In the paper "Algebraization of formal moduli II" Artin proves the existence of "modification" and "dilatation". Are these opertaions valid on algebraic stacks??
I am attaching the particular page in the paper. The definitions are in the first page itself.
This was too long for a comment.
The main theorem of Artin's paper applies for Deligne-Mumford stacks of finite type over a field or over an excellent Dedekind domain $S$: for a finite type Deligne-Mumford stack $X'$ over $S$ (with separated and finite type diagonal), for a $1$-morphism $Y'\hookrightarrow X'$ that is representable by closed immersions, for a $1$-morphism $f_Y:Y'\to Y$ to a finite type, Deligne-Mumford $S$-stack $Y$ such that $f_Y$ is representable by proper morphisms, for the associated "formal completion" $\mathcal{X}'$ of $X'$ along $Y'$ (if memory serves, Iwanari was going to work out the theory of formal stacks), for a formal dilatation $\mathfrak{f}:\mathfrak{X}'\to \mathfrak{X}$ reducing to $f_Y$, there exists a unique dilatation of stacks $f:X'\to X$ fitting into the commutative diagram.
The point is, you can reduce this to Artin's theorem for algebraic stacks spaces using étale covers. Using facts about coarse moduli spaces and the local theory of Deligne-Mumford stacks, for every geometric point of $Y'$, there exists a $1$-morphism from an algebraic space $\widetilde{Y}\to Y$ that is representable by étale morphisms and whose image contains the geometric point, and there exists a $1$-morphism from an algebraic space $\widetilde{X}'\to X'$ that is representable by étale morphisms and that restricts over $Y'$ to the $2$-fiber product $\widetilde{Y}'=\widetilde{Y}\times_Y Y'$. So now you can pullback the formal dilatation to the formal neighborhood of $\widetilde{Y}'$ in $\widetilde{X}'$. Apply Artin's theorem to get $\widetilde{f}:\widetilde{X}'\to \widetilde{X}$. Finally use the uniqueness in Artin's theorem to prove the descent condition to construct a $1$-morphism $f:X'\to X$.
I do not see how to make this argument work for algebraic stacks, precisely because it is difficult to arrange a compatible "atlas" of the stack $Y$ and the stack $X'$. Perhaps an expert on stacks can say whether this is known (I am a little skeptical).