Talk:Complex affine space

Proposal to merge

In complex algebraic geometry, a "complex space" usually means something completely different (a geometric space obtained by gluing complex manifolds with singularity). This article discusses Cⁿ. I therefore suggest we merge it to complex number. Meantime, we should start an article on "complex space" in algebraic geometry (Draft:complex space). -- Taku (talk) 13:56, 26 January 2015 (UTC)

• Oppose. Many disciplines use the idea of a complex space, and I'm sure they all think their idea is rather different from every other discipline's. The idea of a manifold is also closely linked. The present article is very short and I would prefer to see it become a disambiguation page, able to identify commonalities and differences between the various disciplines. For example, what are the commonalities and differences between number theory and complex algebraic geometry in the way that each relates to the idea of a complex Hilbert space? Somebody who has learned about complex space in one discipline could benefit strongly from such a comparative primer on understanding the term when encountered in other disciplines. — Cheers, Steelpillow (Talk) 14:50, 26 January 2015 (UTC)
• Oppose. I don't think a merge is the right approach here. This article presently concerns n-dimensional complex coordinate space, so I think it should probably be moved to a more descriptive title. Presumably we can say some non-trivial things about that in its own right: as n-dimensional affine space in algebraic geometry, as the coordinate space for complex manifolds, as the carrier of the basic finite-dimensional complex/unitary representations of various groups, etc. None of these topics are really appropriate for the main complex number article though. Sławomir Biały (talk) 14:59, 1 February 2015 (UTC)

Withdrawal: The consensus seems clear and the reasons given are valid. Also, I found the "complex space" article in the sense in algebraic geometry already exists. I have thus taken the laziest solution of putting a hatnote, leaving the matter to the other editors. Also, following Sławomir's suggestion, I have moved the page to "complex affine space" (I didn't think the move was controversial.) -- Taku (talk) 03:36, 4 February 2015 (UTC)

Is it affine?

Looking at this article I do not understand why is it called "complex affine space" rather than (say) "complex vector space" or even "complex Cartesian space"? Boris Tsirelson (talk) 08:00, 4 February 2015 (UTC)

It is affine as $\mathbb {A} ^{n}$ over the complex numbers, in contrast to "complex projective space" $\mathbb {P} ^{n}$. A "complex vector space" would not be a right title since the structure of a vector space does not include a choice of basis.

Wow... As far as I understand, the structure of an affine space does not include a choice of basis, nor of the origin (zero) point. At least, this way it is treated in our "affine space" article, isn't it? Boris Tsirelson (talk) 19:55, 4 February 2015 (UTC)

Well, we were clearly talking about different concepts using the same term "affine"; whence the confusion. I was using "affine space" in the sense in algebraic geometry; i.e., the set of n-tuples of elements in the base field. To clarify, I have started affine space (algebraic geometry). -- Taku (talk) 23:10, 4 February 2015 (UTC)

I see. Nice. Boris Tsirelson (talk) 12:13, 5 February 2015 (UTC)

Hilbert spaces

How do complex affine spaces relate to complex Hilbert spaces? — Cheers, Steelpillow (Talk) 12:31, 30 September 2015 (UTC)

A complex affine space is a Hilbert space with the usual (complex) inner product giving a structure of Hilbert space. -- Taku (talk) 03:21, 4 October 2015 (UTC)

This article does not mention inner products but the article on Hilbert Space does. To define something as "a Hilbert space with [an] inner product giving a structure of Hilbert space" is a tautology, it does not clarify anything for me. Do you mean that a complex affine space and a complex Hilbert space are the same thing? — Cheers, Steelpillow (Talk) 09:52, 4 October 2015 (UTC)

No and the article should mention the standard inner product. It's not a tautology; I just meant a complex affine space together with the standard inner product is a Hilbert space. In fact, every complex Hilbert space of finite dimension is isomorphic to a complex affine space (the article now has a paragraph on this.) -- Taku (talk) 00:21, 5 October 2015 (UTC)

Thank you. However I remain puzzled. If a complex Hilbert space is a complex affine space together with its inner product then how can it be isomorphic to the complex affine space (presumably without the latter's inner product)? Can you give an example of a complex Hilbert space that is not a complex affine space, and an example of a complex affine space that is not a complex Hilbert space?

Also, you have edited the section on the complex line to say that it comprises two real dimensions. This is patently absurd, its whole motivation is that one dimension is imaginary and therefore not real. (The Argand diagram exists in two real dimensions, but that is a different thing). What is your justification for your edit?

— Cheers, Steelpillow (Talk) 11:12, 5 October 2015 (UTC)

I mean: isomorphic to the complex affine space that is equipped with the standard inner product (without an inner product it's not a Hilbert space). I think it is standard to assume the complex affine space is equipped with the standard inner product (just as it is natural to assume an Euclidean space is equipped with the usual metric). Since the complex affine space that is equipped with the standard inner product is a Hilbert space, there is no example of 'Cⁿ that is not a Hilbert space (unless you equip it with a non-standard metric). As for a non-example in the other way, let V be a complex Hilbert space of dimension n with the inner product denoted by $\langle ,\rangle$. Consider $\beta (x,y)=\langle Tx,Ty\rangle$ with some isomorphism $T:V\to V$ of vector spaces. Unless T is unitary, this gives an inner product that is not the same as $\langle ,\rangle$.

As for my other edit, I think the paragraph in question is problematic; I don't think there is such a thing as imaginary dimension (or maybe I just haven't heard it.) I will add a tag requesting clarification. -- Taku (talk) 23:34, 5 October 2015 (UTC)

So in this respect I think we should be saying that a complex affine space is an example of a complex Hilbert space with a certain isomorphism condition. Personally, I would like to see a section in one of the articles explaining this, much as you have here, together with an outline of any important consequences (if there are any).

If one talks of two dimensions, having real and imaginary coordinates respectively, it seems invidious not to talk of the latter as an imaginary dimension. For example a Calabi-Yau manifold in string theory is often said to comprise six dimensions. Yet mathematically, the manifold has three complex dimensions. It is untenable to talk of a "real dimension with imaginary coordinates" - such a phrase is conflating philosophy with mathematics in an utterly non-rigorous way: the "real" being a philosophical notion dragged in to avoid conflict with the strictly mathematical usage of "imaginary". While the physicist may muddle along as he pleases, the pure mathematician is left with the notion of an imaginary dimension. Then again, if the imaginary aspect were merely a property of the coordinate system, one would expect a non-metric construct such as a polygon to behave the same regardless. But it does not: a complex polygon behaves very differently from a real one. Anyway, I have edited the section to try and avoid the "d" word. — Cheers, Steelpillow (Talk) 07:20, 6 October 2015 (UTC)

Up to an isomorphism, I don't think there is anything interesting about the distinction between complex affine spaces and Hilbert spaces. One way to think is this is a matter of an automorphism of a complex affine space, which is certainly an interesting topic (though a complex projective space is perhaps a more natural setup.) I don't view coordinates or dimensions are interchangeable; a dimension is just one way to measure a size of the space, which can be defined and interpreted In a completely coordinate-free way. Hence, "imaginary dimension" just doesn't sound right. The term "imaginary coordinate" makes a perfect (usual) sense; so I think the new version is ok. -- Taku (talk) 01:37, 7 October 2015 (UTC)

If the distinction is so trivial, should the two articles be merged? — Cheers, Steelpillow (Talk) 08:29, 7 October 2015 (UTC)

No. I don't think the geometry of a Hilbert space of "finite-dimension" is really seriously studied, precisely because it's basically that of the complex affine space. The article Hilbert space is mainly about a Hilbert space of infinite-dimension; many topological subtleties disappear in the finite-dimensional case. -- Taku (talk) 10:30, 7 October 2015 (UTC)

Now I'm getting confused again. Your latest comment implies that the infinite-dimension case is a significant distinction. But the presence or lack of infinity is not a matter of homeomorphism, which you previously said was the only significant distinction. Whatever the definitive criteria for distinction, it all seems to point to the idea that the complex affine spaces form a subset of the more broadly-defined Hilbert spaces. Would that be wrong? — Cheers, Steelpillow (Talk) 14:31, 7 October 2015 (UTC)

No that's correct, since a complex affine space is an example of a Hilbert space. But, conversely, a complex Hilbert space of finite-dimension is isomorphic to a complex affine space. In other words, up to isomorphisms (not homeomorphisms) of Hilber spaces, there is no interesting distinction between complex affine spaces and complex Hilbert spaces of finite-dimension. But an infinite-dim Hilbert spaces do behave very differently. -- Taku (talk) 04:46, 9 October 2015 (UTC)

Thank you. I think I understand now. I'll try and cross-link the two articles. — Cheers, Steelpillow (Talk) 11:28, 12 October 2015 (UTC)

I have now added a brief section on complex affine spaces to the examples at Hilbert space. I hope I got it right. — Cheers, Steelpillow (Talk) 11:05, 13 October 2015 (UTC)

Requested move 13 October 2015

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: NOT MOVED. Consensus is that the article contents now meet the article title. SilkTork ^{✔Tea time} 18:42, 6 November 2015 (UTC)

