Projective line over a ring

In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A with 1, the projective line P(A) over A consists of points identified by homogeneous coordinates. Let U be the group of units of A; pairs (a,b) and (c,d) from A × A are related when there is a u in U such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U(a,b).

P(A) = { U(a,b) : aA + bA = A }, that is, U(a,b) is in the projective line if the ideal generated by a and b is all of A.

The projective line P(A) is equipped with a group of homographies. The homographies are expressed through use of the matrix ring over A and its group of units V as follows: If c is in Z(U), the center of U, then the group action of matrix ${\begin{pmatrix}c&0\\0&c\end{pmatrix}}$ on P(A) is the same as the action of the identity matrix. Such matrices represent a normal subgroup N of V. The homographies of P(A) correspond to elements of the quotient group V / N .

P(A) is considered an extension of the ring A since it contains a copy of A due to the embedding E : a → U(a,1). The multiplicative inverse mapping u → 1/u, ordinarily restricted to the group of units U of A, is expressed by a homography on P(A):

U(a,1){\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U(1,a)\thicksim U(a^{-1},1).

Furthermore, for u,v ∈ U, the mapping a → uav can be extended to a homography:

{\begin{pmatrix}u&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}v&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}u&0\\0&v\end{pmatrix}}.

U(a,1){\begin{pmatrix}v&0\\0&u\end{pmatrix}}=U(av,u)\thicksim U(u^{-1}av,1).

Since u is arbitrary, it may be substituted for u⁻¹. Homographies on P(A) are called linear-fractional transformations since

U(z,1){\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U(za+b,zc+d)\thicksim U((zc+d)^{-1}(za+b),1).

Instances

Finite rings have finite projective lines. The projective line over GF(2) has three elements: U(0,1), U(1,0), and U(1,1). Its homography group is the permutation group on these three.^[1]^:29

The ring Z/3Z, or GF(3), has the elements 1, 0, and −1; its projective line has the four elements U(1,0), U(1,1), U(0,1), U(1,−1) since both 1 and −1 are units. The homography group on this projective line has 12 elements, also described with matrices or as permutations.^[1]^:31

The projective line over the ring of integers Z includes points U(m,n) where n and m are relatively prime. The homography group on this projective line is the modular group. Its congruence subgroups serve as homography groups on projective lines over Z/nZ.^[2]^[3]

The projective line over a division ring results in a single auxiliary point ∞ = U(1,0). Examples include the real projective line, the complex projective line, and the projective line over quaternions. These examples of topological rings have the projective line as their one-point compactifications. The case of the complex number field has the Möbius group as its homography group.

The projective line over the dual numbers was described by Josef Grünwald in 1906.^[4] This ring includes a nonzero nilpotent n satisfying nn = 0. The plane { z = x + yn : x,y ∈ R } of dual numbers has a projective line including a line of points U(1, xn), x ∈ R.^[5] Isaak Yaglom has described it as an "inversive Galilean plane" that has the topology of a cylinder when the supplementary line is included.^[6]^:149–53 Similarly, if A is a local ring, then P(A) is formed by adjoining points corresponding to the elements of the maximal ideal of A.

The projective line over the ring M of split-complex numbers introduces auxiliary lines { U(1,x(1 + j)) : x ∈ R } and { U(1,x(1 − j)) : x ∈ R }. Using stereographic projection the plane of split-complex numbers is closed up with these lines to a hyperboloid of one sheet.^[6]^{: 174–200}^[7] The projective line over M may be called the Minkowski plane when characterized by behaviour of hyperbolas under homographic mapping.

Chains

The real line in the complex plane gets permuted with circles and other real lines under Mobius transformations, which actually permute the canonical embedding of the real projective line in the complex projective line. Suppose A is an algebra over a field F, generalizing the case where F is the real number field and A is the field of complex numbers. The canonical embedding of P(F) into P(A) is

U_{F}(x,1)\mapsto U_{A}(x,1),\quad U_{F}(1,0)\mapsto U_{A}(1,0).

