Proposition 4

The group $\mathbf{SL}(n, K)$ is dense in $\mathbf{SL}(n, A)$.

Let G be the closure of $\mathbf{SL}(n, K)$ in $\mathbf{SL}(n, A)$; as K is dense in A (Proposition 3), G contains all the matrices of the form $I + a . E_{ij}$ for $i \neq j$ and $a \in A$. For all $p \in P$ and all $\lambda \in \hat{K}_p$, let $\lambda(p)$ be the restricted adèle $x = (x_q)_{q \in P}$ such that $x_p = \lambda$ and $x_q = 0$ for $q \neq p$; the above shows that G contains the matrices $I + \lambda(p)E_{ij}$ for $i \neq j$. But we know that the matrices of the form $I + \lambda E_{ij}$ for $\lambda \in \hat{K}_p$ generate the group $\mathbf{SL}(n, \hat{K}_p)$ (Algebra, Chapter 111). For every matrix $U \in \mathbf{SL}(n, A)$ let $U_p$ denote the canonical image of $U$ in $\mathbf{SL}(n, \hat{\mathbf{K}}_p)$; it is therefore seen that, for all $p \in P$, $G$ contains the matrices $U \in \mathbf{SL}(n, A)$ such that $U_q = I$ for all $q \# p$. Since $G$ is a group, it also contains all the matrices $U \in \mathbf{SL}(n, A)$ such that $U_p = I$ except for a finite number of $p \in P$; now, the definition of the topology on $A$ shows immediately that the set of these matrices is dense in $\mathbf{SL}(n, A)$.