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Linearizaton of group stack actions and the Picard group of the moduli of $$SL_r/\mu_s$$-bundles on a curve. (English) Zbl 0936.14024
Summary: In this paper, we consider morphisms of algebraic stacks $$\mathcal X \to \mathcal Y$$ which are torsors under a group stack $$\mathcal G$$. We show that line bundles on $$\mathcal Y$$ correspond exactly with $$\mathcal G$$-linearized line bundles on $$\mathcal X$$ (with a suitable definition of a $$\mathcal G$$-linearization). We use this fact to determine the precise structure of the Picard group of the moduli stack of $$G$$-bundles on an algebraic curve when $$G$$ is any group of type $$A_n$$.

##### MSC:
 14H60 Vector bundles on curves and their moduli 14C22 Picard groups 14H10 Families, moduli of curves (algebraic) 14L30 Group actions on varieties or schemes (quotients)
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##### References:
 [1] BEAUVILLE (A.) , LASZLO (Y.) , SORGER (C.) . - The Picard group of the moduli of G-bundles on a curve , preprint alg-geom/9608002, to appear in Compositio Math. arXiv | Zbl 0976.14024 · Zbl 0976.14024 [2] BREEN (L.) . - Bitorseurs et cohomologie non abélienne , in ‘The Grothendieck Festschrift I’, Progr. Math., Birkhäuser, t. 86, 1990 , p. 401-476. MR 92m:18019 | Zbl 0743.14034 · Zbl 0743.14034 [3] DELIGNE (P.) . - Théorie de Hodge III , Publ. Math. IHES, t. 44, 1974 , p. 5-78. Numdam | MR 58 #16653b | Zbl 0237.14003 · Zbl 0237.14003 [4] FALTINGS (G.) . - Stable G-bundles and Projective Connections , JAG, t. 2, 1993 , p. 507-568. MR 94i:14015 | Zbl 0790.14019 · Zbl 0790.14019 [5] GROTHENDIECK (A.) . - Géométrie formelle et géométrie algébrique , FGA, Sém. Bourbaki, t. 182, 1958 / 1959 , p. 1-25. Numdam | Zbl 0229.14005 · Zbl 0229.14005 [6] LAUMON (G.) , MORET-BAILLY (L.) . - Champs algébriques , preprint Université Paris-Sud, 1992 . · Zbl 0945.14005 [7] LASZLO (Y.) , SORGER (C.) . - The line bundles on the moduli of parabolic G-bundles over curves and their sections , Ann. École Normale Supérieure, 4e série, t. 30, 1997 , p. 499-525. Numdam | MR 98f:14007 | Zbl 0918.14004 · Zbl 0918.14004 [8] MAC LANE (S.) . - Categories for the working mathematician , GTM, Springer-Verlag, t. 5, 1971 . MR 50 #7275 | Zbl 0232.18001 · Zbl 0232.18001
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