A chain is the image of P(F) under a homography on P(A). Four points lie on a chain if and only if their cross-ratio is in F. Karl von Staudt exploited this property in his theory of "real strokes" [reeler Zug].^[8]

Point-parallelism

Two points of P(A) are parallel if there is no chain connecting them. The convention has been adopted that points are parallel to themselves. This relation is invariant under the action of a homography on the projective line. Given three pair-wise non-parallel points, there is a unique chain that connects the three.^[9]

Modules

The projective line P(A) over a ring A can also be identified as the space of projective modules in the module $A\oplus A$ . An element of P(A) is then a direct summand of $A\oplus A$ . This more abstract approach follows the view of projective geometry as the geometry of subspaces of a vector space, sometimes associated with the lattice theory of Garrett Birkhoff^[10] or the book Linear Algebra and Projective Geometry by Reinhold Baer. In the case of the ring of rational integers Z, the module summand definition of P(Z) narrows attention to the U(m,n), m coprime to n, and sheds the embeddings which are a principle feature of P(A) when A is topological. The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition.

In an article "Projective representations: projective lines over rings"^[2] the group of units of a matrix ring M₂(R) and the concepts of module and bimodule are used to define a projective line over a ring. The group of units is denoted by GL(2,R), adopting notation from the general linear group, where R is usually taken to be a field.

The projective line is the set of orbits under GL(2,R) of the free cyclic submodule R(1,0) of R × R. Extending the commutative theory of Benz, the existence of a right or left multiplicative inverse of a ring element is related to P(R) and GL(2,R). The Dedekind-finite property is characterized. Most significantly, representation of P(R) in a projective space over a division ring K is accomplished with a (K,R)-bimodule U that is a left K-vector space and a right R-module. The points of P(R) are subspaces of P(K, U × U) isomorphic to their complements.

Cross-ratio

A homography h that takes three particular ring elements a, b, c to the projective line points U(0,1), U(1,1), U(1,0) is called the cross-ratio homography. Sometimes^[11]^[12] the cross-ratio is taken as the value of h on a fourth point x : (x,a,b,c) = h(x).

To build h from a, b, c the generator homographies

{\begin{pmatrix}0&1\\1&0\end{pmatrix}},{\begin{pmatrix}1&0\\t&1\end{pmatrix}},{\begin{pmatrix}u&0\\0&1\end{pmatrix}}

are used, with attention to fixed points: +1 and −1 are fixed under inversion, U(1,0) is fixed under translation, and the "rotation" with u leaves U(0,1) and U(1,0) fixed. The instructions are to place c first, then bring a to U(0,1) with translation, and finally to use rotation to move b to U(1,1).

Lemma: If A is a commutative ring and b − a, c − b, c − a are all units, then

{\frac {1}{b-c}}+{\frac {1}{c-a}}

is a unit.

proof: Evidently ${\frac {b-a}{(b-c)(c-a)}}={\frac {(b-c)+(c-a)}{(b-c)(c-a)}}$ is a unit, as required.

Theorem: If $(b-c)^{-1}+(c-a)^{-1}$ is a unit, then there is a homography h in G(A) such that

h(a) = U(0,1), h(b) = U(1,1), and h(c) = U(1,0).

proof: The point $p=(b-c)^{-1}+(c-a)^{-1}$ is the image of b after a was put to 0 and then inverted to U(1,0), and the image of c is brought to U(0,1). As p is a unit, its inverse used in a rotation will move p to U(1,1), resulting in a, b, c being all properly placed. The lemma refers to sufficient conditions for the existence of h.

