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 P1(A) over A consists of points identified by projective coordinates. Let A× be the group of units of A; pairs Template:Nowrap and Template:Nowrap from Template:Nowrap are related when there is a u in A× such that Template:Nowrap and Template:Nowrap. This relation is an equivalence relation. A typical equivalence class is written Template:Nowrap.
Template:Nowrap, that is, Template:Nowrap is in the projective line if the one-sided ideal generated by a and b is all of A.
The projective line P1(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(A×), the center of A×, then the group action of matrix <math>\left(\begin{smallmatrix}c & 0 \\ 0 & c \end{smallmatrix}\right)</math> on P1(A) is the same as the action of the identity matrix. Such matrices represent a normal subgroup N of V. The homographies of P1(A) correspond to elements of the quotient group Template:Nowrap.
P1(A) is considered an extension of the ring A since it contains a copy of A due to the embedding Template:Nowrap. The multiplicative inverse mapping Template:Nowrap, ordinarily restricted to A×, is expressed by a homography on P1(A):
- <math>U[a,1]\begin{pmatrix}0&1\\1&0\end{pmatrix} = U[1, a] \thicksim U[a^{-1}, 1].</math>
Furthermore, for Template:Nowrap, the mapping Template:Nowrap can be extended to a homography:
- <math>\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}. </math>
- <math>U[a,1]\begin{pmatrix}v&0\\0&u\end{pmatrix} = U[av,u] \thicksim U[u^{-1}av,1].</math>
Since u is arbitrary, it may be substituted for u−1. Homographies on P1(A) are called linear-fractional transformations since
- <math>U[z,1] \begin{pmatrix}a&c\\b&d\end{pmatrix} = U[za+b,zc+d] \thicksim U[(zc+d)^{-1}(za+b),1].</math>
InstancesEdit
Rings that are fields are most familiar: The projective line over GF(2) has three elements: Template:Nowrap, Template:Nowrap, and Template:Nowrap. Its homography group is the permutation group on these three.<ref name=Rankin>Template:Citation</ref>Template:Rp
The ring [[Modular arithmetic#Integers modulo m|ZTemplate:Hsp/Template:Hsp3Z]], or GF(3), has the elements 1, 0, and −1; its projective line has the four elements Template:Nowrap, Template:Nowrap, Template:Nowrap, Template:Nowrap since both 1 and −1 are units. The homography group on this projective line has 12 elements, also described with matrices or as permutations.<ref name=Rankin/>Template:Rp For a finite field GF(q), the projective line is the Galois geometry Template:Nowrap. J. W. P. Hirschfeld has described the harmonic tetrads in the projective lines for q = 4, 5, 7, 8, 9.<ref>Template:Cite book</ref>
Over discrete ringsEdit
Consider Template:Nowrap when n is a composite number. If p and q are distinct primes dividing n, then Template:Angle bracket and Template:Angle bracket are maximal ideals in Template:Nowrap and by Bézout's identity there are a and b in Z such that Template:Nowrap, so that Template:Nowrap is in Template:Nowrap but it is not an image of an element under the canonical embedding. The whole of Template:Nowrap is filled out by elements Template:Nowrap, where Template:Nowrap and Template:Nowrap, A× being the units of Template:Nowrap. The instances Template:Nowrap are given here for n = 6, 10, and 12, where according to modular arithmetic the group of units of the ring is Template:Nowrap, Template:Nowrap, and Template:Nowrap respectively. Modular arithmetic will confirm that, in each table, a given letter represents multiple points. In these tables a point Template:Nowrap is labeled by m in the row at the table bottom and n in the column at the left of the table. For instance, the point at infinity Template:Nowrap, where v is a unit of the ring.
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The extra points can be associated with Template:Nowrap, the rationals in the extended complex upper-half plane. The group of homographies on Template:Nowrap is called a principal congruence subgroup.<ref>Template:Citation</ref>
For the rational numbers Q, homogeneity of coordinates means that every element of P1(Q) may be represented by an element of P1(Z). Similarly, a homography of P1(Q) corresponds to an element of the modular group, the automorphisms of P1(Z).
Over continuous ringsEdit
The projective line over a division ring results in a single auxiliary point Template:Nowrap. 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 C has the Möbius group as its homography group.
The projective line over the dual numbers was described by Josef Grünwald in 1906.<ref name="Grünwald">Template:Citation</ref> This ring includes a nonzero nilpotent n satisfying Template:Nowrap. The plane Template:Nowrap of dual numbers has a projective line including a line of points Template:Nowrap.<ref name=CS>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.</ref> Isaak Yaglom has described it as an "inversive Galilean plane" that has the topology of a cylinder when the supplementary line is included.<ref name=Yaglom79>Template:Citation</ref>Template:Rp Similarly, if A is a local ring, then P1(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 Template:Nowrap and Template:Nowrap Using stereographic projection the plane of split-complex numbers is closed up with these lines to a hyperboloid of one sheet.<ref name=Yaglom79/>Template:Rp<ref name=Benz73>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 Template:Isbn Template:MathSciNet</ref> The projective line over M may be called the Minkowski plane when characterized by behaviour of hyperbolas under homographic mapping.