---

Complex affine space → Complex coordinate space – In my opinion, the content of the article complex affine space should be moved to complex coordinate space. Affine spaces do not have a linear structure, let alone a preferred coordinate system. They also lack an inner product. So the current article title, while it does refer to a thing in mathematics, does not match the thing described in the article. The move requires administrator tools. Sławomir
Biały 14:15, 13 October 2015 (UTC) --Relisted. Tiggerjay (talk) 06:42, 22 October 2015 (UTC)

Slawomir, do you withdraw the move request now? Boris Tsirelson (talk) 17:24, 18 October 2015 (UTC)

• Oppose The discussion is now obsolete. Major changes were made to the article during this discussion, and now it corresponds well to its current name (and does dot correspond to the new name proposed). Boris Tsirelson (talk) 15:26, 6 November 2015 (UTC) Support (again): I tried to understand the position of the opponent (Taku), but unsuccessfully. The move is a cheap solution. Boris Tsirelson (talk) 14:53, 17 October 2015 (UTC) Abstain: The move is not the best solution of such terminological problem. A better solution, via equivalent definitions of mathematical structures, is possible. See comments near the bottom. Boris Tsirelson (talk) 06:07, 15 October 2015 (UTC) Support: When an affine space has an inner product given on its difference space, it is called a Euclidean affine space (in the real case; not sure about the complex case). [User:Tsirel|Boris Tsirelson]] (talk) 15:06, 13 October 2015 (UTC)

  But an inner product on an affine space is nonsense. There is no linear structure for the inner product to be "bilinear" with respect to. An inner product on the underlying vector space would make sense (and that would be a Euclidean space, for which we already have an article). But it's not even clear that that's what this article is trying to be. Sławomir
  Biały 11:10, 15 October 2015 (UTC)

  Who wrote nonsense? I wrote "inner product given on its difference space". On the affine space itself, it gives Euclidean metric (not norm). Boris Tsirelson (talk) 14:02, 15 October 2015 (UTC)

  Sorry, I missed that phrase in your post. Anyway, the article as currently written is nonsense, because it does describe an "inner product" defined on the "affine space" itself. Sławomir
  Biały 14:57, 15 October 2015 (UTC)

  Sure. Should we adapt the title to the content, or the content to the title? The article 'Real_coordinate_space" contains the section "afine space" that specifies (too sketchy) the group of automorphisms. We could do so here. It is itself not a crime, to use something (this time, the origin and the inner product) not invariant under the isomorphisms, provided that this non-invariance is noted boldly (for now it is not, indeed). In other words: the affine space can be upgraded (not uniquely) to a vector space, and further, to a finite-dim Hilbert space. Some people like to think this way: start with a favorite model, and then downgrade it as needed (in the spirit of Erlangen program). Let them, provided that this downgrade is (or will be...) noted boldly. Topological space has at least 7 definitions; also here we could give more than one. Boris Tsirelson (talk) 15:25, 15 October 2015 (UTC)

• Support: What is described is an affine space, but, generally, an affine space does not have an origin nor canonical coordinates. D.Lazard (talk) 17:00, 13 October 2015 (UTC)

  Comment 1: Presently Complex coordinate space redirects to Several complex variables. This is not a problem for the move.

  Comment 2: Real coordinate space is about the real analogue of this article. Thus the move would increase the coherency of WP

  Comment 3: Affine space (algebraic geometry) is about the analogue over any field. I suggest to move it to Coordinate space. This article already exists, but it has "multiple issues" since a long time, and I guess that its terminology is WP:OR. In any case, this is not a common terminology. Thus this move would not destroy anything relevant. This kind of concerns against Coordinate space appears clearly in its talk page. Quondum has tried, some time ago to improve this page. Would he agree with the suppression of the present content of this article? If not, it would be possible to keep a part of the present content as a section (that is merging instead of moving).

  Comment 4: I was uncomfortable with the fact that the "affine space" that is used to define algebraic varieties (in many articles) are not exactly what mathematicians, including algebraic geometers call affine space. Replacing "affine space" by "coordinate space" in these article would be a strong improvement. Also, such a change of terminology would be more natural for the definition of the coordinate ring of an algebraic set. D.Lazard (talk) 17:00, 13 October 2015 (UTC)

I'm wondering what the source of affine space (algebraic geometry) could be? The n-dimensional affine space really ought to be defined as the spectrum of a free n-dimensional commutative algebra. This has a canonical affine structure, but no canonical choice of coordinates. Coordinates presumably correspond to a choice of generators of the algebra. Sławomir
Biały 19:26, 13 October 2015 (UTC)

The sources would be Hartshorne and Mumford's red book (I've just checked them); Hartshorne defines the affine space at the very first paragraph of the very first section of the very first chapter as kⁿ (with k an algebraically closed field); Mumford assumes the readers know the affine space = kⁿ. Thus, the article is consistent with reliable sources (regardless of the editors' feelings.)

By the way, there is a name for an algebra freely generated by n elements: a polynomial ring (no isomorphism involves since it is just a definition.) -- Taku (talk) 20:37, 13 October 2015 (UTC)

Taku, do you agree or disagree that the spectrum of a polynomial ring is an affine space? (Or does being an affine space require some extra structure, like a choice of coordinates?) Sławomir
Biały 20:57, 13 October 2015 (UTC)

I submit that Hartshorne is engaging in what the Bourbaki folks would call an abuse de langage. Hodge and Pedoe define projective space very carefully, avoiding coordinates, and then define affine space by fixing a prime at infinity. Perhaps a better title for the article that currently occupies affine space (algebraic geometry), would be affine space (commutative algebra), to refer to the ringed space $(k^{n},k[x_{1},\dots ,x_{n}])$. The ultimate article should make it clear that one is not talking about the set $k^{n}$, but the ringed space. Also, it's not clear to me whether this ring is the polynomial algebra or the ring of polynomial functions (so k[x] is taken "modulo Frobenius"). Presumably these are the kinds of questions that an article should be able to answer, and maybe the answer depends on exactly what one is doing. But a silly article calling "affine space" as "kⁿ, the set of n-tuples of elements in k" is (1) misleading, (2) wrong, even for the purposes of algebraic geometry, and (3) not really all that informative anyway. Sławomir
Biały 21:54, 13 October 2015 (UTC)

It's likely that the definitions we're using are not historical accurate; Hodge–Pedoe is clearly written in a style that is not comtemporary. (I can't follow their presentations and I don't think I'm alone.) This is a bit unfortunately and I'm certainly for adding notes to the relevant articles.

As for more technical matter, it is certainly important to distinguish the set kⁿ and the possible algebraic-variety structure on it. I agree that the spectrum of a polynomial ring is the affine space (and affine space (algebraic geometry) does say this.) As you said, this definition is really optimal since one can get a coordinate-free definition of an affine space easily: it's just the spectrum of the ring of polynomial functions; this ought to be mentioned in affine space (algebraic geometry). -- Taku (talk) 23:35, 13 October 2015 (UTC)

Well for contemporary algebraic geometry, the standard is EGA. Grothendieck defines an affine space by its functor of points. For affine spaces over a ring, it turns out that this functor has a specific model, namely the n-dimensional coordinate space. As a functor of points, it's given by a universal construction, so "the" functor is only uniquely defined up to isomorphism. But that's probably a distraction and not worth arguing about. The real guts of the thing is the affine space of a locally free sheaf over a ring (see EGA I, 9.5.2). There the functor of points for a locally free sheaf E over a scheme S is $\operatorname {Hom} _{O_{S}-Alg}(\operatorname {Sym} (E^{*}),O_{S})$. In particular, this makes sense if S is the trivial scheme Spec(k), and E is a k vector space. The key point: k has only one affine space of each dimension up to isomorphism (!). Sławomir
Biały 00:40, 14 October 2015 (UTC)

One way to define a projective space is by the universal construction: Pⁿ is characterized by maps to it; i.e., line bundles together with generating sections (Global Proj touches on this by it's written too abstarctly for me). I suppose we can do something similar for the affine space; that certainly fits to affine space (algebraic geometry). Wikipedia does not have much on functor of points except the "Global Proj" section. We do now have morphism of schemes#Morphisms as points, which should be a good start.

