# Cohomology of unipotent group schemes

## TL;DR

This paper explores the cohomology of unipotent group schemes, linking universal classes to explicit cohomology classes of Frobenius kernels and analyzing the cohomology of specific unipotent groups like the Heisenberg group.

## Contribution

It establishes explicit cohomology classes for Frobenius kernels of linear algebraic groups and investigates the relationship between inverse limits and rational cohomology.

## Key findings

- Universal classes determine explicit cohomology classes of Frobenius kernels.
- The relationship between inverse limit cohomology and rational cohomology is clarified.
- Detailed analysis of the cohomology of Frobenius kernels of the Heisenberg group.

## Abstract

We verify that universal classes in the cohomology of $GL_N$ determine explicit cohomology classes of Frobenius kernels $G_{(r)}$ of various linear algebraic groups $G$ . We consider the relationship of $\varprojlim_r H^*(U_{(r)},k)$ to the rational cohomology $H^*(U,k)$ of many unipotent algebraic groups $U$. The second half of this paper investigates in detail the cohomology of Frobenius kernels $(U_3)_{(r)}$ of the Heisenberg group $U_3 \subset GL_3$.