The Complexity of a Flat Groupoid

Abstract

Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid $R\rightrightarrows X$ with finite stabilizer to be the length of the canonical sequence of the finite map $R\to X{\times }_{X/R}X$, where $X/R$ is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient $X\to X/R$ and a theorem on the existence of the quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of quotients by groupoids.