Ree群的上同调与例外同构核
On the cohomology of the Ree groups and kernels of exceptional isogenies
摘要：
设$G$是特征为$p>0$的代数闭域$k$上的单连通代数群.设$σ: G \rightarrow G$是$G$的满射自同态，使得不动点集$G(σ)$是Suzuki或Ree群.然后，让$G_σ$表示$σ.$的模式理论核使用Jantzen和Bendel-Nakano-Pillen的方法，我们计算Frobenius核的系数在诱导模$H^{1}(G_σ, H^{0}(λ))$中的$1$-上同调，以及Frobenius核的系数在简单模中的$1$-上同调，$H^{1}(G_σ, L(λ))$用于Suzuki和Ree群。此外，我们改进了已知的界限识别扩展的$F_4$型Ree群的代数群。
英文摘要：
Let $G$ be a simple, simply connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$. Let $σ: G \rightarrow G$ 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, $H^{1}(G_σ, H^{0}(λ))$, and the $1$-cohomology for the Frobenius kernels with coefficients in the simple modules, $H^{1}(G_σ, L(λ))$ for the Suzuki and Ree groups. Moreover, we improve the known bounds for identifying extensions for the Ree groups of type $F_4$ with the ones for the algebraic group.
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