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We prove an analog of Delignes 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 $(k[dd]/(dd^2),1ot 1 + ddot dd)$. 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 $H^i_{rm{Sw}}(mathcal{O}(A),K)$, $ige 1$, of the function algebra $mathcal{O}(A)$ of $A$.
We prove that every finite symmetric integral tensor category $mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber functor to $text{sVec}$. This proves Ostriks conjecture cit
This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of the curre
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_
We study exact sequences of finite tensor categories of the form $Rep G to C to D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $Gamma$ and mutual actions by permutations $rhd: Gamma times G to G$ and $
We describe graded commutative Gorenstein algebras ${mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)cong{mathcal E}_{n}(p)$, where $mathsf{Ver}_{p^{n+1}}$ are the new symmet