Examples of Sweedler cohomology in positive characteristic

TL;DR

This paper calculates the Sweedler cohomology of function algebras on (Z/2)^r over fields of characteristic 2, revealing significant differences from characteristic zero and extending results to characteristic p.

Contribution

It provides explicit calculations of Sweedler cohomology in positive characteristic and introduces a variant of known results for lazy cohomology in this setting.

Findings

Sweedler cohomology differs markedly in characteristic 2 from characteristic 0.

Explicit calculations of cohomology groups for (Z/2)^r.

A new variant of lazy cohomology results in characteristic p.

Abstract

In this paper we provide a detailed calculation of the Sweedler cohomology of the algebra of functions on (Z/2)^r, in all degrees, over a field of characteristic 2. The result is strikingly different from the characteristic 0 analog. Then we show that there is a variant in characteristic p of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group.