Editing Projective line over a ring (section)

== Cross-ratio ==
A homography ''h'' that takes three particular ring elements ''a'', ''b'', ''c'' to the projective line points {{nowrap|''U''[0, 1]}}, {{nowrap|''U''[1, 1]}}, {{nowrap|''U''[1, 0]}} is called the '''cross-ratio homography'''. Sometimes<ref>{{citation |first1=Gareth |last1=Jones |first2=David |last2=Singerman |date=1987 |title=Complex Functions |pages=23, 24 |publisher=[[Cambridge University Press]] }}</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 (journal)|Simon Stevin]] 42:97–111 {{MathSciNet|id=0266032}}</ref> the [[cross-ratio]] is taken as the value of ''h'' on a fourth point {{nowrap|1=''x'' : (''x'', ''a'', ''b'', ''c'') = ''h''(''x'')}}.

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 point (mathematics)|fixed point]]s: +1 and −1 are fixed under inversion, {{nowrap|''U''[1, 0]}} is fixed under translation, and the "rotation" with ''u'' leaves {{nowrap|''U''[0, 1]}} and {{nowrap|''U''[1, 0]}} fixed. The instructions are to place ''c'' first, then bring ''a'' to {{nowrap|''U''[0, 1]}} with translation, and finally to use rotation to move ''b'' to {{nowrap|''U''[1, 1]}}.

Lemma: If ''A'' is a [[commutative ring]] and {{nowrap|''b'' − ''a''}}, {{nowrap|''c'' − ''b''}}, {{nowrap|''c'' − ''a''}} are all units, then {{nowrap|(''b'' − ''c'')<sup>−1</sup> + (''c'' − ''a'')<sup>−1</sup>}} 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 {{nowrap|(''b'' − ''c'')<sup>−1</sup> + (''c'' − ''a'')<sup>−1</sup>}} is a unit, then there is a homography ''h'' in G(''A'') such that 
: {{nowrap|1=''h''(''a'') = ''U''[0, 1]}}, {{nowrap|1=''h''(''b'') = ''U''[1, 1]}}, and {{nowrap|1=''h''(''c'') = ''U''[1, 0]}}.

Proof: The point {{nowrap|1=''p'' = (''b'' − ''c'')<sup>−1</sup> + (''c'' − ''a'')<sup>−1</sup>}} is the image of ''b'' after ''a'' was put to 0 and then inverted to {{nowrap|''U''[1, 0]}}, and the image of ''c'' is brought to {{nowrap|''U''[0, 1]}}. As ''p'' is a unit, its inverse used in a rotation will move ''p'' to {{nowrap|''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 {{nowrap|1=(''x'', ''a'', ''b'', ''c'') = −1}}. Such a quadruple is a [[projective harmonic conjugate#Galois tetrads|harmonic tetrad]]. Harmonic tetrads on the projective line over a [[finite field]] GF(''q'') were used in 1954 to delimit the projective linear groups {{nowrap|PGL(2, ''q'')}} for ''q'' =  5, 7, and 9, and demonstrate [[accidental isomorphism]]s.<ref>{{citation |first1=Jean |last1=Dieudonné |authorlink=Jean Dieudonné |date=1954 |title=Les Isomorphisms exceptionnals entre les groups classiques finis |journal=[[Canadian Journal of Mathematics]] |volume=6 |pages=305–315 |doi=10.4153/CJM-1954-029-0 }}</ref>