Rational cohomology of an algebra need not be detected by Frobenius kernels

TL;DR

This paper presents an example where the rational cohomology of an algebra in characteristic two cannot be detected by Frobenius kernels, challenging assumptions about cohomology detection methods.

Contribution

It provides a specific example of an algebra with an SL_2 action in characteristic two, showing nontrivial cohomology classes not detected by Frobenius kernels.

Findings

Existence of a polynomial algebra with unusual SL_2 action in characteristic two

Identification of a nontrivial class in H^1(SL_2,A)

Demonstration that Frobenius kernels do not detect all cohomology

Abstract

Friedrich Knop has given an example of a polynomial algebra A in characteristic two with an unusual SL_2 action. We exhibit a nontrivial class in H^1(SL_2,A), not detected by Frobenius kernels.