A universal \'etale lift of a proper local embedding

TL;DR

This paper introduces a universal étale lift for proper local embeddings of Deligne--Mumford stacks, providing a canonical, functorial construction that simplifies the morphism by replacing it with a closed embedding.

Contribution

It constructs a universal étale, universally closed morphism associated to any finite local embedding of Deligne--Mumford stacks, enabling canonical replacements of such morphisms.

Findings

The stack $F_{Y/X}$ is étale and universally closed over $X$.

The construction is functorial and compatible with base change.

It provides a canonical way to replace local embeddings with closed embeddings.

Abstract

To any finite local embedding of Deligne--Mumford stacks $g: Y\to X$ we associate an \'etale, universally closed morphism $F_{Y/X}\to X$ such that for the complement $Y^2_X$ of the image of the diagonal $Y \to Y\times_XY$, the stack $F_{Y^2_X/Y}$ admits a canonical closed embedding in $F_{Y/X}$, and $F_{Y/X}\times_XY$ is a disjoint union of copies of $F_{Y^2_X/Y}$. 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 $Y^2_X$ 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 $F_{Y^2_X/Y}\to F_{Y/X}$.