On cohomology in symmetric tensor categories in prime characteristic
David Benson, Pavel Etingof

TL;DR
This paper explores the structure of cohomology algebras in symmetric tensor categories over fields of prime characteristic, proposing a conjecture relating Ext groups to explicitly described Gorenstein algebras, supported by computational and theoretical evidence.
Contribution
It introduces a conjecture linking Ext groups in new symmetric tensor categories to graded Gorenstein algebras and provides partial theoretical and computational evidence for this conjecture.
Findings
Identifies a Koszul construction for parameters in Ext algebras.
Provides computational evidence for small n values.
Shows the Ext algebra has the same Krull dimension as the conjectured algebra.
Abstract
We describe graded commutative Gorenstein algebras over a field of characteristic , and we conjecture that , where are the new symmetric tensor categories recently constructed in \cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,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 . We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in with a homogeneous system of parameters in . These parameters have…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Homotopy and Cohomology in Algebraic Topology
