Faithful-forgetful separability transfer

Exact statement

Let $C\to B$ be finite superalgebras in $\operatorname{Rep}^fH$ . If $B\otimes_C B\to B$ has an even bimodule section after forgetting $H$ , and $C$ is internally absolutely flat, then $B$ is internally absolutely flat.

StatusProved and hostile-audited within FPRD Lab

External reviewNo external or specialist review is recorded; the proof has passed a hostile internal audit.

Context

The separability section need not respect the group action. It is used only after applying the exact faithful forgetful functor.

Hypotheses and scope

C is internally absolutely flat.
The underlying superalgebra extension is separable by an even B-bimodule section.

Proof or evidence

After forgetting equivariance, tensor product over B is a natural retract of tensor product over C on every equivariant short exact sequence. Retracts preserve exactness, and faithful forgetting reflects it.

Faithful-forgetful retract proof →

Verification notes

The audit removed the stronger and unnecessary assertion that internal C-flatness implies ordinary flatness against arbitrary nonequivariant modules.

Limitations

The conclusion concerns internal modules and exact sequences in the equivariant ind-category.
No equivariant separability section is asserted.

Open work

Seek tensor-category review of the arbitrary-ind-module scope.