Topologies de Grothendieck, descente, quotients

TL;DR

This paper discusses the existence of quotients of schemes under group actions, exploring Grothendieck topologies, descent theory, and specific quotient cases with improved proofs using algebraic spaces.

Contribution

It provides new proofs and insights into the existence of scheme quotients, including specific cases and the use of algebraic spaces for simplification.

Findings

Established existence theorems for scheme quotients

Improved proofs for key results in SGA 3

Analyzed quotients of algebraic groups and affine schemes

Abstract

In this note, we present a few existence theorems for the quotient of a scheme by the action of a group. The first two sections are devoted to Grothendieck topologies and descent theory. The third one is dealing with quotients: we first give direct and (almost) complete proofs for the main existence results of SGA 3, expos\'e V. Then we discuss some specific situations: the quotient of an algebraic group over a field by a subgroup, the quotient of a group by the normalizer of a smooth subgroup, and quotients of affine schemes by free actions of diagonalizable groups. From place to place, the original proofs have been slightly improved (e.g. with the use of algebraic spaces). This note grew out of lectures given by the author in the CIRM (Luminy) during the Summer School "Sch\'emas en groupes" in 2011.