Eight colors illustrate the projective line over Galois field GF(7)

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 projective 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 ${\displaystyle {\begin{pmatrix}c&0\\0&c\end{pmatrix}}}$ 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):

${\displaystyle U[a,1]{\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U[1,a]\thicksim U[a^{-1},1].}$

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

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

${\displaystyle U[a,1]{\begin{pmatrix}v&0\\0&u\end{pmatrix}}=U[av,u]\thicksim U[u^{-1}av,1].}$

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

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

Instances

Six colors illustrate the projective line over Galois field GF(5)

Rings that are fields are most familiar: 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 For a finite field GF(q), the projective line is the Galois geometry PG(1, q). J. W. P. Hirschfeld has described the harmonic tetrads in the projective lines for q = 4, 5, 7, 8, 9.^[2]

Modules

The projective line P(A) over a ring A can also be identified as the space of projective modules in the module ${\displaystyle A\oplus A}$. An element of P(A) is then a direct summand of ${\displaystyle A\oplus A}$. 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^[8] 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 that are a principal 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"^[9] 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^[10][11] 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

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

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

${\displaystyle {\frac {1}{b-c}}+{\frac {1}{c-a}}}$ is a unit.

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

Theorem: If ${\displaystyle (b-c)^{-1}+(c-a)^{-1}}$ 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 ${\displaystyle p=(b-c)^{-1}+(c-a)^{-1}}$ 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.

One application of cross ratio defines the projective harmonic conjugate of a triple a, b, c, as the element x satisfying (x, a, b, c) = −1. Such a quadruple is a harmonic tetrad. Harmonic tetrads on the projective line over a finite field GF(q) were used in 1954 to delimit the projective linear groups PGL(2, q) for q = 5, 7, and 9, and demonstrate accidental isomorphisms.^[12]

Chains

The real line in the complex plane gets permuted with circles and other real lines under Möbius 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

${\displaystyle U_{F}[x,1]\mapsto U_{A}[x,1],\quad U_{F}[1,0]\mapsto U_{A}[1,0].}$

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].^[13]

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 Möbius. In 1902 Theodore Vahlen contributed a short but well-referenced paper exploring some linear fractional transformations of a Clifford algebra.^[15] The ring of dual numbers D gave Josef Grünwald opportunity to exhibit P(D) in 1906.^[4] Corrado Segre (1912) continued the development with that ring.^[5]

Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.^[16] In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.^[17] 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,^[7] which included the homogeneous coordinates taken from M.

See also

• Euclid's orchard

Notes and references

• Sky Brewer (2012) "Projective Cross-ratio on Hypercomplex Numbers", Advances in Applied Clifford Algebras, DOI 10.1007/s00006-12-0335-7.
• I. M. Yaglom (1968) Complex Numbers in Geometry.

Further reading

• G. Ancochea (1941) "Le théorèm de von Staudt en géométrie projective quaternionienne", Journal für Mathematik, Band 184, Heft 4, SS. 193–8.
• N. B. Limaye (1972) "Cross-ratios and Projectivities of a line", Mathematische Zeitschrift 129: 49–53, MR0314823.
• B.V. Limaye & N.B. Limaye (1977) "The Fundamental Theorem for the Projective Line over Commutative Rings", Aequationes Mathematica 16:275–81. MR0513873.
• B.V. Limaye & N.B. Limaye (1977) "The Fundamental Theorem for the Projective Line over Non-Commutative Local Rings", Archiv der Mathematik 28(1):102–9 MR0480495.
• Marcel Wild (2006) "The Fundamental Theorem of Projective Geometry for an Arbitrary Length Two Module", Rocky Mountain Journal of Mathematics 36(6):2075–80.

External links

• Mitod Saniga (2006) Projective Lines over Finite Rings (pdf) from Astronomical Institute of the Slovak Academy of Sciences