover (in your choice of topology) there are isomorphisms $p^{-1}(U)={\Bbb A}^n \times U$ --- just like you said. (2) A torsor for a vector bundle, i.e., like (1) but with the added condition that the transition functions are affine-linear.

In the situation of (2), it's a vector bundle exactly when there's a section (like any torsor). A simple non-vector-bundle-example is the complement of the diagonal in ${\Bbb P}^1 \times {\Bbb P}^1$, projecting onto one of the factors.

For (1), I don't know any general (non-trivial) criterion for such a thing to be a vector bundle. (Maybe because the group $Aut({\Bbb A}^n)$ is so complicated...) A simple non-example is the 2nd-order jet scheme $\mathrm{Hom}(\mathrm{Spec}(k[t]/(t^3)),{\Bbb P}^1) \to {\Bbb P}^1$. The fibers are ${\Bbb A}^2$, and there's a section, but it's not linear. I suppose one test is whether the sheaf of $O_X$-algebras $p_*O_Y$ admits a grading generated in degree one. (This fails for the jet schemes, though there is a natural grading by scaling $t$.)

Answer 2

Dear rvk, as you and Dave said, an affine bundle is a morphism $p:Y\to X$ such that for some open cover $(U_i)$ of $X$ there are isomorphisms $p^{-1}(U_i) \simeq {\Bbb A}^n \times U_i$. However the question arises: a cover for what topology? $\:$ In the cases you evoke, étale and Zariski, I am happy to report that it doesn't matter in the case of affine-linear transition functions.

In other words an affine bundle locally trivial in the étale topology is already locally trivial in the Zariski topology. The reason is that the affine group $Aff_n (k)$ is "special" in the terminology introduced by Serre here. Serre first proves that $GL_n(k)$ is special (Théorème 2) and then that an extension of a special group by a special group is special (Lemme 6). The specialness of $Aff_n (k)$ (which he doesn't mention) then follows easily from (Proposition 14).

Remark 1: It is difficult to overestimate the importance of Serre's article. It might be considered the birth certificate of étale topology under the guise of "fibrés localement isotriviaux".

Remark 2 : Groups are rarely special ( Serre knows what words mean!) . For example $PGL_N(k)$ is not special and so bundles with typical fiber $\mathbb P^n$ which are locally trivial in the étale topology needn't be locally trivial in the Zariski topology. This is the subject of Brauer groups of schemes, as envisioned by Grothendieck.