Second cohomology groups for algebraic groups and their Frobenius kernels

TL;DR

This paper computes second cohomology groups for algebraic groups and their Frobenius kernels over fields of characteristic 2, completing previous work for characteristics greater than or equal to 3.

Contribution

It provides explicit calculations of second cohomology groups for various Frobenius kernels and algebraic group schemes in characteristic 2, extending prior results.

Findings

Computed $ ext{H}^2(U_1,k)$, $ ext{H}^2(B_r, ext{λ})$, $ ext{H}^2(G_r,H^0( ext{λ}))$, and $ ext{H}^2(B, ext{λ})$ for characteristic 2.

Extended the cohomology computations to characteristic 2, completing the picture for $p eq 2$.

Provided explicit cohomology group descriptions for simple simply connected algebraic groups.

Abstract

Let $G$ be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic $p > 0$. Let $T$ be a maximal split torus in $G$, $B \supset T$ be a Borel subgroup of $G$ and $U$ its unipotent radical. Let $F: G \rightarrow G$ be the Frobenius morphism. For $r \geq 1$ define the Frobenius kernel, $G_r$, to be the kernel of $F$ iterated with itself $r$ times. Define $U_r$ (respectively $B_r$) to be the kernel of the Frobenius map restricted to $U$ (respectively $B$). Let $X(T)$ be the integral weight lattice and $X(T)_+$ be the dominant integral weights. The computations of particular importance are $\h^2(U_1,k)$, $\h^2(B_r,\la)$ for $\la \in X(T)$, $\h^2(G_r,H^0(\la))$ for $\la \in X(T)_+$, and $\h^2(B,\la)$ for $\la \in X(T)$. The above cohomology groups for the case when the field has characteristic 2 one computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen for $p \geq 3$ \cite{BNP2}.