Geodesic
The cohomology rings of moduli stacks of principal bundles over curves
Documenta mathematica, Tome 15 (2010), pp. 423-488.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We prove that the cohomology of the moduli stack of $G$-bundles on a smooth projective curve is freely generated by the Atiyah--Bott classes in arbitrary characteristic. The main technical tool needed is the construction of coarse moduli spaces for bundles with parabolic structure in arbitrary characteristic. Using these spaces we show that the cohomology of the moduli stack is pure and satisfies base-change for curves defined over a discrete valuation ring. Thereby we get an algebraic proof of the theorem of Atiyah and Bott and conversely this can be used to give a geometric proof of the fact that the Tamagawa number of a Chevalley group is the number of connected components of the moduli stack of principal bundles.
Classification : 14D23, 14F20, 14D20
Mots-clés : principal bundle, parabolic bundle, moduli space, stack, cohomology, Tamagawa number 
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     title = {The cohomology rings of moduli stacks of principal bundles over curves},
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     volume = {15},
     year = {2010},
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Heinloth, Jochen; Schmitt, Alexander H.W. The cohomology rings of moduli stacks of principal bundles over curves. Documenta mathematica, Tome 15 (2010), pp. 423-488. https://geodesic-test.mathdoc.fr/item/DOCMA_2010__15__a22/