As for EGA, again it's too abstract. You forgot 9.5.1., in which the affine n-space over a ring R is defined as Rⁿ. So what we have here is consistent here too. Despite the terminology what they are considering seems what is called an affine bundle, a fiber bundle with fibers isomorphic to an affine space. -- Taku (talk) 04:57, 14 October 2015 (UTC)

Well, I think we all would agree that Rⁿ is an affine space. If R is a ring, it might even make sense to apply the definite article, as in "the affine n-space". But I don't think it is appropriate for affine space (algebraic geometry) to describe this case exclusively, especially not when there are other examples, even in EGA. Per EGA 9.5.2, the spectrum of the symmetric algebra of $V^{*}$, for V a vector space over k is also an affine space. Yet you insist that we must call it an "affine bundle" instead, which I strongly disagree with. This is an affine space in the usual sense of mathematics everywhere: affine space. It is only a "bundle" over a single point, in the same sense that all spaces in algebraic geometry are "bundles" over some terminal object. This usage is supported, for instance, by Borel's Linear algebraic groups (section 7.1). And anyway, even if you were to insist that in algebraic geometry "affine space" only means "coordinate space" (which I disagree with), I really think it is wrong for this marginal usage to leak out into other articles such as the present one, which apparently has nothing to do with algebraic geometry. That's clearly an error. So, I'd appreciate it if you'd adjust your vote accordingly. Sławomir
Biały

There is some miscommunication. I do agree that Spec of the symmetric algebra of $V^{*}$ is an affine space and that is indeed mentioned in affine space (algebraic geometry); the article has been modified to make this clearer. The reference to "affine bundle" was because, from what I can tell, that notion is what they want to consider. For that, the universal construction, which allows a base to be general, is an appropriate language. But using the term "affine space" to refer to both an affine space (which admits a vector space model) and an affine bundleis is confusing; just as calling a vector bundle a vector space is confusing, even if it makes some sense. (please see also my reply below as well). -- Taku (talk) 21:55, 14 October 2015 (UTC)

• Comment: I'm responding to a mention of me. I'm essentially in agreement with everything said in this thread so far. I am generally in favour of defining a concept clearly and in the manner of modern terminology. I do not know what content suppression D.Lazard refers to, though I do feel that the mention of the inner product is presently unduly prominent, regardless of the proposed renaming. I leave the choices in the very capable hands of those showing an interest. —Quondum 17:47, 13 October 2015 (UTC)
• Oppose: I think the proposed move would constitute the OR; as I replied to Sławomir, the article title is consistent with the reliable sources. It's certainly inconsistent with the definition at affine space but is consistent with affine space (algebraic geometry). There are two articles on "affine space" since the terminology is inconsistent (which is just the nature of life.) -- Taku (talk) 20:37, 13 October 2015 (UTC)

However, there is a deep question that underlies this terminology question: the simplest definition of an affine algebraic variety and of its coordinates ring involves polynomials, and thus affine coordinates in the underlying affine space. Although all the theory may be developed in a coordinate free way, this is too technical to be done in an elementary presentation. This is the reason for which the good introductions, such as Mumford's and Hartshorne's, use a coordinate dependent presentation, and then prove (or omit to prove, if this is not important in what follows) that an affine change of coordinates induces an automorphism of the coordinate ring, and thus that everything is invariant under affine transformations and affine changes of coordinates. IMO, this is the reason for which, by shortcut and/or abuse of language, Mumford's and Hartshorne's (and many others) talk of Kⁿ as the affine space of dimension n over the field K. I am quite sure that they did not intend to change the usual definition of an affine space (that they knew and used in other works). They simply simplified the definitions for a pedagogical purpose. IMO, this is the following sentence which is OR (and wrong): "In algebraic geometry, the affine space is Kⁿ." An example: algebraic groups, such as the orthogonal group or the affine group, are affine algebraic varieties (by definition of algebraic group) which are generally defined coordinate free. Thus the Kⁿ that is used is not uniquely defined. D.Lazard (talk) 21:51, 13 October 2015 (UTC)

Hartshorne is known for engaging in some tinkering with definitions for pedagogical purposes: the famous example is the accusation that his definition of "coherent sheaf" isn't quite correct (which has some merit but...) Of course, he is not alone and I think Wikipedia should join the rest; that's what the editorial policy of Wikipedia dictates after all. I certainly agree that the coordinate-free presentation is needed in some contexts; especially in the discussion of algebraic groups; e.g., the orthogonal group O(V) should be defined as a (Zariski-)closed subgroup of GL(V) as opposed to GL_n. The article affine space (algebraic geometry) should be expanded to having discussion on this matter (affine space without coordinates). -- Taku (talk) 23:23, 13 October 2015 (UTC)

I don't see how moving to a less ambiguous title is OR. In fact, what seems to be OR is calling the present article "complex affine space", when sources such as Berger's Geometry clearly mean something else. Does Hartshorne use the exact phrase "complex affine space"? I didn't think so. What's OR is to claim that the title is consistent with Hartshorne's definition (especially not when there is some contention over whether Hartshorne actually intended to redefine affine spaces, and when even in algebraic geometry there are countermanding sources like EGA, Hodge and Pedoe, Borel, etc). Also, Berger does use the exact phrase "complex affine space". And he does not mean the subject of this article. He means an affine space that is complex. Finally, it's ridiculous to suggest that the title "complex coordinate space" is, in itself OR. This term is used by a large number of sources, one of which is the famous book Characteristic classes, by Milnor and Stasheff. Sławomir
Biały 10:58, 14 October 2015 (UTC)

(Please see below for my reply. -- Taku (talk) 01:04, 15 October 2015 (UTC))

• Comment. The correct (whatever that means) definition is: X is an affine space over an algebraically closed Spec k if X is a torsor over G_aⁿ. It is easy to show that such an object is always isomorphic as a torsor to G_aⁿ = Spec k[x₁, ..., x_n], where the latter is given the trivial torsor structure over itself. In general, X is an affine bundle over S if X is a torsor (with respect to an appropriate topology, like fppf) over the base change (G_a)_S, but over general bases, there are uniqueness questions for torsors (see, e.g., Severi–Brauer variety for the case of projective spaces). I don't know what happens for Spec k with k not algebraically closed off the top of my head; surely this is some simple thing in Galois cohomology? Ozob (talk) 05:22, 14 October 2015 (UTC)
• Comment. Is this a problem of affine space, specifically? Surely, not; much more generally, this is a problem of equivalent definitions of mathematical structures. When defining a (unique up to isomorphism) mathematical structure, we typically have a choice: either

give a favorite model and specify automorphisms; or

a "favorite-less" (in particular, coordinate-less) definition.

What is better? Tastes differ. Why take one side? Boris Tsirelson (talk) 05:48, 14 October 2015 (UTC)

• Further comment: The raison d'etre of Wikipedia is to clarify confusing terminology, among the others. Boris Tsirelson nailed the issue here. If something is isomorphic to an affine space (in some specific context), do you call it an affine space? I think so; or more specifically that is the matter of a model (e.g., an infinite projective space is a "model" for the classifying space of line bundles.) Affine space (algebraic geometry) has been edited to clarify this.

As for the title of this article: no Hartshorne does not use the exact phrase "complex affine space" and Berger in Geometry I does use "complex affine space" (Exercise 2.8.7.) but Berger here simply meant an affine space over the complex numbers. This article is not about this and the can be fixed by a hat note. (Milnor–Stasheff is old and isn't good for the terminology reference.) So, I still maintain the current article title is ok. (see below) A better term is probably "complex space" but is unfortunately in conflict with "complex complex" (complex-analytic space) in complex algebraic geometry. Taku (talk) 01:04, 15 October 2015 (UTC)

• The large majority of sources for the term "complex affine space" are for an affine space that is complex. Examples: [1], [2], https://books.google.com/books?id=SAscCAAAQBAJ&pg=PA233&dq=%22complex+affine+space%22&hl=en&sa=X&ved=0CCkQ6AEwAmoVChMI8Lywg6nEyAIVjf2ACh3GMA9x], [3], [4], [5] (by I M Gelfand), [6] (by John Milnor), [7]. So I find the claim that the algebraic geometry meaning is the primary one (even assuming it is correct, which is debatable) to be very questionable. And anyway, there's no need for a hatnote if we move the article to a title that is not ambiguous. Sławomir
  Biały 11:08, 15 October 2015 (UTC)

  Correct me if I'm wrong, but aren't they are basically sources from topology and differential geometry? "Affine space" here (that is Aⁿ) is used in the sense in algebraic geometry not in the sense in affine space. I can be open to renaming; my understanding still is that "complex coordinate space" is nonstandard; cannot be found in Hartshorne, Mumford, EGA, etc. -- Taku (talk) 04:02, 16 October 2015 (UTC)

  Hartshorne, Mumford, EGA, etc., are not the final arbiters on what is "standard" terminology in mathematics. But even if they were, the term "complex affine space" does not actually appear in any of these sources either, so claiming that "complex coordinate space" is non-standard per these sources is a red herring. Furthermore, it has been established that the present article is not consistent, even with the use of the term "affine space" in algebraic geometry (Hodge and Pedoe, EGA, Borel). The term "affine space" does not mean some very different thing in algebraic geometry than it does in other branches of geometry. I have shown that "complex coordinate space" is not original research. There are sources that use this term. One can find the term "coordinate space" even in algebraic geometry! See, for example, V.I. Danilov "Algebraic varieties and schemes" (trans. D. Corday), in I.R. Shafarevich Algebraic geometry I, Springer. Here he discusses the coordinate space Kⁿ and the coordinate affine space $\mathbb {A} ^{n}$.

