# Finite symmetric tensor categories with the Chevalley property in characteristic $2$

## TL;DR

This paper classifies finite symmetric tensor categories with the Chevalley property in characteristic 2, showing they are equivalent to representation categories of certain finite group schemes and computing related cohomology groups.

## Contribution

It extends Deligne's theorem to characteristic 2, establishing a classification of symmetric tensor categories with the Chevalley property and computing associated cohomology groups.

## Key findings

- Categories admit symmetric fiber functors to a specific tensor category
- Existence of a unique finite group scheme representing the category
- Computed cohomology groups for group algebras and function algebras

## Abstract

We prove an analog of Deligne's theorem for finite symmetric tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$ admits a symmetric fiber functor to the symmetric tensor category $\mathcal{D}$ of representations of the triangular Hopf algebra $(k[\dd]/(\dd^2),1\ot 1 + \dd\ot \dd)$. Equivalently, we prove that there exists a unique finite group scheme $G$ in $\mathcal{D}$ such that $\mathcal{C}$ is symmetric tensor equivalent to $\Rep_{\mathcal{D}}(G)$. Finally, we compute the group $H^2_{\rm inv}(A,K)$ of equivalence classes of twists for the group algebra $K[A]$ of a finite abelian $p$-group $A$ over an arbitrary field $K$ of characteristic $p>0$, and the Sweedler cohomology groups $H^i_{\rm{Sw}}(\mathcal{O}(A),K)$, $i\ge 1$, of the function algebra $\mathcal{O}(A)$ of $A$.