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Hi,

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?

Thanks!

flag	asked Aug 4 2011 at 19:08 Kevin Lin 12.9k●2●33●115

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See theorem 4.6.2.1 on page 29 of above. – Donu Arapura Aug 4 2011 at 19:20

Moosbrugger · Accepted Answer · 2011-08-04 21:05:21Z UTC

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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: http://math.harvard.edu/theses/senior/wang/wang.pdf.

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

edited Aug 4 2011 at 21:05

answered Aug 4 2011 at 19:19

Moosbrugger
945●3●10

Thanks Moosbrugger, this is a really nice document. – Kevin Lin Aug 5 2011 at 4:30

Jason Starr · Answer 2 · 2011-08-04 20:32:57Z UTC

This is in LM-B. It is Théorème 4.6.2.1 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.

Reference for moduli stack of principal G-bundles?

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