  Also, I have argued that this is consistent with usage elsewhere on wikipedia (cf. real coordinate space, coordinate space), so that the principle of least surprise suggests that article title would be best, unless you want to counter-propose moving all of those articles elsewhere. Sławomir
  Biały 11:47, 17 October 2015 (UTC)

  Of course, Hartshorne and Mumford are not the only references on algebraic geometry, but they are more standard than the others. Shafarevich is known for sometimes using somehow non-standard terms; perhaps because he is Russian. Mathematically, that's nothing wrong with it, but it's not a good reference on terminology; as far as the terminology is concerned we have a bias towards English-language sources. I've just looked at Vakil's [foundations of algebraic geometry, which has increasingly become one of standard references and he uses the exact term "complex affine n-space" (pg. 104.). In pg 164 of Red Book, Mumford (sort of) defines "affine spaces" as schemes isomorphic to Aⁿ. Personally, I have no strong preference. The evidence is in favor of "complex affine space" as opposed to "complex coordinate space" (I'm still open to the compromise "complex n-space".) -- Taku (talk) 22:14, 17 October 2015 (UTC)

  Several issues: (1) why are "references on algebraic geometry" the only references to consider? This article is not called "complex affine spaces in algebraic geometry" is it? "Affine space" has a well established meaning throughout mathematics, even in algebraic geometry (as sources like Borel's "Linear algebraic groups", Milne's "Algebraic geometry", Grothendieck's "Elements de Geometrie Algebrique" amply show). You have shown no evidence in any of this discussion of appreciating this fact, nor indeed do you show any evidence of understanding the definition. (2) Also, to point out what seems to be unclear to you: if we define an affine space as a scheme isomorphic to Aⁿ, then that is not the same thing as defining an affine space as the n-fold Cartesian product of k with itself. Perhaps you are familiar with a similar situation in linear algebra. An n-dimensional vector space is isomorphic to kⁿ. But of course, we don't say that kⁿ is the only n-dimensional vector space. It has distinguished coordinate functions on it. Many sources may define something like real n-space as $\mathbb {R} ^{n}$, for instance. This is precisely that kind of situation, where sources define affine n-space. Some sources (such as Milne, Borel, Hodge-Pedoe, and Shafarevich) are a little more careful about foundational issues like whether there are coordinates. Others, like Mumford's "Red Book", say that affine spaces are those isomorphic to the standard one. (3) In spite of point (1), I've already given plenty of very standard algebraic geometry references. You dismissed Hodge and Pedoe's seminal treatise on the subject, as well as the definition that appears in the aforementioned section of EGA. Also, check out Milne's "Algebraic geometry" or Borel's "Linear algebraic groups", where they do things in a coordinate-invariant way. The belief that in algebraic geometry an "affine space" means "a set with n linear coordinate functions defined on it" simply does not agree with sources, nor indeed with any kind of reasonable common sense. (4) The term "affine space" already has an established meaning throughout mathematics. The use in algebraic geometry is not distinguished in this regard (although there are things that are of greater interest to algebraic geometry, such as the Quillen-Suslin theorem and vanishing theorems for all flavors of cohomology groups.) See, for example, Marcel Berger's book on the foundations of geometry ("Geometry"), Dan Pedoe's Dover classic "Geometry", Ito's "Encyclopedic Dictionary of Mathematics", HSM Coxeter's "Introduction to geometry", etc.

  Good grief. It's time to just drop it, Taku. Sławomir
  Biały 00:15, 18 October 2015 (UTC)

  Clearly, we need a tie-breaker; we just don't agree on the reading of the literature. Why, for instance, Berger's "Geometry" more relevant than say Vakil's notes? (Both use the exact phrase "complex affine space") There is still just not enough evident to support the "complex coordinate space" is more standard than "complex affine space". I have also already proposed the compromised solution: "complex n-space". Anyway, we just can't agree and so we should let the others decide. (I don't insist on my position, but I just don't agree with the reading of the literature with you.) -- Taku (talk) 04:13, 18 October 2015 (UTC)

  (1) I never claimed that "Berger is more relevant than Vakil's notes". (Yet I would certainly argue that Berger's source is a substantially more WP:RS compliant source than Vakil's lecture notes.) I've given you a list of very high quality references, both from within algebraic geometry and without. Berger was certainly on that list. These references are HSM Coxeter "Introduction to geometry", Marcel Berger "Geometry", Alexander Grothendieck "Elements de geometrie algebrique", JS Milne "Algebraic geometry", Armand Borel "Linear algebraic groups". I've also given the output of a Google books search which shows pretty definitively that when the exact phrase "complex affine space" is used in the literature, it means a complex space that is affine, rather than the n-fold Cartesian product of $\mathbb {C}$ with itself, together with the n canonical projections into $\mathbb {C}$.

  (2) Vakil also does not actually support this article, as you have misleadingly said it does for some reason. He defines the complex affine space as the spectrum of a polynomial ring. I agree with this definition, yet I disagree with the definition in the present article! The spectrum of a polynomial ring carries a canonical affine structure, but not a canonical linear structure. (The set of (unital) algebra homomorphisms is closed under affine linear combinations, but not arbitrary linear combinations.) There is no canonical identification of this Spec with $\mathbb {C} ^{n}$. For example, the latter has a linear structure while the former does not. Sławomir
  Biały 14:18, 18 October 2015 (UTC)

• Proposal: How about complex n-space? Since the problem is the crash of terminology, that name would solve the problem. -- Taku (talk) 01:30, 15 October 2015 (UTC)

  Comment. This article was originally named Complex space and was later changed to Complex affine space. Personally I see no value in the n either, it would be better to simply change it back and, if need be, realign the content back to the original intent. IMHO the topic needs an overview article and this was originally supposed to be it. All the confusions can then be explained together, not just diced, sliced and scattered about in more specialist topics. The article would then provide a structured set of links to those topics. — Cheers, Steelpillow (Talk) 08:23, 15 October 2015 (UTC)

  Alternatively, since this article has long left its origins behind, I could restore the page at Complex space (currently a redirect) to its state of 20 November 2013], immediately before the series of edits which led to its move, and begin adding links to the various specialised varieties. Does anybody object to that? — Cheers, Steelpillow (Talk) 11:24, 15 October 2015 (UTC)

  While nothing is wrong with complex n-space, complex coordinate space would already follow other Wikipedia naming conventions (e.g., real coordinate space, coordinate space) to refer to the analog over other fields. Also, complex coordinate space is less ambiguous. Sławomir
  Biały 10:57, 15 October 2015 (UTC)

  Complex space is a very problematic title, because the word "space" has multiple meanings. (E.g., a complex manifold, complex variety, complex vector space, as well as the ostensible subject of this article). Sławomir
  Biały 12:23, 15 October 2015 (UTC)

  Which is exactly what makes it so important to have an introductory article that summarises and links to all these different kinds of complex space. You can think of it as a disambiguation page if you like. — Cheers, Steelpillow (Talk) 13:42, 15 October 2015 (UTC)

  But it seems like that would then be a different article, totally unlike this one. I have no objection to a disambiguation page (or whatever the kind of article that describes related uses of a term). But it shouldn't be about the complex coordinate space, with its inner product, etc. That's a separate topic, in its own right. Sławomir
  Biały 13:49, 15 October 2015 (UTC)

  Yes it would. That is exactly the point I have been trying to make. A long time ago this article began life at Complex space. As it was expanded the focus of its content was changed and it was moved here. I am simply asking whether we should refocus it back and move it back again, or recognise its new focus, move it somewhere else and independantly re-create the original introductory article at its original location. — Cheers, Steelpillow (Talk) 14:00, 15 October 2015 (UTC)

We are in a complete agreement that "complex space" is not a good article title (hence, the initial move to the current article title). As far as I can tell, there is no much issue with the article content-wise. Cⁿ does come with the standard inner product, and having a link to Hilbert space is also perfectly reasonable. The only issue here is a "differential-geometry"-minded editor find the article title to be not to his liking. I think I understand the feeling; whence, my proposal complex n-space. It has been noted that the algebraic-geometry sense need not be primary one, but I disagree. This article should discuss the structure sheaf; that would give more of algebraic-geometry feel. Complex geometry, in my humble opinion, is closer to algebraic geometry than to differential geometry. -- Taku (talk) 04:10, 16 October 2015 (UTC)

Taku, please, clarify your position. Do you agree that whatever is called "affine" must be invariant under affine transformations (that is, shifts and linear transformations)? Or do you claim that this is the dif-geom point of view, not the alg-geom point of view? Boris Tsirelson (talk) 09:52, 16 October 2015 (UTC)

Sounding like a politician, but it's not a yes/no question, I think. Given a closed subvariety X in some Aⁿ (i.e., an affine variety), one can consider the set of all tangent lines to X through some fixed point P of Aⁿ (tangency can be made sense once we have intersection number). The set is nonempty if and only if P lies in X. When nonempty, the "set" can be given a structure of A^m (with m the dimension of X when P is a smooth point) and the resulting variety is the tangent space to X at P. For me (and in fact in general?), this is an example of an affine space. (It has a structure of a vector space coming from A^m and so we can talk about inner product (or Hermitian metric) if the base field is the complex field.)

