Hi,

assume we have an algebraic stack $A$ over $Sch(\mathbb{Z})$ which is quasi-compact and with separated diagonal. Assume that I have a stack $B$ which is obtained by rigidifying $A$ by a subgroup of the inertia group. Assume moreover that $A\rightarrow B$ is faithfully-flat and smooth over $\mathbb{Z}[\frac{1}{d}]$ and that the diagonal of $B$ is finite. Question : is $B$ proper over $\mathbb{Z}$, or at least over $\mathbb{Z}[\frac{1}{d}]$ ?? If so is it trivial to prove it?