Exercise 21, § 2

Let A be a discrete valuation ring (Commutative Algebra, Chap. VI, § 3, no. 6), m its maximal ideal, $\gamma$ a generator of m and K the field of fractions of A. Let G be the group $\mathbf{SL}(2, K)$, B the subgroup of G consisting of the matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $a, b, d \in A$ and $c \in m$ (with $ad - bc = 1$) and N the subgroup consisting of the matrices belonging to G and having only one non-zero entry in each row and column.

a) Show that $T = B \cap N$ is normal in $N$ and that $W = N/T$ is an infinite dihedral group generated by the classes $s$ and $s'$ of the matrices $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $\begin{pmatrix} 0 & \gamma \\ -\gamma^{-1} & 0 \end{pmatrix}$, respectively.

b) Show that $(G, B, N, S)$ (with $S = \{s, s'\}$) is a Tits system.

c) Let $H = \mathbf{SL}(2, A)$ be the subgroup of $G$ consisting of the matrices with coefficients in $A$. Show that $(H, B \cap H, N, \{s\})$ is a Tits system. Compare with Exerc. 18 e).

d) Let $\hat{A}$ be the completion of $A$ and let $\hat{G}, \hat{B}, \hat{N}, \hat{T}$ be the groups defined as above but replacing $A$ by $\hat{A}$. Show that the injection of $G$ into $\hat{G}$ defines an isomorphism from the building $I$ associated to $(G, B)$ (Exerc. 10) to the building $\hat{I}$ associated to $(\hat{G}, \hat{B})$. Let $(I, \mathscr{U})$ (resp. $(\hat{I}, \hat{\mathscr{U}})$) be the structured building associated to $(G, B, N)$ (resp. $(\hat{G}, \hat{B}, \hat{N})$) (Exerc. 12): show that $j(\mathscr{U}) \subset \hat{\mathscr{U}}$, but that $j(\mathscr{U}) \neq \hat{\mathscr{U}}$ if $A \neq \hat{A}$ (remark that the apartments $\hat{\mathscr{U}}$ (resp. $\mathscr{U}$) correspond bijectively to the conjugates of $T$ (resp. $\hat{T}$) by $G$ (resp. $\hat{G}$).