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 $GL_n(k)$?
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$\begingroup$ Well it's a $X$-scheme which locally on $X$ looks like $A^n \times X$. It's almost the same as a vector bundle, because in a vector bundle the linear structure belongs to the data. $\endgroup$– Martin BrandenburgCommented Feb 6, 2011 at 8:04
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3 Answers
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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 $t$.)
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$\begingroup$ Just as an additional data point, I think have only seen the meaning (2). $\endgroup$ Commented Feb 6, 2011 at 11:25
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1$\begingroup$ I agree with AByer on common usage. The other useful test is that, given a vector bundle $V$, the torsors for $V$ are classified by $H^1(X, V)$. For example (exercise!), in the $\mathbb{P}^1 \times \mathbb{P}^1$ example, the vector bundle is $\mathcal{O}(-2)$ and the affine bundle Dave gives corresponds to a nontrivial element of $H^1(X, \mathcal{O}(-2)) \cong k$. $\endgroup$ Commented Feb 6, 2011 at 14:39
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$\begingroup$ I prefer (2) as well, but Fulton's Intersection Theory uses (1) -- and it seems others (e.g. Georges!) do as well. $\endgroup$ Commented Feb 6, 2011 at 19:16
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$\begingroup$ Dear Dave, no, no, I'm with you , AByer and David on prefering (2) and that is what I mean with my notation $Aff_n(k)$ . I'm sorry that my post was ambiguous on this point : I've just edited it and adopted your excellent terminology "affine-linear". It is amusing that you mention Fulton: his meaning (1) for "affine bundle" had also struck me as unusual, but on the other hand it makes more general his surjectivity theorem for the morphism of Chow groups induced by the projection of an affine bundle . $\endgroup$ Commented Feb 7, 2011 at 0:53
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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.
answered Feb 6, 2011 at 16:17
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$\begingroup$ Thanks for this! (I hadn't known all solvable groups were special -- only thinking of the classification for semisimple ones, I forgot not every group is semisimple...) $\endgroup$ Commented Feb 6, 2011 at 19:26
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$\begingroup$ Dear Georges: I have a few times decided to try to understand what the Brauer group is, but not even grasped the definition of an Azumaya algebra. Your last remark makes the whole thing sound very reasonable and geometric -- could you perhaps say a few words about what $\mathrm{PGL}_n$-torsors have to do with the Brauer group, or recommend a readable reference? $\endgroup$ Commented Feb 6, 2011 at 20:15
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$\begingroup$ Dan, if you will accept an interruption, my own recommendation would be the first parts of Deligne's Arcata lectures in SGA4 1/2. $\endgroup$– inkspotCommented Feb 6, 2011 at 23:01
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$\begingroup$ Dear Dan, my favourite reference is without doubt Gille and Szamuely's book [1] Central Simple Algebras And Galois Cohomology. It is written in a very user-friendly style and I think you can fairly quickly jump to Chapter 5, where Severi-Brauer schemes are introduced and linked to $PGL_n(k)$ . [1]; books.google.fr/… (to be continued) $\endgroup$ Commented Feb 6, 2011 at 23:11
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$\begingroup$ (continuation) There is also a a nice survey by Jahnel of these questions that you can find here: uni-math.gwdg.de/jahnel/Arbeiten/Bpal.pdf $\endgroup$ Commented Feb 6, 2011 at 23:20
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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.