History

August Ferdinand Möbius investigated the Möbius transformations between his book Barycentric Calculus (1827) and his 1855 paper "Theorie der Kreisverwandtschaft in rein geometrischer Darstellung". Karl Wilhelm Feuerbach and Julius Plücker are also credited with originating the use of homogeneous coordinates. Eduard Study in 1898, and Élie Cartan in 1908, wrote articles on hypercomplex numbers for German and French Encyclopedias of Mathematics, respectively, where they use these arithmetics with linear fractional transformations in imitation of those of Mobius. In 1902 Theodore Vahlen contributed a short but well-referenced paper exploring some linear fractional transformations of a Clifford algebra.^[13] The ring of dual numbers D gave Josef Grünwald opportunity to exhibit P(D) in 1906. Corrado Segre (1912) continued the development with that ring.

Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.^[14] In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in his paper "Stereographic projection and the linear fractional group of transformations of quaternions". In 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian. There he uses P(D) to describe line geometry in the Euclidean plane and P(M) to describe it for Lobachevski's plane. Yaglom's text A Simple Non-Euclidean Geometry appeared in English in 1979. There in pages 174 to 200 he develops Minkowskian geometry and describes P(M) as the "inversive Minkowski plane". The Russian original of Yaglom's text was published in 1969. Between the two editions, Walter Benz (1973) published his book which included the homogeneous coordinates taken from M.

Notes and references

1 2 Robert Alexander Rankin (1977) Modular forms and functions, Cambridge University Press ISBN 0-521-21212-X
1 2 A Blunck & H Havlicek (2000) "Projective representations: projective lines over rings", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 70:287–99, MR 1809553. This article uses an alternative definition of projective line over a ring that restricts elements of the projective line over Z to those of the form U(m,n) where m and n are coprime.
↑ Metod Saniga, Michel Planat, Maurice R. Kibler, Petr Pracna (2007) "A classification of the projective lines over small rings", Chaos, Solitons & Fractals 33(4):1095–1102, MR 2318902
↑ Josef Grünwald (1906) "Über duale Zahlen und ihre Anwendung in der Geometrie", Monatshefte für Mathematik 17: 81–136
↑ Corrado Segre (1912) "Le geometrie proiettive nei campi di numeri duali", Paper XL of Opere, also Atti della R. Academia della Scienze di Torino, vol XLVII.
1 2 Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, ISBN 0387-90332-1, MR 520230
↑ Walter Benz (1973) Vorlesungen über Geometrie der Algebren, §2.1 Projective Gerade über einem Ring, §2.1.2 Die projective Gruppe, §2.1.3 Transitivitätseigenschaften, §2.1.4 Doppelverhaltnisse, Springer ISBN 0-387-05786-2 MR 353137
↑ Karl von Staudt (1856) Beträge zur Geometrie der Lage
↑ Walter Benz, Hans-Joachim Samaga, & Helmut Scheaffer (1981) "Cross Ratios and a Unifying Treatment of von Staudt’s Notion of Reeller Zug", pp 127–50 in Geometry – von Staudt’s Point of View, Peter Plaumann & Karl Strambach editors, Proceedings of NATO Advanced Study Institute, Bad Windsheim, July/August 1980, D. Reidel, ISBN 90-277-1283-2, MR 0621313
↑ Birkhoff and Maclane (1953) Survey of modern algebra, pp 293–8, or 1997 AKP Classics edition, pp 312–7
↑ Gareth Jones and David Singerman (1987) Complex Functions, pp 23,4 Cambridge University Press
↑ Joseph A. Thas (1968/9) "Cross ratio of an ordered point quadruple on the projective line over an associative algebra with at unity element" (in Dutch) Simon Stevin 42:97–111 MR 0266032
↑ Theodore Vahlen, Theodor (1902) "Über Bewegungen und complexe Zahlen", Mathematische Annalen 55:585–93
↑ Arthur Conway (1911) "On the application of quaternions to some recent developments of electrical theory", Proceedings of the Royal Irish Academy 29:1–9, particularly page 9

Sky Brewer (2012) "Projective Cross-ratio on Hypercomplex Numbers", Advances in Applied Clifford Algebras, DOI 10.1007/s00006-12-0335-7.
P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A 51:67–85.
I. M. Yaglom (1968) Complex Numbers in Geometry.