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If $Y \to X$ is a finite etale map of schemes, then there exists a finite Galois morphism $Z \to X$ (i.e. it's a $Aut(Z/X)$-torsor) that factors as $Z \to Y \to X.$ The case when $X$ is normal is easy to prove: take the galois closure of function field of $Y$ and then take the normalization of $X$ in it. The general case is proved in Murre's book.

My question is, is this true for stacks? Namely if $Y \to X$ is a representable finite etale map of algebraic stacks, can it be refined to a representable Galois morphism $Z \to X?$

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    $\begingroup$ You must intend X and Y connected. The method of SGA1 (Exp. V, section 4, axioms G1--G6) is inspired by tensor product of extn field against itself over ground field and looking for max-degree factor fields instead of "normalizing in a Galois closure" as us mortals do with normal Y. There's good notion of "open and closed substack" and finite etale maps are open and closed, so any finite etale over X is finite disjoint union of connected clopen substacks and Aut groups are finite. Thus, SGA1 Galois formalism applies; see consequence (g) in that part of SGA1 for the connected Galois cover. QED $\endgroup$
    – BCnrd
    Commented Apr 17, 2010 at 2:49
  • $\begingroup$ To Brian: Thank you! This should be an answer rather than a comment. I read that part of SGA1 but still have some questions. The category of finite etale covers of a stack should be a 2-category, right? So all 'limits' and 'colimits' should be '2-limits' and so on, and I don't know if the same proof still work. Also for (G2), one has to be careful when taking the quotient, and I don't know how to define this quotient. Thank you again! $\endgroup$
    – shenghao
    Commented Apr 22, 2010 at 19:18
  • $\begingroup$ Shenghao: Discussing finite etale covers of a stack (or even finite morphisms, let alone affine morphisms, to a stack) is just a big game with descent theory for qcoh sheaves. There's no need to go off into the land of 2-limits and such general nonsense. Ditto for the quotient procedures. All that one is using about stacks if their encoding of systematic use of descent theory. There's no need for some hypergeneral theory of quotients; it is all very concrete stuff as far as descent theory goes. $\endgroup$
    – BCnrd
    Commented May 14, 2010 at 1:52

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