Geodesic
The Complexity of a Flat Groupoid
Documenta mathematica, Tome 23 (2018), pp. 1157-1196
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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⇉X with finite stabilizer to be the length of the canonical sequence of the finite map R→X×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→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.
DOI : 10.4171/dm/644
Classification : 14A20, 14L15, 14L30
Mots-clés : groupoids, group schemes, quotients, algebraic spaces, effective epimorphisms, descent 
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     author = {Matthieu Romagny and David Rydh and Gabriel Zalamansky},
     title = {The {Complexity} of a {Flat} {Groupoid}},
     journal = {Documenta mathematica},
     pages = {1157--1196},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/644},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/644/}
}
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Matthieu Romagny; David Rydh; Gabriel Zalamansky. The Complexity of a Flat Groupoid. Documenta mathematica, Tome 23 (2018), pp. 1157-1196. doi: 10.4171/dm/644

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