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 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.

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.

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 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.

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 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.