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Corollary. There is a canonical bijection 0(MG;X ) ! 1 Proof. 1 (iv). 10. The line bundles on the moduli stack of G-bundles Let G be a connected simple complex group, X be a connected smooth projective complex curve. 1. 1. The line bundles on the innite Grassmannian. 2. A natural line bundle. 3 c) of LGe and its restriction to L+ Ge. 5 we have a canonical character G L+ Ge p! 1 +G e ! 2 b) : L[ G m; m hence a line bundle L 1 on the homogeneous space QGe . 1. A line in the innite Grassmannian.
Such that and induce the same map in cohomology. Proof. Choose a representative Ke of the cohomology of F and remark that induces an isomorphism e in the derived category Dcb(S ) Rp F ! Rp (F _ ) ! K e [ 1] Ke ! which is still symmetric (this follows from the symmetry of and standard properties of Grothendieck-Serre duality). 5 This is compatible with the usual signs: the dual of K is supported in degrees 1 and 0; when translated to the right by 1, the dierential acquires a 1 sign.
The section E is non trivial; if there is t0 2 S such that H 0 (X; Et0 E ) 6= 0 then E 6= ;. 2. The pfaan line bundle. Suppose char(k) 6= 2 in this subsection. X . We will view as F _ such that = _ , where F _ = Hom (F; q ! ). an isomorphism F ! 1. Lemma. If K is a representative of the cohomology of F , then K [ 1] is a representative of the cohomology of F _ . Here5 K [ 1] denotes the complex supported in degrees 0 and 1 0 ! K 1 ! K 0 ! 0: Proof. In the derived category Dc (S ), we have Rp (RHom (F; q !