40.1 Introduction

This chapter is devoted to advanced topics on groupoid schemes. Even though the results are stated in terms of groupoid schemes, the reader should keep in mind the $2$-cartesian diagram

40.1.0.1
$$\label{more-groupoids-equation-quotient-stack} \vcenter { \xymatrix{ R \ar[r] \ar[d] & U \ar[d] \\ U \ar[r] & [U/R] } }$$

where $[U/R]$ is the quotient stack, see Groupoids in Spaces, Remark 76.19.4. Many of the results are motivated by thinking about this diagram. See for example the beautiful paper [K-M] by Keel and Mori.

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