An arithmetic valuative criterion for proper maps of tame algebraic stacks

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing Qp-points to Fp-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a Qp-point will specialize to an Fpn-point for some n. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang–Nishimura theorem holds for tame stacks.