Put in another way, in algebraic geometry as I understand, you don't study the concept like tangent spaces from the point of view of affine transformation. (It's very telling that the article only lists diff-geo textbooks as references.) It's possible this is not accurate, historically or philosophically (but that's another issue). -- Taku (talk) 23:21, 16 October 2015 (UTC)

OK, let us consider the tangent space. Yes, it can be given a structure of A^m; but not uniquely! Have you a preferred coordinate system on it? I guess, not; they all are equally useful. Then, it is not quite A^m; rather, it can be upgraded to A^m by arbitrary choice of coordinates. Right? Boris Tsirelson (talk) 06:30, 17 October 2015 (UTC)

As Boris points out, tangent spaces are not A^m (per your definition). Also, it is original research to claim that affine spaces have inner products defined on them. That's definitely not in Hartshorne, Mumford, or EGA. I defy you to find any source defining an inner product on the set of points in an affine space. Sławomir
Biały 12:01, 17 October 2015 (UTC)

But my point was that, a priori, the set of tangent lines is just a set; not even algebraic variety, not even an affine space in either sense affine space or affine space (algebraic geometry). It, however, has the intrinsic structure as the Zariski tangent space (which involves neither the choice of embedding or the choice of coordinates.) I agree it can be given the structure of a vector space without the origin; but that's not what you do in algebraic geometry; you just observe the calculus-style construction of the tangent space coincides with a more algebraic-construction (namely, Zariski tangent space). -- Taku (talk) 22:22, 17 October 2015 (UTC)

Yes, yes, we know all about the Zariski tangent space. But you appear to be claiming that there is a unique isomorphism of the Zariski tangent space to the n-fold Cartesian product of k with itself (because for you an "affine space" is an n-fold Cartesian product of k with itself, right?) So, tell me what the isomorphism is. What are the n coordinate functions that give the Zariski tangent space the structure of a space of n-tuples? Sławomir
Biały 00:22, 18 October 2015 (UTC)

No I have not claimed there is a unique canonical isomorphism between the Zariski tangent space to k^m (as we both know there is no such a thing.) What are we debating about anyway? My point still stands: like in the example of a tangent space, one doesn't seem to use the "affine space" in the sense described in affine space. As I understand, the article seems to follow the sprit of Berger's "Geometry" in defining affine space; nothing wrong with that, of course. But it doesn't mean his way of viewing the affine space (e.g, affine transformation and affine group) is the one and only one way. At least in algebraic geometry, as far as I understand, the affine space simply means Aⁿ or any other variety isomorphic to it. In some situation, it is important to be careful about the choice of coordinates or embedding; in some other situation, one adopts the attitude suppressing such a choice; the tangent space is a good example. I'm certain that we don't disagree on any mathematical substances. But it feels you are demanding to adopt the certain point of view even that's not needed. I know we can conceive of a geometric object: a vector space without the origin; but the point I was trying to make through the example was that you don't need that concept and accordingly is not in use, at least in algebraic geometry (say in Mumford's red book; he considers both calculus-type tangent space and Zariski tangent space.) -- Taku (talk) 04:31, 18 October 2015 (UTC)

Taku, this article says that the affine space is the set of n-tuples of $\mathbb {C}$ with itself. Until my recent corrections affine space (algebraic geometry) said that affine space is the set of n-tuples with elements from k. Now you're saying that this isn't what you meant? I agree that in algebraic geometry, it is often not important to be fussy about coordinates or embeddings. Yes, one is concerned with varieties isomorphic to affine coordinate space. Yet I have no idea why you think it is appropriate to gloss over the "isomorphic to" aspect of it. Indeed, if we were to define affine space as $k^{n}$ (without the "isomorphic to") then this algebraic variety has a canonical group structure, and it's equipped with a canonical set of n morphisms into k. This group has an identity element, etc. Whether one means a variety with this structure or not is something that is very relevant, for algebraic or any other kind of geometry. (I assume you have probably heard of algebraic groups, even though it is clear you haven't read Armand Borel's seminal textbook on them.) So I have no idea why you continue to defend a definition that is so manifestly wrong. Sławomir
Biały 14:02, 18 October 2015 (UTC)

"as far as I (Taku) understand, the affine space simply means Aⁿ or any other variety isomorphic to it." Nice... but you cannot say "isomorphic" until you say which structure do you mean! And you insistently reject any structure, but retain the right to say "isomorphic". How so? To be isomorphic to Aⁿ as a unitary vector space is one thing. As a vector space - another thing. As an affine space - still another. As a commutative group - still another. And the final step: as a structure-less set - means, to have the same cardinality. And the other end of the spectrum: as a Cartesian space - means, to have a preferred coordinate system. Boris Tsirelson (talk) 14:56, 18 October 2015 (UTC)

I'm afraid this might be original research. But I don't think, in this particular instance, there is any meaningful difference between the definitive definition and the definition up to isomorphism. Imagine two distinct (affine) lines A¹ inside A². A tricky question: are there two lines? Or just one line? Of course, the correct answer is that the line is embedded into A² at two different positions; there is no "intrinsic way" to distinguish the two but only the extrinsic way. If a "space" looks like an affine space, it must be an affine space; otherwise, how do you call such a space? I get the pendantic difference between the two definitions but in practice the context makes it clear what one means (i.e., there is no need to insist distinguishing the two definitions.) For example, in the above example, it's better (and standard) to introduce the two embeddings $i_{1},i_{2}:\mathbf {A} ^{1}\hookrightarrow \mathbf {A} ^{2}$. In other words, it all depends on the context. -- Taku (talk) 00:05, 19 October 2015 (UTC)

• Oppose. The proposed is ambiguous with Complex plane. If there is indeed a problem with the current title, try again for something more precise. --SmokeyJoe (talk) 03:18, 21 October 2015 (UTC)

It seems I understand

I guess, Taku means that, in algebraic geometry, the object is (by default) an algebraic variety, and isomorphisms are interpreted accordingly. Now, Aⁿ is indeed an algebraic variety, and its automorphisms appear to be exactly the affine transformations (shifts and linear transformations). Right? If so, then the two structures, "algebraic variety isomorphic to Aⁿ" and "n-dim affine space" (in the "old" sense) are deducible from each other. It means, we really have two equivalent definitions of a mathematical structure. Thus, no conflict between alg-geom and dif-geom points of view. Just two equivalent definitions, one of them being indeed of alg-geom style. So, what is the problem?? Which definition to write first, and which - second? Hope, not. I saw enough such disputes on Talk:Monty Hall problem, and will never return there. But they are not mathematicians; let them... Boris Tsirelson (talk) 16:56, 18 October 2015 (UTC)

Oops, no; that is just wrong! From mathoverflow: "the automorphism group of A² includes all the maps of the form (x,y)↦(x,y+f(x)) where f is any polynomial."

Therefore the two structures are not equivalent; the alg-geom one has more automorphisms; it means, the alg-geom structure is deducible from the "old" one, but not the opposite. But then, having two points a, b and c of a Zariski tangent space, we cannot construct a+b-c. Right?

Now, we have a touchstone. What do reliable sources in alg-geom say? If they say that a+b-c is well-defined (on Zariski tangent space), then they mean the "old" definition. Otherwise – the "new" one. Boris Tsirelson (talk) 17:14, 18 October 2015 (UTC)

Actually, the two definitions are completely equivalent (possibly modulo some technical quibbles about polynomials versus polynomial functions in fields of prime characteristic, and whether the field is algebraically closed). Thekey point is that, although the concept of a linear functional is not well defined on an affine space, the concept of an affine functional is. One can therefore obtain the algebra of polynomial on an affine space as the function algebra generated by the affine functionals. Note that, over an algebraically closed field, the points if the affine space are in one-to-one correspondence (naturally) with unital algebra homomorphisms into the ground field. (Youre probably familiar with a similar situation in functional analysis, e.g., the Gelfand correspondence.) Algebraic geometry usually defines an affine space as the spectrum of a polynomial algebra. This spectrum carries an affine structure in the usual sense. Sławomir
Biały 17:30, 18 October 2015 (UTC)

No, sorry, the two definitions are provably NOT equivalent! A map of the form (x,y)↦(x,y+f(x)) (with nonlinear f) does not preserve affine functionals (and I mean just the field of complex numbers). That is, you cannot define affine functionals on the tangent space, if you forget any structure beyond the structure of algebraic variety. Boris Tsirelson (talk) 17:49, 18 October 2015 (UTC)

Hm.. yes that seems pretty comvincing. The affine structure of the "affine space" of algebraic geometry is something extra it would seem. (That having been said, I don't see any evidence that affine spaces are regarded up to isomorphism in the category of algebraic varieties in alg e brain geomtery, any more than affine spaces are taken up to diffeomorphism in differential geometry.) This seems to be a retrospective justification of the terminology that we've come up with ourselves. Sławomir
Biały 19:22, 18 October 2015 (UTC)

Sure. But I came to this hypothetic terminology, trying desperately to find a reasonable interpretation of Taku's position. That is, trying to answer myself my question to him: what kind of isomorphism (to Aⁿ) does he mean? Now we see that I did not succeed (again). (See my "summary" below.) Аnother nail in the coffin? Boris Tsirelson (talk) 19:51, 18 October 2015 (UTC)

I am very far from being an expert in alg-geom, but it seems to me, experts treat the tangent space as a vector space; for example, here, lect.8. (And in fact, dif-geom treat it also as a vector space, not just affine space; basically, the given point becomes the origin.) Boris Tsirelson (talk) 17:53, 18 October 2015 (UTC)

Thus, it seems to me that algebraic geometers do not treat Zariski tangent space as  just an algebraic variety. Not only a+b-c is well-defined, but also a+b is. Boris Tsirelson (talk) 17:58, 18 October 2015 (UTC)

The tangent space aside, I summarize: if someone interprets "affine space" as "algebraic variety isomorphic to Aⁿ", then he must agree that there is no notion of "affine function on an affine space". Boris Tsirelson (talk) 18:28, 18 October 2015 (UTC)

Some comments about above discussion

I have not had the time in the above discussion nor to read it in details. However, I can provide some comments that, I hope could be useful.

