A $K$-affine structure on $B^{sm}$ compatible with the given ${\bf Z}$-affine structure is a sheaf $Aff_{K}$ of abelian groups on $B^{sm}$, an exact sequence of sheaves

$$0\to K^{\times}\to Aff_{K}\to(T^{\ast})^{{\bf Z}}\to 0\,\,,$$

together with a homomorphism $\Phi$ of this exact sequence to the exact sequence of sheaves of abelian groups

$$0\to{{\bf R}}\to Aff_{{\bf Z}}\to(T^{\ast})^{{\bf Z}}\to 0\,\,,$$

such that $\Phi=id$ on $(T^{\ast})^{{\bf Z}}$ and $\Phi=val$ on $K^{\times}$.