# A Note on the Algebra of Operations for Hopf Cohomology at Odd Primes

## TL;DR

This paper investigates the algebraic structure of operations on the cohomology of cocommutative Hopf algebras over finite fields, revealing fundamental differences at odd primes compared to the prime 2 case.

## Contribution

It demonstrates that for odd primes, the algebra of operations cannot be structured as an object in the Singer category with compatible coproducts, unlike the prime 2 case.

## Key findings

- ${
m f B}(p)$ cannot be an object in the Singer category for odd primes
- Coproducts do not act coherently on generators at odd primes
- Structural differences between prime 2 and odd primes in Hopf cohomology operations

## Abstract

Let $p$ be any prime, and let ${\mathcal B}(p)$ be the algebra of operations on the cohomology ring of any cocommutative $\mathbb{F}_p$-Hopf algebra. In this paper we show that when $p$ is odd (and unlike the $p=2$ case), ${\mathcal B}(p)$ cannot become an object in the Singer category of $\mathbb{F}_p$-algebras with coproducts, if we require that coproducts act on the generators of ${\mathcal B}(p)$ coherently with their nature of cohomology operations