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I'm looking for a reference for the fact that the moduli stack $M_{GL_r,X}$ of $GL_r$-bundles over $X$ is an algebraic (Artin) stack. I'm only interested in the case where $X$ is a curve (for now).

I think this is supposed to be in Laumon-Moret--Bailly's "Champs Algebriques", but my French is not so great and I have been unable to find it in there. If it is actually in there, can you help a non-Francophone out?


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See theorem on page 29 of above. – Donu Arapura Aug 4 '11 at 19:20
up vote 3 down vote accepted

I actually don't think$^{\dagger}$ that this example is in Laumon/Moret-Bailey, but Jonathan Wang's senior thesis is a detailed write up in the style of LMB (and in English!) of this fact:

$^{\dagger}$ Edit: I stand corrected!

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Thanks Moosbrugger, this is a really nice document. – Kevin H. Lin Aug 5 '11 at 4:30

This is in LM-B. It is Théorème on p. 29. A generalization is proved in Max Lieblich's article.

MR2233719 (2008c:14022) Lieblich, Max(1-PRIN) Remarks on the stack of coherent algebras. Int. Math. Res. Not. 2006, Art. ID 75273, 12 pp. 14D20 (14A20)

Wang's senior thesis is also a well-written source.

Edit: The reference for Laumon and Moret-Bailly was already posted by Donu. Sorry for missing that.

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I see that Donu already posted the reference! The generalization in Lieblich's article is Theorem 2.1.1 (in English). – Jason Starr Aug 4 '11 at 20:34
No problem. Also, I think Behrend's thesis (from his webpage) does some of this. – Donu Arapura Aug 4 '11 at 21:15

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