On cohomology in symmetric tensor categories in prime characteristic

We describe graded commutative Gorenstein algebras $\mathcal{E}_n(p)$ over a field of prime characteristic $p$, and we conjecture that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}} (\mathbb{1},\mathbb{1}) \cong \mathcal{E}_{n}(p)$, where $\mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently constructed by the current authors, with Ostrik, and also by Coulembier. We investigate the combinatorics of these algebras, and the relationship with Minc’s partition function, as well as possible actions of the Steenrod operations on them.

Evidence for the conjecture includes a large number of computations for small values of $n$. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in $\mathcal{E}_n(p)$ with a homogeneous system of parameters in $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}} (\mathbb{1},\mathbb{1})$. These parameters have degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1\leqslant i \leqslant n$. This at least shows that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(\mathbb{1},\mathbb{1})$ is a finitely generated graded commutative algebra with the same Krull dimension as $\mathcal{E}_n(p)$. For $p=2$ we also show that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{2^{n+1}}}(\mathbb{1},\mathbb{1})$ has the expected rank $2^{n(n-1)/2}$ as a module over the subalgebra of parameters.

Keywords

symmetric tensor category, cohomology ring, Gorenstein algebra, Minc’s partition function, Steenrod operation

2010 Mathematics Subject Classification

13H10, 16E30, 55S10

Full Text (PDF format)

Received 24 September 2020

Received revised 19 August 2021

Accepted 19 August 2021

Published 10 August 2022