Geodesic
The Complexity of a Flat Groupoid
Documenta mathematica, Tome 23 (2018), pp. 1157-1196.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $R\rightarrow X\times_{X/R} X$, 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 $X\rightarrow X/R$ 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.
Classification : 14A20, 14L15, 14L30
Keywords: groupoids, group schemes, quotients, algebraic spaces, effective epimorphisms, descent 
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     author = {Romagny, Matthieu and Rydh, David and Zalamansky, Gabriel},
     title = {The {Complexity} of a {Flat} {Groupoid}},
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Romagny, Matthieu; Rydh, David; Zalamansky, Gabriel. The Complexity of a Flat Groupoid. Documenta mathematica, Tome 23 (2018), pp. 1157-1196. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a28/