A universal étale lift of a proper local embedding

Abstract

To any finite local embedding of Deligne--Mumford stacks $g:Y\to X$ we associate an étale, universally closed morphism $F_{Y/X}\to X$ such that for the complement $Y_X^2$ of the image of the diagonal $Y\to Y{\times }_{X}Y$, the stack $F_{Y_X^2/Y}$ admits a canonical closed embedding in $F_{Y/X}$, and $F_{Y/X}{\times }_{X}Y$ is a disjoint union of copies of $F_{Y_X^2/Y}$. The stack $F_{Y/X}$ has a natural functorial presentation, and the morphism $F_{Y/X}\to X$ commutes with base-change. The image of $Y_X^2$ in $Y$ is the locus of points where the morphism $Y\to g(Y)$ is not smooth. Thus for many practical purposes, the morphism $g$ can be replaced in a canonical way by copies of the closed embedding $F_{Y_X^2/Y}\to F_{Y/X}$.