Supertannakian semisimplicity equivalence

Exact statement

For a finite associative algebra object $A$ in a supertannakian category over an algebraically closed field of characteristic zero, absolute flatness, internal $\operatorname{Ext}^1$ -vanishing for finitely presented modules, product decomposition into simple algebra objects, internal semisimplicity, absence of nonzero nilpotent two-sided ideal objects, and rigidity of finitely presented bimodules are equivalent.

StatusProved and hostile-audited within FPRD Lab; no external review

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

Context

Coulembier and Sherman left the characteristic-zero supertannakian case open after proving the tannakian and positive-characteristic cases.

Hypotheses and scope

The base field is algebraically closed and has characteristic zero.
The category is supertannakian.
A is a finite unital associative algebra object; it need not be commutative.

Proof or evidence

A minimal connected-invariant ideal is differentiably simple by FPRD-ST-L01. Kac's normal form gives S tensor-hat Lambda(V); the Grassmann factor is the canonical simple commutative supercenter. Coulembier--Sherman make the center absolutely flat, and FPRD-ST-L02 plus explicit matrix/queer separability transfers flatness to the full algebra. A direct internal-Ext argument supplies the remaining arrow.

Full proof →

Verification notes

The six vulnerable bridges, exact source scope, and priority boundary are recorded in /evidence/supertannakian-semisimplicity/FPRD-supertannakian-publication-audit-2026-09-06.md. One proof gap was repaired; no fatal obstruction remained. Exact computations check the matrix/queer signs but are not the completeness proof.

Limitations

There is no external specialist review.
A bounded search found no earlier resolution but does not establish priority.
The theorem is not stated over non-algebraically-closed fields or for arbitrary ind-algebras.

Open work

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