The genericity theorem for the essential dimension of tame stacks

Let X be a regular tame stack. If X is locally of finite type over a field, we prove that the essential dimension of X is equal to its generic essential dimension; this generalizes a previous result of P. Brosnan, Z. Reichstein and the second author. Now suppose that X is locally of finite type over a 1-dimensional noetherian local domain R with fraction field K and residue field k. We prove that edk Xk ≤ edK XK if X → Spec R is smooth and edk Xk ≤ edK XK + 1 in general.