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Appendix on Galois cohomology

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1968)

Abstract

For a connected reductive group G over F, let Ĝ denote the algebraic group over C, which is the connected component of the Langlands L-group of G. Consider the category of reductive groups over F, whose morphisms are the group homomorphism G → H, which are defined over F. The Borovoi fundamental group π(G) is a functor from the category of reductive groups over F to the category of finitely generated Abelian groups with ΓF -action. The Borovoi fundamental group is defined as follows. For tori T over F we have a canonical isomorphism \(\pi (T) = Hom_{\bar{F}-alg}(G_m, T )\), and \(\hat{T}=\pi(T)\otimes_Z C^\ast\) with induced ΓF -action. For semisimple G over F we have a canonical isomorphism π(G) = π1(G) (algebraic fundamental group). In general, let Gsc be the simply connected covering of the derived group Gder of G. Choose maximal F-tori TscTderT. Then π(G) = π(T)/image(π(Tsc)).

Keywords

  • Algebraic Group
  • Reductive Group
  • Group Homomorphism
  • Canonical Isomorphism
  • Maximal Compact Subgroup

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Rainer Weissauer .

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© 2009 Springer-Verlag Berlin Heidelberg

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Weissauer, R. (2009). Appendix on Galois cohomology. In: Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds. Lecture Notes in Mathematics(), vol 1968. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89306-6_11

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