• In the above discussion, the historical aspect is never considered. IMO this is an error. In fact, during the 20th century, the language and the methods of algebraic geometry have strongly evolved. However, the object of study has not really changed, it is the study of curves, surfaces and so on, that can be defined by polynomial equations. The evolution of the language has been caused firstly for making rigorous the results of 19th century and of Italian school of algebraic geometry. Then the language has further evolved for allowing the proof of difficult conjectures (in the preface of EGA, Grothedieck clearly asserts that he has introducing scheme as a formalism for proving, in the future, Weil's conjectures. These changes of languages have enlarged the class of objects that are viewed as algebraic varieties, but everything that has been viewed once as an algebraic variety remains always an algebraic variety. It results that it is somehow an anachronism to refer to scheme theory for defining an affine space, and even for defining an affine variety.
• The fact that the affine space of algebraic geometry is not reduced to Kⁿ is clear from sentences that algebraic geometers commonly use, such as "putting this singularity at the origin, we have ..." or "let us put the variety in Noether position". Both sentences involve a change of affine coordinates. The fact that affine changes of coordinates are allowed, shows that one works on a general affine space (recall that choosing a system of affine coordinates for an affine space is equivalent to choose an affine isomorphism between it and Kⁿ).
• Contrarily to what has been said, the spectrum of a polynomial ring has not the structure of a (classical) affine space. Only the maximal spectrum of a polynomial ring over an algebraically closed field has this structure (This is Hilbert's Nullstellensatz).
• As the tangent space has a point fixed (the contact point), this is not an affine space, but a vector space (recall that affine spaces are vector spaces for which one forget the origin, and that vector spaces of finite dimensions are obtained by forgetting the canonical basis of Kⁿ.
• Affine spaces are algebraic varieties. Thus, when talking of "isomorphism" one always need to precise which kind of isomorphisms: isomorphisms of affine spaces, isomorphism of variety, birational equivalence, ... Some misunderstandings in above discussion come from an ambiguous use of the word "isomorphism".

D.Lazard (talk) 23:17, 18 October 2015 (UTC)

In light of Boris' example in the previous section, I feel that something is still not quite right with your third bullet on the Nullstellensatz. I think the affine structure on the algebraic variety $\mathbb {A} ^{n}$ comes from equipping the affine coordinate ring $k[x_{1},\dots ,x_{n}]$ with a filtration. In a standard way, we can associate to this filtered algebra a graded algebra in one more generator, which is the ring of homogeneous functions on the projective space. It's not clear whether such an approach is ever used in the literature though. Sławomir
Biały 10:40, 19 October 2015 (UTC)

Reading again this third bullet, it appears that there is an ambiguity about the spectrum. In my post I have considered the spectrum as the set of prime or maximal ideals. If on the contrary, "spectrum" denotes the affine scheme Spec(R) associated to the ring R, then my item becomes a non-sense. In the case of $\operatorname {Spec} (k[x_{1},\dots ,x_{n}])$ one may define easily an affine structure on the set of maximal primes of the form $\langle x_{1}-a_{1},\ldots ,x_{n}-a_{n}\rangle ,$, by using that affine changes of variables induce affine transformations of this set of primes. IMO, this is almost a tautology, as this consists essentially in reading backward the definitions of affine varieties and affine schemes. In this case, the affine space is naturally equipped with affine coordinates, and therefore canonically isomorphic to the affine space kⁿ. For having a general affine space one may proceed as follows: Let us consider a filtered algebra of finite type over k, which may be generated by elements of degree 1, and such that the set of elements of degree 0 is exactly k. Without knowing a set of generators of degree one, one may consider the set of maximal ideals generated by elements of degree at most one, which can clearly be equipped with an affine structure, without choosing any coordinate system (I have not checked the details).

This approach is probably OR, but this may be explained: As algebraic geometry is built above affine spaces, why reconstructing affine spaces from algebraic geometry? D.Lazard (talk) 13:20, 19 October 2015 (UTC)

It seems, here "affine" means: we forget the generators $x_{1},\dots ,x_{n}$ but retain the graduation (or should I say, filtration?). Boris Tsirelson (talk) 14:14, 19 October 2015 (UTC)

Not exactly: "affine space" and "affine transformation" have their standard mathematical meanings. "Affine varieties" has also its standard meaning of algebraic subset of an affine space. The status of "affine scheme" is slightly different; it is generalization of affine varieties that may be associated to any commutative ring, but corresponds to an affine variety only if the ring is an integral domain that is a finitely generated algebra over a field. All these definitions are standard and universally accepted by mathematicians. My point is that the definition of "affine space" is the same in algebraic geometry as in standard geometry. The fact that some authors introduce algebraic geometry by considering only specific affine spaces (kⁿ) is not a change of definition, but a simplification justified by the fact that every affine space is affinely isomorphic to some kⁿ. My preceding post consists in showing that the affine scheme of an affine space may be intrinsically characterized with a filtration (not a graduation), and that one may reconstruct the affine space without introducing coordinates. In other words the definitions of algebraic geometry are fully coherent with the standard definition of affine spaces. D.Lazard (talk) 16:21, 19 October 2015 (UTC)

Nice; thank you. Boris Tsirelson (talk) 16:48, 19 October 2015 (UTC)

It is true that the authors of math textbooks introduce a terminology that ignores historical or uses in other fields. I think we can all agree that in "algebraic geometry" an affine n-space is defined as kⁿ, which is an affine variety and thus comes with the structure sheaf (cf. affine variety#structure sheaf.) Or one can require this holds up to isomorphisms; in some context (e.g., algebraic group) the distinction certainly does matter. But then you be more careful: like saying just "curve" can be very imprecise in some context (say moduli context). You just define the affine space on your writing and the problem is solved. But we shouldn't and don't demand full rigor everywhere and every time. (You can also use the language of scheme theory but of course the result is still equivalent.) What is the issue here whether this is equivalent to the gometric object that is a vector space without the origin. My point was this isn't quite a yes/no question. One can perhaps show the two definitions are equivalent in a "suitable sense" (need to avoid a counterexample mentioned above.) But the point is that, in practice, the question on such an equivalence is typically ignored. As Lazard points out, that might not be a good thing (ignoring history and connection to other fields). Perhaps. And the article should clarify why algebraic geometers use the terms they use. But that's entirely different from demanding the algebraic geometers respect history and the usages in other fields. (I feel this is what I have already said.) -- Taku (talk) 23:52, 19 October 2015 (UTC)

Do affine spaces have inner products on them? Do they have a preferred set of coordinates on them? These aren't minor little fussy things to do with rigor. One would walk away from such an article very badly mislead about what an affine space is (both in algebraic geometry and outside it). Such a view is absolutely indefensible. It would be like insisting that we should rewrite the main article calculus from some weird perspective, like insisting that "all calculus is about infinitesimals". Sławomir
Biały 00:30, 20 October 2015 (UTC)

Taku, when you say "affine n-space is defined as kⁿ", you evidently mean "up to isomorphism" (moreover, you already wrote so; I understand that for you it goes without saying that everything is meant up to isomorphism.) It remains to say, what is an isomorphism. Specify the isomorphisms (in one way or another), and 90% of the misunderstanding will disappear painlessly. In particular, what is the group of automorphisms? Say this, at last; and we'll see immediately, whether "your" affine space is the same as "our", or not the same. So simple! Boris Tsirelson (talk) 04:46, 20 October 2015 (UTC)

• I believe it has already been made clear that Taku's version is not acceptable. He has apparently not acknowledged that here, and apparently does not seem to understand what the issue is. Sławomir
  Biały 22:29, 20 October 2015 (UTC)

  Ok. I admit I don't understand the issue here: what is wrong with mentioning that Cⁿ is an affine space? Do you claim it's not an affine space? The problem remains: why do you claim "affine space" in the sense of say Berger is the only acceptable definition?

  I understand the feeling and argument why you want to insist on "your" definition, but I keep telling you that that's not consistent with the literature: Cⁿ is virtually always referred as an affine space (well at least in algebraic geometry); see for instance exotic affine space, if you prefer it or not. -- Taku (talk) 23:31, 20 October 2015 (UTC)

'why do you claim "affine space" in the sense of say Berger is the only acceptable definition' Fine, let's look at Hodge and Pedoe, or Ito's "Encyclopedic dictionary...", or HSM Coxeter's "Introduction to Geometry", or Bourbaki's "Elements of mathematics". I'm fine with using any of those sources too. I've even added a few of them to the article. While it is certainly true that $\mathbb {C} ^{n}$ is an example of an affine space, it is wrong to put on the right-hand side of a definition, as in "Affine n-space is $\mathbb {C} ^{n}$." This is very misleading, because it is not clear from this sentence what we mean by $\mathbb {C} ^{n}$. Do we mean affine space is this set? As a set, it is equipotent with the real numbers. So would you agree that "Complex affine n-space is $\mathbb {R}$"? I would guess not. Do you mean $\mathbb {C} ^{n}$ as a vector space? If so, it has a linear structure and distinguished origin. (Hint: This is not what algebraic geometers mean.) Do you mean $\mathbb {C} ^{n}$ as a Hermitian inner product space? (Until earlier today, you apparently maintained that this was what you mean.) Do you mean $\mathbb {C} ^{n}$ as a linear space equipped with n preferred coordinate functions? (This is what is called in mathematics a coordinate space, fyi.) I think you have hopefully come to realize in this discussion that you don't actually mean any of these things. You mean complex affine n-space is $\mathbb {C} ^{n}$ with its affine structure! The problem is that saying "Complex affine space is $\mathbb {C} ^{n}$" might make someone feel good that there is some reasonably concrete definition here, but really it explains precisely nothing, because one really needs to know what the algebraic geometer means by the term "affine" to begin with. Only then does a definition like "Affine n-space is $\mathbb {C} ^{n}$" actually make proper sense as a definition. Otherwise one is led to nonsense like a Hermitian metric. If someone who claims to know a lot of geometry literature can make such an obvious error, what hope is there for a reader who is not given any proper geometric context for the statement "Affine n-space is $\mathbb {C} ^{n}$"? Sławomir
Biały 23:55, 20 October 2015 (UTC)