ModulesEdit
The projective line P1(A) over a ring A can also be identified as the space of projective modules in the module Template:Nowrap. An element of P1(A) is then a direct summand of Template:Nowrap. 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<ref>Birkhoff and Maclane (1953) Survey of modern algebra, pp. 293–298, or 1997 AKP Classics edition, pp. 312–317</ref> 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 P1(Z) narrows attention to the Template:Nowrap, m coprime to n, and sheds the embeddings that are a principal feature of P1(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"<ref>Template:Citation. 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 Template:Nowrap where m and n are coprime.</ref> 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 Template:Nowrap, 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 Template:Nowrap of the free cyclic submodule Template:Nowrap of Template:Nowrap. Extending the commutative theory of Benz, the existence of a right or left multiplicative inverse of a ring element is related to P1(R) and Template:Nowrap. The Dedekind-finite property is characterized. Most significantly, representation of P1(R) in a projective space over a division ring K is accomplished with a Template:Nowrap-bimodule U that is a left K-vector space and a right R-module. The points of P1(R) are subspaces of Template:Nowrap isomorphic to their complements.
Cross-ratioEdit
A homography h that takes three particular ring elements a, b, c to the projective line points Template:Nowrap, Template:Nowrap, Template:Nowrap is called the cross-ratio homography. Sometimes<ref>Template:Citation</ref><ref>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 Template:MathSciNet</ref> the cross-ratio is taken as the value of h on a fourth point Template:Nowrap.
To build h from a, b, c the generator homographies
- <math>\begin{pmatrix}0 & 1\\1 & 0 \end{pmatrix}, \begin{pmatrix}1 & 0\\t & 1 \end{pmatrix}, \begin{pmatrix}u & 0\\0 & 1 \end{pmatrix}</math>
are used, with attention to fixed points: +1 and −1 are fixed under inversion, Template:Nowrap is fixed under translation, and the "rotation" with u leaves Template:Nowrap and Template:Nowrap fixed. The instructions are to place c first, then bring a to Template:Nowrap with translation, and finally to use rotation to move b to Template:Nowrap.
Lemma: If A is a commutative ring and Template:Nowrap, Template:Nowrap, Template:Nowrap are all units, then Template:Nowrap is a unit.
Proof: Evidently <math>\frac{b-a}{(b-c)(c-a)} = \frac{(b-c)+(c-a)}{(b-c)(c-a)}</math> is a unit, as required.
Theorem: If Template:Nowrap is a unit, then there is a homography h in G(A) such that
Proof: The point Template:Nowrap is the image of b after a was put to 0 and then inverted to Template:Nowrap, and the image of c is brought to Template:Nowrap. As p is a unit, its inverse used in a rotation will move p to Template:Nowrap, 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 Template:Nowrap. 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 Template:Nowrap for q = 5, 7, and 9, and demonstrate accidental isomorphisms.<ref>Template:Citation</ref>
ChainsEdit
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 P1(F) into P1(A) is
- <math>U_F[x, 1] \mapsto U_A[x, 1] , \quad U_F[1, 0] \mapsto U_A[1, 0].</math>
A chain is the image of P1(F) under a homography on P1(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].<ref>Template:Citation</ref>
Point-parallelismEdit
Two points of P1(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.<ref>Walter Benz, Hans-Joachim Samaga, & Helmut Scheaffer (1981) "Cross Ratios and a Unifying Treatment of von Staudt's Notion of Reeller Zug", pp. 127–150 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, Template:Isbn, Template:MathSciNet</ref>
HistoryEdit
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.<ref>Template:Citation</ref> The ring of dual numbers D gave Josef Grünwald opportunity to exhibit P1(D) in 1906.<ref name="Grünwald"/> Corrado Segre (1912) continued the development with that ring.<ref name=CS/>
Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.<ref>Template:Citation</ref> In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.<ref>Template:Citation</ref> In 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian. There he uses P1(D) to describe line geometry in the Euclidean plane and P1(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 P1(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,<ref name=Benz73/> which included the homogeneous coordinates taken from M.
See alsoEdit
Notes and referencesEdit
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Further readingEdit
External linksEdit
- Mitod Saniga (2006) Projective Lines over Finite Rings (pdf) from Astronomical Institute of the Slovak Academy of Sciences