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Projective line over a ring
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the projective line over a ring is an extension of the concept of <a href="/facts/Projective_line/PhRgf87G">projective line</a> over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. Given a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> <i>A</i> (with 1), the projective line P1(<i>A</i>) over <i>A</i> consists of points identified by <a href="/facts/Projective_coordinates/TG9ebKN4">projective coordinates</a>. Let <i>A</i>× be the <a href="/facts/Group_of_units/GJSBu8Y7">group of units</a> of <i>A</i>; pairs (<i>a</i>, <i>b</i>) and (<i>c</i>, <i>d</i>) from <i>A</i> × <i>A</i> are related when there is a <i>u</i> in <i>A</i>× such that <i>ua</i> = <i>c</i> and <i>ub</i> = <i>d</i>. This relation is an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a>. A typical <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> is written <i>U</i>[<i>a</i>, <i>b</i>]. </p><p>P1(<i>A</i>) = { <i>U</i>[<i>a</i>, <i>b</i>] | <i>aA</i> + <i>bA</i> = <i>A</i> }, that is, <i>U</i>[<i>a</i>, <i>b</i>] is in the projective line if the <a href="/facts/Right_ideal/vCvytduX">one-sided ideal</a> generated by <i>a</i> and <i>b</i> is all of <i>A</i>. </p><p>The projective line P1(<i>A</i>) is equipped with a <a href="/facts/Homography/wCkNzUjd">group of homographies</a>. The homographies are expressed through use of the <a href="/facts/Matrix_ring/oj5HBr8B">matrix ring</a> over <i>A</i> and its group of units <i>V</i> as follows: If <i>c</i> is in Z(<i>A</i>×), the <a href="/facts/Center_(group_theory)/gqAg6B8I">center</a> of <i>A</i>×, then the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">group action</a> of matrix ( c 0 0 c ) {\displaystyle \left({\begin{smallmatrix}c&0\\0&c\end{smallmatrix}}\right)} on P1(<i>A</i>) is the same as the action of the identity matrix. Such matrices represent a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> <i>N</i> of <i>V</i>. The homographies of P1(<i>A</i>) correspond to elements of the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> <i>V</i> / <i>N</i>. </p><p>P1(<i>A</i>) is considered an extension of the ring <i>A</i> since it contains a copy of <i>A</i> due to the embedding <i>E</i> : <i>a</i> → <i>U</i>[<i>a</i>, 1]. The <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> mapping <i>u</i> → 1/<i>u</i>, ordinarily restricted to <i>A</i>×, is expressed by a homography on P1(<i>A</i>): </p> U [ a , 1 ] ( 0 1 1 0 ) = U [ 1 , a ] ∼ U [ a − 1 , 1 ] . {\displaystyle U[a,1]{\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U[1,a]\thicksim U[a^{-1},1].} <p>Furthermore, for <i>u</i>,<i>v</i> ∈ <i>A</i>×, the mapping <i>a</i> → <i>uav</i> can be extended to a homography: </p> ( u 0 0 1 ) ( 0 1 1 0 ) ( v 0 0 1 ) ( 0 1 1 0 ) = ( u 0 0 v ) . {\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}}.} U [ a , 1 ] ( v 0 0 u ) = U [ a v , u ] ∼ U [ u − 1 a v , 1 ] . {\displaystyle U[a,1]{\begin{pmatrix}v&0\\0&u\end{pmatrix}}=U[av,u]\thicksim U[u^{-1}av,1].} <p>Since <i>u</i> is arbitrary, it may be substituted for <i>u</i>−1. Homographies on P1(<i>A</i>) are called linear-fractional transformations since </p> U [ z , 1 ] ( a c b d ) = U [ z a + b , z c + d ] ∼ U [ ( z c + d ) − 1 ( z a + b ) , 1 ] . {\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].}