Talk:Projective line over a ring

history

Hi Rgdboer, Why do you think the history section should appear at the end of the article? It is more customary to place it close to the beginning. Tkuvho (talk) 17:16, 12 December 2010 (UTC)

Hello Tkuvho, thank you for starting the discussion. I agree that integrating the history of a topic often improves its readability. Sometimes citing the references amounts to the history. The subject of inversive ring geometry has fewer references than does Inversive geometry, the subtopic corresponding to the complex number field. In that article I provided the references found in Inversive geometry#Transformation theory, and they give some inkling of history. As for the "History" found in this article, I wrote it as a cushion between the facts of the subject as presented, and the various journal citations; it didn't have the flow needed for an introductory section.Rgdboer (talk) 00:44, 14 December 2010 (UTC)

Possible Rewording

"linear-fractional transformations of a Clifford algebra"

Wouldn't that just mean "conformal transformations"? 166.147.103.108 (talk) 16:49, 15 October 2011 (UTC) Collin237

The phrase you mention refers to Theodor Vahlen's work which does not mention conformality but does mention Clifford's work with algebra. The citation does not expand Vahlen's work to the property you suggest.Rgdboer (talk) 21:35, 15 October 2011 (UTC)

An error?

It is said that the identity is the unique linear-fractional transformations sending the ordered triple 0,1,infinity onto itself. This is not true: for any u invertible, the map q->uqu^{-1} is a linear-fractional transformation that does the job. — Preceding unsigned comment added by 134.157.87.231 (talk) 16:33, 21 November 2012 (UTC)

Thank you for the note. The ring geometry is to generalize Mobius transformation#Specifying a transformation by three points. The statement is only that the map for p,q,r to 0,1,∞ is unique. Your example goes to a frequently stated "Fundamental theorem of projective geometry".Rgdboer (talk) 21:33, 21 November 2012 (UTC)

Your note has resulted in a change of title to Transitivity from "Cross-ratio theorems" since sharp 3-transitivity is required for an invariant cross-ratio. More suggestions to improve the article are welcome.Rgdboer (talk) 02:17, 22 November 2012 (UTC)

New title

This article used to be titled Inversive ring geometry. An editor indicated that such title does not meet Notability requirements for the Encyclopedia. Investigating, the current title was found to be the subject of several articles in reviewed mathematical literature. The old article was moved under the current title, with an appropriate rewrite of the introduction. One might notice that inverse (mathematics) has several meanings, detracting from its use as a discriminative descriptor. The article continues to express a branch of transformation geometry as the structural aspect of a ring suggests many mappings, as indicated in the sources.Rgdboer (talk) 21:25, 1 March 2013 (UTC)

The new title definitely generates multiple relevant hits in googlescholar, so I believe this to be a great improvement over the old title. Rschwieb (talk) 22:12, 1 March 2013 (UTC)

Classic definition

The definition P(A) = {U(a,b): aA + bA = A } has been restored. This definition is most common. The alternative using the group of units from the matrix ring over A is mentioned where appropriate.Rgdboer (talk) 22:45, 1 April 2013 (UTC)

To those only familiar with a projective line over a field or even a division ring, an unexpected feature is that many of the "homogeneous vectors" (a,b) may be omitted from the space, meaning that the familiar feature of the whole space of "vectors" being partitioned into equivalence classes U(a,b) (points of the line) does not apply, and that the space of "vectors" is not a vector space in the sense of being closed under addition and (ring) scalar multiplication. Wording in both the lead and in the body that makes this more salient would be an improvement. —Quondum 03:45, 22 February 2015 (UTC)

There are no vectors in this article, only ordered pairs. Yes, some ordered pairs are excluded when they do not generate the whole ring. Your concern is not clear to me.Rgdboer (talk) 20:25, 22 February 2015 (UTC)

My concern is that, the way it is presented, it takes time to realize which concepts to throw out when generalizing from a projective line over a field or division ring; one would expect this to be made more obvious. The title, the initial description using homogeneous coordinates and the use of what is indistinguishable from scalar multiplication (albeit restricted to scalars that are units) positively invites the interpretation of the ordered pair as an element of a module. For example, the sentence "Given a ring A with 1, the projective line P(A) over A consists of points identified by homogeneous coordinates" gives no hint up to that point that there is not necessarily a point associated with an arbitrary nonzero pair. Let's just say that until I realized that the space is not to be treated as a vector space as in the normal projective line (and I only realized this once I'd looked at the case of integers), it was rather confusing trying to make sense of it. As a side-point, it seems pretty trivial to include (probably) all pairs, along with a full free module structure, by simply defining all points that are any scalar multiple of an existing pair to be part of the same equivalence class – that is to say, we can define homogeneous coordinates and a projective line over a module almost indistinguishably from the usual version using a vector space, restricting the set of scalar multipliers far less (e.g. to include all but zero divisors rather than only units). This should produce the identical object. —Quondum 23:26, 22 February 2015 (UTC)

The following text is found at projective line:

An arbitrary point in the projective line P¹(K) may be given in homogeneous coordinates by a pair

$[x_{1}:x_{2}]$

of points in K which are not both zero.

The stipulation "which are not both zero" is generalized here with aA + bA = A for a ring A. A link to equivalence relation has been put in the Projective line article since points are equivalence classes. Verifying the properties of an equivalence relation is trivial in these articles so details are not given though some readers may need to fill them in to comprehend the articles.Rgdboer (talk) 21:06, 23 February 2015 (UTC)

Definitions

If R is a principal ideal domain, then every R-valued point of the scheme P^1 over R is an equivalence class (a:b) of pairs (a,b) of elements of R generating the unit ideal, modulo scalar multiplication by units. But this is false for more general commutative rings. For instance, it is false for $R=\mathbb {Z} [{\sqrt {-5}}]$. If this article is going to continue to exist, perhaps the simplest solution may be to require R to be a PID.

Also, is a homography an element of GL_2(R), an element of GL_2(R)/R*, or an R-valued point of the group scheme PGL_2 over a commutative ring R? The article needs to be clear about this, and to be clear about which kind of rings this is being defined for. Ebony Jackson (talk) 22:33, 16 November 2013 (UTC)

Thank you for raising important issues. The second one has been treated. The problem of loss of unique factorization as in that quadratic integer ring has me studying the article ideal class group#Examples of ideal class groups. As mentioned there, this subject can stand some attention.Rgdboer (talk) 02:03, 21 November 2013 (UTC)