I admit it is unfortunate that people (especially algebraic geometries) use the term "complex affine space" when the better term is "complex coordinate space"; they should study "exotic complex coordinate space" as opposed to "exotic affine space". I keep telling you that I understand your feeling. But we're not here to correct the literature. Like a politician, I've been deliberately dodging the question of the definition of affine space since, as I said, it depends on the context. I don't disagree with Berger's definition of an affine space; I never did. But that doesn't give me a right to change the standard usage: to many, "complex affine space = Cⁿ" (whatever that means; like is the topology Zariski? complex? all depends on the context.)

The readers want to know whether democracy is good or not. Similarly, they want to know whether the complex affine space is Cⁿ. Giving a lecture instead of giving a short answer doesn't make us popular.

One more thing: I have never claimed to know a lot of geometry; for instance, I've never read Berger's geometry (it's not my style). If the others can point out my misreading of the literature (for me "affine space" appears more popular than "coordinate space"), I'm happy to reconsider my position (why do I keep sounding like a politician.) -- Taku (talk) 00:52, 21 October 2015 (UTC)

"I don't disagree with Berger's definition of an affine space; I never did. But that doesn't give me a right to change the standard usage" I disagree very strongly with this characterization of the disagreement. Have you not been paying attention. I have conclusively shown that you have the matter completely backwards. There is already a standard usage of "affine space" in geometry. If you don't believe me, you can consult any number of standard reference textbooks like Hodge and Pedoe, Bourbaki, or Coxeter. I even invited you to consult one of the sources you suggested had become "standard", Vakil's notes, and see precisely what he has to say on the matter. I've given you ample opportunity to do all of that, and yet you persist in framing this dispute as "Berger's definition versus the standard one". No, it's the standard definition versus Takuya's definition. Sławomir
Biały 10:20, 21 October 2015 (UTC)

So... Being asked to clarify his position (especially, the isomorphisms used), Taku replies, basically, that this matter remains unclear in some literature, and therefore must remain unclear on Wikipedia. Wow! Quite a new way to do mathematics! Indeed, "sounding like a politician". O tempora o mores! Boris Tsirelson (talk) 05:50, 21 October 2015 (UTC)

Or not new... "nothing new under the sun" (Ecclesiastes); "nothing my generation ever came across (at any rate in English) had the sharp bracing precision the student gets today" (A Mathematician's Miscellany by Littlewood, page 70). Now, back to the 19th century? Boris Tsirelson (talk) 06:24, 21 October 2015 (UTC)

File:Page 1 of Hartshorne.jpeg

Page 1 of Hartshorne

.

Sławomir, what is the "standard" depends on one's field; Berger's (or Coxter etc) style of geometry is definitely not typical to some "geometers"; I have already mentioned the case of exotic affine space where "affine space" really means "coordinate space" in your terminology. I took the screenshot of page 1 of Hartshorne (the same definitions also appear in Mumford and Vakil, although Vakil's in terms of schemes.). There is no "my definition"; the references I follow just don't give me a room to interpret them in a different way. Anyway, we need a third-party to break the tie; I myself will refrain from editing the article. -- Taku (talk) 23:42, 21 October 2015 (UTC)

1. In case I am very mistaken, Boris and Daniel are "third" and "fourth" parties. There is no tie. You lose.

2. There is some point at which rejecting references as "not standard" becomes very questionable. Is Hodge and Pedoe standard? Is Bourbaki standard? Is Ito's "Encyclopedic dictionary of mathematics" standard? Is Milne's "Algebraic geometry" standard? Is Borel's "Linear algebraic groups" standard? Are Vakil's notes regarded as standard (you earlier said they were the new standard, but now apparently have changed your opinion on that matter after discovering that they don't support your idiosyncratic point of view)? Is Shafarevich standard? If none of these count as "standard" to you, then at a minimum I expect that you should explain in detail why we shouldn't rely on the definition given in these sources. So far, you have dismissed no less than seven sources as "Berger's definition", and now "Coxeter's definition", even though Berger and Coxeter (as far as I know) are not responsible for the works of Borel, Bourbaki, Ito, Milne, Shafarevich, Vakil. At one point, you even conceded that Hartshorne does some things wrong, just to simplify the exposition. It is totally mystifying why you would concede this point, and yet now Hartshorne gets a pass, while a pile of other, ostensibly standard sources (most of which are much more careful than Hartshorne), would be lumped into a pile of crank mathematics. It looks to me as though any source that does not agree exactly with Hartshorne is being dubbed "nonstandard". This looks like a fallacy of overwhelming exception.

3. You haven't responded to the objection of what $\mathbb {C} ^{n}$ means in the sentence "Complex affine n-space by definition is $\mathbb {C} ^{n}$." Do we mean the set $\mathbb {C} ^{n}$. That is, a set with cardinality equal to the continuum. This would make the following sentence also true: "Complex affine n-space by definition is $\mathbb {R}$", for example. Do you mean the additive group $\mathbb {C} ^{n}$? Do you mean the vector space $\mathbb {C} ^{n}$? Do you mean the set $\mathbb {C} ^{n}$ equipped with n coordinate functions? The problem with this sentence, as I have already said, is that it is a non-definition. One needs to know what structure $\mathbb {C} ^{n}$ is equipped with in order for this sentence to make proper sense. The answer, is $\mathbb {C} ^{n}$ equipped with its affine structure. Dr. Lazard, himself a distinguished algebraic geometer, already gave ample explanation that this is in fact what algebraic geometers mean. Any reasonable person would already have been persuaded by the discussion in Hodge and Pedoe, which is just basic background in algebraic geometry (far before one would encounter the likes of Hartshorne). I can only conclude that you do not fit that description. Sławomir
Biały 00:20, 22 October 2015 (UTC)

1. I don't think Boris and Daniel are with you. Or maybe I'm mistaken.

2. Again we disagree on the reading of the literature; we've been there.

3. Again depends on the context. The clearest case is when we use the language of schemes: in that case, the complex affine n-space is by definition $\operatorname {Spec} \mathbb {C} [x_{1},\dots ,x_{n}]$, whose set of the closed points is $\mathbb {C} ^{n}$. (cf. Vakil). I think we both agree on this. Sometimes, out of laziness or convention or whatever, this scheme is also denoted by $\mathbb {C} ^{n}$. This can cause an ambiguity (do we mean a closed point or possibly closed point?); so we have to be careful. In any case, to some people (including me), this is the only complex affine space. Depending on the situation, it might be preferable to call any scheme isomorphic as scheme to it also a "complex affine space"; a sort of variation on definitions typical in practice. In the context of complex-algebraic geometry, it is better and standard to equip the set $\mathbb {C} ^{n}$ with the usual complex topology and then with the sheaf of ring of holomorphic functions. The resulting ringed space is called the complex n-space or complex affine n-space (to distinguish it from the complex projective space). Clearly, these definitions are coordinate-dependent and there is a way to do this in a coordinate-free way (use the symmetric algebra of a dual vector space.). I suppose this is what you claim "my definition"? For me, this looks pretty standard in algebraic geometry and so that's why I said we need tie-breaker. -- Taku (talk) 00:54, 22 October 2015 (UTC)

"in that case, the complex affine n-space is by definition $\operatorname {Spec} \mathbb {C} [x_{1},\dots ,x_{n}]$, whose set of the closed points is $\mathbb {C} ^{n}$". No, it is not. There is a bijection of the points of $\operatorname {Spec} \mathbb {C} [x_{1},\dots ,x_{n}]$ with $\mathbb {C} ^{n}$, but actually writing this bijection depends on choices (generators of the affine coordinate ring). So there is a huge difference between Vakil's definition as the spectrum of a ring and the one you want to present as the standard one.

"it might be preferable to call any scheme isomorphic as scheme to it also a 'complex affine space'" This would not be in accordance with the standard meaning of "affine space" in mathematics. As Boris already noted, the scheme does not carry an affine structure. That's something extra.

I seriously think that if you want to understand what "affine" means in algebraic geometry, you should look at Hodge and Pedoe. Look at some actual geometry. Think about curves, and what the contrast of "affine" is to "projective". Try to understand the discussion that is now in the lead of this article.

Trust those of us that have commented here that "affine" has a completely standard meaning in geometry, and that your extrapolation from Hartshorne does not agree with this standard meaning. I've given sources in geometry, and even some very standard sources in algebraic geometry that bear this out. You're more than welcome to try to see the big picture. But saying a huge pile of sources is not relevant because they don't agree with whatever narrow area of mathematics you happen to live in is deep in WP:CHEESE territory.

