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So, wikipedia has a page about "affine space", where such a space is an object (set, topological space, blah) with a free transitive (blah, continuous, blah) action by the additive group of a free module.

At least when the base ring is an algebraically closed field, we can identify the top. sp. of $\mathbf{A}^n_k$, with such an affine space (I don't care enough to check whether or not the topology is compatible). (The point sets agree by the Nullstellensatz)

In the case that k is a field but not alg. closed, is there any way to make sense of the discrepancy between the definitions, or is calling the affine variety $\mathbf{A}^n_k$ "affine n-space" just a relic of the ultraclassical algebraically closed case where the identification actually makese sense?

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    Don't you just have to change the corresponding group? A_n^k is acted on by an algebraic group, blah, etc.2010-11-09
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    Well, A^n_k as a classical variety only makes sense when we've got k algebraically closed, right? So in the general case, you have some points floating about that don't appear as points in the k-vector space of the same dimension (basically because the nullstellensatz doesn't give the bijection we would expect in the ac case). Don't hesitate to correct me if I've gotten ahead of myself though.2010-11-09
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    Once you pick an identity there's an obvious group scheme structure on A_n^k (is it right to call this G_a^n?), and this is the group scheme for which A_n^k is a torsor. This seems like the right generalization to me. I don't think it's natural to ask for an action by a group rather than a group scheme.2010-11-09
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    Sure, but now we're already talking about our schemes as functors of points! (or at least it seems that way). Could you describe the action of G_a^n for me? Maybe I'm not seeing it.2010-11-09
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    Notice that "affine space" is also a name for an incidence structure (involving the flats in the space, &c), and I am pretty sure that the affine scheme over any field, with the set of flats in the affine scheme, properly defined, is an affine space in that sense.2010-11-09

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Affine space means a vector space in which the origin is not preferred. Its automorphisms are the group of affine linear transformations over $k$ (the usual linear transformations with respect to some (arbitrary) choice of origin, together with translations).

In algebraic geometry, one often encodes geometric objects in slightly complicated ways, and the preferred method of encoding is time-dependent, e.g. in that what Weil meant by a variety, and what it typically meant now, are not literally the same mathematical object, although the are both supposed to refer to the same Platonic reality.

Thus, one shouldn't think about $\mathbb A^n_k$ too literally in terms of its points when comparing with the Wikipedia discussion of affine space (since what its points are will depend on what foundations you use!). It does encode the geometric concept of affine space (as in the linked Wikipedia article) in the context of algebraic geometry. For example, its automorphisms are precisely the group of affine linear transformations over $k$. [Correction: As Mariano points out in a comment below, there are many non-linear automorphisms of $\mathbb A^n$; what I should have written is that the automorphisms of $\mathbb A^n$ which extend to $\mathbb P^n$ --- so these are the automorphisms of $\mathbb A^n$ that extend to the hyperplane at infinity in some reasonable sense --- are precisely the affine linear automorphisms.]

If you do insist on making the connection precise on the level of points, then one way to say it is that if $\Omega$ is any extension field of $k$, then the $\Omega$-valued points of $\mathbb A^n_k$ are naturally an $n$-dimensional affine space over $\Omega$.

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    Dear Matt E, Very nice, that explains it! One quick question though. What is the natural action that equips $A^n_k(L)$ with that structure?2010-11-09
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    Wel, we do not know the polynomial automorphisms of affine space, really... (Very, very recently Ivan Shestakov and Ualbai U. Umirbaev showed that the polynomial ring in 3 variables does have wild automorphisms; in two variables things are more normal)2010-11-09
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    The reference is [I. P. Shestakov and U. U. Umirbaev. The tame and the wild automorphisms of polynomial rings in three variables. J. Amer. Math. Soc. 17 (2004), no. 1, 197--227; MR2015334 (2004h:13022)]2010-11-09
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    @Mariano: Dear Mariano, I was worried about this blunder after I wrote my answer; thanks for your link and comment/correction. (What I was thinking, and should have said, is that the affine transformations are the *linear* automorphism of $\mathbb A^n$, where to avoid being circular, *linear* means automorphisms that extend to automorphisms of $\mathbb P^n$.) (And hopefully I'm not blundering again!)2010-11-10