On the cohomology of the Ree groups and kernels of exceptional isogenies

2021/08/13 by Aura-Cristiana Radu, Radu, Aura-Cristiana
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2108.06291

Abstract

Let G be a simple, simply connected algebraic group over an algebraically closed field k of characteristic p>0. Let σ: G → G be a surjective endomorphism of G such that the fixed point set G(σ) is a Suzuki or Ree group. Then, let G_σ denote the scheme-theoretic kernel of σ. Using methods of Jantzen and Bendel-Nakano-Pillen, we compute the 1-cohomology for the Frobenius kernels with coefficients in the induced modules, H¹(G_σ, H⁰(λ)), and the 1-cohomology for the Frobenius kernels with coefficients in the simple modules, H¹(G_σ, L(λ)) for the Suzuki and Ree groups. Moreover, we improve the known bounds for identifying extensions for the Ree groups of type F₄ with the ones for the algebraic group.

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