Since you've shown no evidence of understanding the discussion, I won't hold my breath. But please don't insist that there is a "dispute", when it is just Takuya versus everyone else (including most sources). That is disruption. WP:STICK, seriously. Sławomir
Biały 01:17, 22 October 2015 (UTC)

Ok. "Is" here really means "is canonically identified with"; but the point is there is no choice of coordinates involves: every maximal ideal of the polynomial ring $\mathbb {C} [x_{1},\dots ,x_{n}]$ is of the form $(x_{1}-a_{1},\dots ,x_{n}-a_{n})$ and so there is a "canonical bijection" between the set of maximal ideals in the ring and the set-theoretic points of $\mathbb {C} ^{n}$. I think the issue is personal one here; I don't appear to appreciate the type of the geometry you do. Guess I don't. As I said before, I can't follow Hodge and Pedoe; especially their use of "generic point" is very foreign to me and I need to translate their expositions to the language I can understand. (To confess, I actually really need to read some parts off Hodge and Pedoe since I might be able to reproduce some of their formula in my day-time research and their "old" language is really bothering me, but that's another story.) In any case, we're not having a productive discussion any more; I'm tired and we need to ask inputs from the third party. -- Taku (talk) 01:52, 22 October 2015 (UTC)

"every maximal ideal of the polynomial ring $\mathbb {C} [x_{1},\dots ,x_{n}]$ is of the form $(x_{1}-a_{1},\dots ,x_{n}-a_{n})$". I agree that every maximal ideal is "of this form", but it is a fallacy to believe that you can read off the list of n complex numbers $(a_{1},\dots ,a_{n})$ just from the ideal itself. The same ideal can be written in different ways. For instance,

$(x_{1}-a_{1},\dots ,x_{n}-a_{n})=(x_{1}-x_{2}^{2}-a_{1}+a_{2}^{2},x_{2}-a_{2},\dots ,x_{n}-a_{n}).$

So, I would claim that the list of numbers $(a_{1}-a_{2}^{2},a_{2},\dots ,a_{n})$ is the "right" list of n numbers. Why is your view right, and mine is wrong? The association is not canonical. It requires extra structure (specifically, a preferred set of generators of the affine coordinate ring). Sławomir
Biały 02:14, 22 October 2015 (UTC)

No the map is this: $(a_{1},\dots ,a_{n})\mapsto (x_{1}-a_{1},\dots ,x_{n}-a_{n})$; I call this a canonical map, you don't? (Of course,you have to check it is bijective. You can also use Hilbert's nullstellensatz; but amounts to the same.) -- Taku (talk) 02:18, 22 October 2015 (UTC)

So, you're claiming that, associated to any maximal ideal in a finitely generated free associative algebra over the complex numbers, there is a list of n complex numbers that requires no extra structure to define? Just to be clear... Sławomir
Biały 02:20, 22 October 2015 (UTC)

After you fixed a basis, yes. Do you agree the above map is a canonical map? -- Taku (talk) 02:23, 22 October 2015 (UTC)

A basis of what? Let's follow Vakil's definition closely. Affine n-dimensional space is a topological space X, together with a sheaf of complex-valued functions on X. This sheaf has the property that it is generated by its global sections (it is an affine scheme), and the algebra A of global sections is a free polynomial algebra of dimension n. You're claiming that there is a natural correspondence between maximal ideals in A and n-tuples of complex numbers. I say, show me the map. This seems to involve extra structure that the topological space, and sheaf of functions defined on it, don't know about. You may as well claim that an n-dimensional vector space is always "canonically" isomorphic to $\mathbb {C} ^{n}$. What you're saying seems to be about as true as that. Sławomir
Biały 02:39, 22 October 2015 (UTC)

A free algebra basis: a choice of algebra isomorphism $R\simeq \mathbb {C} [_{1},\cdots ,x_{n}]$ (highly non-canonical) where $x_{i}$ are alg. independent. Anyway, again, do you agree the above map is canonical or not? If not, there is another way: for each $P=(a_{1},\dots ,a_{n})$, let ${\mathfrak {m}}_{P}$ the kernel of the evaluation map $f\mapsto f(P)$. Then consider $P\mapsto {\mathfrak {m}}_{P}$; ok, in fact, this one is better since this map also gives a canonical identification between a vector space V and the set of closed points in $\operatorname {Spec} \mathbb {C} [V]$, Spec of the ring of polynomial functions on V. -- Taku (talk) 02:58, 22 October 2015 (UTC)

But the whole point is that there are many such mappings. Nothing distinguishes yours as the correct one. There is certainly nothing about the pair $(X,A)$ that tells you how to associate an n-tuple of complex numbers to a (closed) point of X. Yet here you are, happily saying that Vakil's definition is completely equivalent to $A^{n}=\mathbb {C} ^{n}$, and that there is only one way to associate a point of X to a point of $\mathbb {C} ^{n}$. But no, it's clearly not equivalent. You need to invoke some extra structure. That is my point. Sławomir
Biały 03:22, 22 October 2015 (UTC)

Sorry, I'm completely lost: do you claim there is no canonical way to identify a vector space V and the set of closed points in $\operatorname {Spec} \mathbb {C} [V]$? If so, this is something I don't understand at all (and no one else can understand). -- Taku (talk) 06:54, 22 October 2015 (UTC)

To me, the most reasonable meaning of "canonical" would be an "identification" map that factors through the automorphism group of Spec C[V], right? So suppose that $F:Spec\mathbb {C} [V]\to V$ is your "canonical" identification. In order for F to be "canonical", any automorphism $\phi :Spec\mathbb {C} [V]\to Spec\mathbb {C} [V]$ will leave F invariant, so $F\circ \phi =F$. Thus, the test for whether there is a "canonical" identification is whether your identification map is invariant under the automorphism group of the affine scheme $Spec\mathbb {C} [V]$. So, you tell me: is there a canonical way to identify the set of closed points of Spec C[V] with V? Sławomir
Biały 11:01, 22 October 2015 (UTC)

I am quite puzzled as to why this discussion is ongoing. I gave the correct definition of an affine space above and am surprised that nobody seemed to pay attention.

By "point" I will mean T-valued point for a fixed but otherwise arbitrary scheme T. Let k be a field. Let V be an n-dimensional vector space over k (n finite). Let X be a variety over k. We say that a morphism + : V × X → X is an action of V on X if it satisfies the usual commutative diagrams (or equivalently, on points it satisfies 0 + x = x and (v + w) + x = v + (w + x)). The action is free if it has trivial stabilizers; on points, this means that if v + x = x, then v = 0. The action is transitive if it has a single orbit; if x and y are two points, then there exists a v such that v + x = y. Define X to be a V-torsor if the action + is free and transitive. (This combination of properties is said to make the action simply transitive.) Finally, X is an affine space if it is a V-torsor that admits a k-rational point.

Two affine spaces are isomorphic if they are isomorphic as torsors; and two V-torsors are isomorphic if they are isomorphic by an isomorphism φ that commutes with their actions. That is to say, on points φ satisfies v + φ(x) = φ(v + x). The point is the torsor structure is exactly what is meant in classical geometry by a synthetic affine space. It makes an affine space into a space on which one has the usual linear structure except for the basepoint, and once one specifies the basepoint, one has trivialized the torsor and so gotten a vector space just as one expects. Ozob (talk) 04:42, 22 October 2015 (UTC)

I'm sure this is the correct definition of an affine space. But apparently the issue is something else; I don't know what it is myself. -- Taku (talk) 06:54, 22 October 2015 (UTC)

Ok, so you finally now agree that the article is correct! I guess we also agree that there is no further need for a POV tag. Sławomir
Biały 11:04, 22 October 2015 (UTC)

Thank you, Ozob. I am not sure I understand this definition completely, which is of course my problem. But I do understand that its automorphism group is the affine group; this is what I need to know here. Boris Tsirelson (talk) 08:51, 22 October 2015 (UTC)

The definition might be more familiar if I had called X a principal homogeneous space instead of a torsor. In the present case, they are the same thing. However the torsor terminology is quite common in algebraic geometry, where frequently the whole setup is hugely generalized (V and X are replaced by sheaves with respect to some other topology, like fppf) to support number-theoretic applications. Bjorn Poonen's notes, "Rational points on algebraic varieties", discuss this. Ozob (talk) 13:53, 22 October 2015 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

POV tag warring

Hey guys. cool it! If a strict admin dropped by you could both earn blocks for edit warring. I really appreciate the effort you guys are putting in to this topic, but I guess emotions are running a bit high now. Please accept my invitation to a nice cup of tea and a sit down (metaphorically).

My view FWIW: Taku has a strong PoV. However it is not supported by other editors in these discussions and so is probably not enough to justify a PoV dispute tag. But really, it doesn't matter if the wrong thing is posted for a few days while we work out our differences. — Cheers, Steelpillow (Talk) 10:10, 22 October 2015 (UTC)

Complex space

There is a discussion at Talk:Complex space. I am posting here because this article began life as Complex space and was later re-purposed, and the current use of Complex space as a disambig page grew out of discussions on this talk page. — Cheers, Steelpillow (Talk) 17:54, 27 October 2015 (UTC)