Exercise 7.5.6

Proof.

(a)

The group $\mathit{GL}(2,F)$, whose elements are the matrices $M=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$ such that $\mathit{ad}-\mathit{bc}\ne 0$, acts on $F^2$, identified to the matrix columns of order 2, by the action defined by $$(x^{\prime },y^{\prime })=M\cdot (x,y)⟺\left(\begin{array}{c}\hfill x^{\prime }\hfill \\ \hfill y^{\prime }\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \mathit{ax}+\mathit{by}\hfill \\ \hfill \mathit{cx}+\mathit{dy}\hfill \end{array}\right)$$

Indeed, if we write $X=\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)$, then $I\cdot X=X,M\cdot (N\cdot X)=(\mathit{MN})\cdot X$.

The relation $\mathcal{}R$ defined on $F^2\setminus \{(0,0)\}$ by

$$(x,y)\mathcal{}R(x^{\prime },y^{\prime })⟺\exists <!--\; FUNCTION\; APPLICATION\; -->\lambda \in F^{\text{∗}},x^{\prime }=\mathit{\lambda x},y^{\prime }=\mathit{\lambda y}$$

is an equivalence relation. The quotient set is the projective line ${\mathbb{P}}_1(F)$. Write $[x,y]$ the class of $(x,y)$ for the relation $\mathcal{}R$, in other words the projective point with homogen coordinates $(x,y)$.

If $(x,y)\mathcal{}R(x^{\prime },y^{\prime })$, then $M\cdot (x,y)\mathcal{}RM\cdot (x^{\prime },y^{\prime })$. Moreover $M\cdot (x,y)\ne (0,0)$ if $(x,y)\ne (0,0)$, so we can define the action on a projective point $P=[x,y]$ by $M\cdot [x,y]=M\cdot (x,y)$, where $(x,y)$ is any representative of the class $P$. This is again an action of the group $\mathit{GL}(2,F)$ on the set ${\mathbb{P}}_1(F)$.

The map $f:{\mathbb{P}}_1(F)\to \hat{F}=F\cup \{\infty \}$, defined for $X=[x,y]$ by $f([x,y])=x∕y$ if $y\ne 0$, $f([x,0])=\infty$ otherwise, is well defined, and this is a bijection, whose inverse $f^{-1}=g$ is defined by $g(x)=[x,1],g(\infty )=[1,0]$.

By representing the projective point by its coordinate $z\in F\cup \{\infty \}$, we define for $M=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$, $M\cdot z=f(M\cdot f^{-1}(z))$. Explicitly, for $z\in F\setminus \{-d∕c\}$

$$M\cdot z=f\left(M\cdot [z,1]\right)=f([\mathit{az}+b,\mathit{cz}+d])=\frac{\mathit{az}+b}{\mathit{cz}+d}$$

and also

$$M\cdot (-d∕c)=\infty ,M\cdot \infty =a∕c$$

The group $\mathit{GL}(2,F)$ acts on $\hat{F}$: for all $z\in \hat{F}$, and all $M,N\in \mathit{GL}(2,F)$, $I.z=z$ and

$$\begin{array}{llll}\hfill M\cdot (N\cdot z)& =f(M\cdot f^{-1}(f(N\cdot f^{-1}(z)))& \hfill & \\ \hfill & =f(M\cdot (N\cdot f^{-1}(z)))=f(\mathit{MN}\cdot f^{-1}(z))& \hfill & \\ \hfill & =(\mathit{MN})\cdot z& \hfill & \end{array}$$

We resume this in the following proposition:

Proposition. The action defined for every $M=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\in \mathit{GL}(2,F)$and for every $z\in \hat{F}=F\cup \{\infty \}$by

$$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\cdot z=\frac{\mathit{az}+b}{\mathit{cz}+d}(z\in F\setminus \{-d∕c\})$$

$$M\cdot (-d∕c)=\infty ,M\cdot \infty =a∕c(\mathrm{if}c\ne 0),$$

$$M\cdot \infty =\infty (\mathrm{if}c=0),$$

is a (left) action of the group $\mathit{GL}(2,F)$on $F\cup \{\infty \}$: for all $z\in \hat{F}$, and for all $M,N\in \mathit{GL}(2,F)$,

(i)

$I\cdot z=z$

(ii)

$M\cdot (N\cdot z)=(\mathit{MN})\cdot z$

(b)

If $\lambda \in F^{\text{∗}}$, and $z\in F$, $(\mathit{\lambda I})\cdot z=\frac{\mathit{\lambda z}+0}{0.z+\lambda }=z$, so $(\mathit{\lambda I})\cdot \infty =\infty$. The elements of $C=F^{\text{∗}}{I}_2$ act trivially on $\hat{F}$. The quotient group $\mathit{PGL}(2,F)=\mathit{GL}(2,F)∕C$, where $C=F^{\text{∗}}{I}_2=\{\mathit{\lambda I},\lambda \in F^{\text{∗}}\}$, acts on $\hat{F}$.

Indeed the action is well defined: two elements $M,N$ of a same class modulo $C$ satisty $M=\mathit{\lambda N},\lambda \in F^{\text{∗}}$, thus $M\cdot z=(\mathit{\lambda N})\cdot z=N\cdot ((\lambda I_2)\cdot z)=N\cdot z$. We can so define the action by $[M]\cdot z=M\cdot z$, where $[M]$ is the class of $M$ in $\mathrm{PGL}(2,F)$. Then the relations (i)(ii) are always true

(i)

$[I]\cdot z=z$

(ii)

$[M]\cdot ([N]\cdot z)=([M][N])\cdot z$