The complexity of a flat groupoid

TL;DR

This paper introduces a new measure called the complexity of a flat groupoid with finite stabilizer, analyzing its implications for descent theory and quotient constructions in algebraic geometry.

Contribution

It defines the complexity of flat groupoids and proves descent and quotient theorems for those of complexity at most 1, advancing understanding of groupoid quotients.

Findings

Defined the complexity of flat groupoids with finite stabilizer.

Proved descent theorem along the quotient for groupoids of complexity ≤ 1.

Established quotient theorem for groupoids by a normal subgroupoid.

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 of 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.