On cohomology in symmetric tensor categories in prime characteristic (Q2107540)

This paper is about two classes of algebras, each indexed over a prime \(p\) and a non-negative integer \(n\): \begin{itemize} \item[1.] \(\mathcal{E}_n(p)\) is a graded Gorenstein algebra over a field of characteristic \(p\), and \item[2.] \(\operatorname{Ext}_{\mathsf{Ver}_{p^{n+1}}}(\mathbf{1},\mathbf{1})\) is the cohomology ring of the unit object \(\mathbf{1}\) in the symmetric tensor abelian category \(\mathsf{Ver}_{p^{n+1}}\), which were constructed in [\textit{K. Coulembier}, Compos. Math. 157, No. 7, 1584--1609 (2021; Zbl 1471.18020)] and [\textit{D. Benson} et al., Duke Math. J. 172, No. 1, 105--200 (2023; Zbl 07653252)]. \end{itemize} The authors conjecture, that these classes coincide, namely \(\mathcal{E}_n(p) \cong \operatorname{Ext}_{\mathsf{Ver}_{p^{n+1}}}(\mathbf{1},\mathbf{1})\). They give some evidence for this conjecture: \begin{itemize} \item It holds for \(n \leq 1\). \item For low degrees and small \(p\) and \(n\), the dimension and the algebra structure agree; this was checked using the computer algebra package \textsc{Magma}. \item The only possible Steenrod operations on \(\mathcal{E}_n(p)\) have the same properties as the Steenrod operations on \(\operatorname{Ext}_{\mathsf{Ver}_{p^{n+1}}}(\mathbf{1},\mathbf{1})\). \end{itemize}