Semisimple algebras as products of simple algebras and simple algebras as absolutely flat

Determine whether semisimple algebras in the ind-completion of an arbitrary tensor category are necessarily products of simple algebras, and whether artinian simple algebras in such ind-completions are necessarily absolutely flat.

Background

The paper studies whether the classical equivalence among semisimplicity, decomposition into products of simple algebras, and absolute flatness persists for artinian algebras in ind-completions of tensor categories. General structural results establish several implications, but do not settle the two reverse implications in complete generality.

The unresolved issues are particularly important for commutative algebras in symmetric tensor categories, where the authors subsequently prove affirmative results under well-fiberedness and provide counterexamples outside moderate growth.

References

Nonetheless some results demonstrate that in complete generality the answer deviates from the classical case. More concretely, in Section~\ref{sec:genstuff} we show that some implications survive, see Theorems~\ref{prop:consolidate} and~\ref{prop:consolidate2}, and that the main open questions are whether semisimple algebras need to be products of simple algebras and whether artinian simple algebras need to be absolutely flat.

— Absolutely flat algebras in tensor categories  (2608.13137 - Coulembier et al., 13 Aug 2026) in Section 1, Introduction

Therefore, we do not know whether $R$ is an example of a non-artinian simple commutative algebra, or an example of a non-absolutely flat artinian simple algebra.

— Absolutely flat algebras in tensor categories  (2608.13137 - Coulembier et al., 13 Aug 2026) in Remark following Proposition \ref{lem:counter}, Section 3.3, “Superexponential growth”

Theorem~\ref{thm noncomm case} covers all well-fibered categories apart from one class, which we leave as an open question. \begin{question}\label{question:super} If $C$ is a supertannakian category in characteristic 0 and $A\inC$, are the conditions of Theorem \ref{prop:consolidate} all equivalent? \end{question}

— Absolutely flat algebras in tensor categories  (2608.13137 - Coulembier et al., 13 Aug 2026) in Question \ref{question:super}, Section 5.4, “The noncommutative case for finite algebras”