Publication audit: semisimplicity under odd symmetry

Date: 6 September 2026

Mode: PUBLISH

Decision: pass – the candidate proof survives after one necessary proof repair and one categorical clarification

External review: none

Priority status: a bounded search found no earlier resolution; no priority claim is made

1. Audited theorem

Let $k$ be algebraically closed of characteristic zero, let $\mathscr C$ be a supertannakian category over $k$, and let $A\in\mathscr C$ be a finite associative algebra object. Then the six conditions in Coulembier–Sherman, Theorem 3.17, are equivalent:

1. absolute flatness;
2. internal $\operatorname{Ext}^1$-vanishing for every finitely presented left module;
3. finite product decomposition into simple algebra objects;
4. internal semisimplicity;
5. absence of nonzero nilpotent two-sided ideal objects; and
6. rigidity of the category of finitely presented bimodules.

The two added arrows are

$$(5)\Longrightarrow(1),\qquad (2)\Longrightarrow(5).$$

This answers Coulembier–Sherman Question 5.16 affirmatively, subject to the explicit no-external-review qualification.

2. The proof repair found by the audit

The candidate packet incorrectly passed directly from an ideal $J\triangleleft I$ of a minimal ambient ideal $I$ to minimality of $I$. An ideal of $I$ need not be an ideal of $A$. The repair is exact.

If $J\ne0$ is invariant under every derivation of $I$, then it is stable under $\operatorname{Lie}(H)$, hence under the connected supergroup $H$. Its ambient saturation

$$AJA$$

is a nonzero $H$-stable two-sided ideal of $A$ contained in $I$. Minimality gives $AJA=I$. Since $I^2=I$ and $IA=AI=I$,

$$I=I^3=I(AJA)I=IJI\subseteq J.$$

Thus $J=I$, so $I$ is differentiably simple in Kac’s sense. This is the only genuinely missing proof step found by the audit.

3. The categorical clarification

The candidate’s separability paragraph was correct in direction but stated too loosely. Internal flatness over the supercenter does not need to imply ordinary flatness against arbitrary nonequivariant modules.

Let $U$ be the exact faithful forgetful functor from equivariant modules to super vector spaces. A nonequivariant separability section for $C\to B$ makes

$$U(M\mathbin{\widehat\otimes}_B-)$$

a natural retract of

$$U(M\mathbin{\widehat\otimes}_C-)$$

on the category of equivariant $B$-modules. Absolute flatness of $C$ makes the second functor exact on every equivariant short exact sequence. Its retract is exact, and $U$ reflects exactness. This proves internal $B$-flatness for arbitrary ind-modules without claiming more.

4. Hostile-audit ledger

5. Corrected structural proof

Pass to $\operatorname{Rep}^fG$ using the same finite-generation reduction as Coulembier and Sherman, and let $H=G^\circ$. A nilpotent $H$-invariant ideal would produce a nilpotent $G$-invariant ideal by summing its finitely many component translates inside the ordinary Jacobson radical. Thus a minimal nonzero $H$-invariant ideal $I$ satisfies $I^2=I$.

The saturation argument in Section 2 makes $I$ differentiably simple. Kac’s Proposition 5.1.1 gives

$$I\cong S\widehat\otimes\Lambda(V).$$

Its identity is a central $H$-fixed idempotent of the ambient algebra, so induction decomposes $A$ into $H$-simple factors of this form. The component group permutes the factors.

For each factor $B$,

$$C=Z_s(B)=\Lambda(V)$$

is canonical, commutative, invariant, and simple as an algebra object. Coulembier and Sherman’s Corollary 6.6 makes $C$ absolutely flat. Wall’s matrix/queer classification makes the simple factor $S$ separable, and the faithful-forgetful retract transfers absolute flatness from $C$ to $B$, then to $A$.

For the other new arrow, if $R$ is a nilpotent invariant ideal and $M=A/R$, internal $\operatorname{Ext}^1$-vanishing lifts the identity of the underlying $M$ to an even $A$-linear section $s:M\to A$. With $e=s(1+R)$,

$$e^2=e,\qquad e\equiv1\pmod R,\qquad Re=0.$$

The idempotent $1-e$ lies in the nilpotent ideal $R$, hence vanishes; then $R=Re=0$.

6. Exact controls

The publication verifier checks, with exact rational arithmetic:

• 144 matrix-unit centrality identities for fixed-column separability elements across the small matrix superalgebras $M(p|q)$, $0\le p,q\le2$;
• the averaged matrix separability element independently;
• the queer separability element;
• the full supercenter of $Q(1)\widehat\otimes\Lambda(k^n)$ for $0\le n\le3$, with the expected Grassmann dimensions.

These computations catch sign and indexing mistakes. Completeness of the global theorem comes from the written structural proof, not from finite testing.

7. Sources and intellectual credit

• Kevin Coulembier and Alexander Sherman, Absolutely flat algebras in tensor categories: open question, established implication diagram, supertannakian reduction, and commutative absolute-flatness theorem.
• Victor G. Kac, Lie Superalgebras: differentiably-simple-superalgebra normal form.
• Richard E. Block, Determination of the Differentiably Simple Rings with a Minimal Ideal: classical method adapted by Kac.
• Akira Masuoka, Harish–Chandra pairs for algebraic affine supergroup schemes, and Alexander Sherman, Two geometric proofs…: connected supergroup integration framework.
• C. T. C. Wall, Graded Brauer groups: matrix/queer classification context.

The combination of these ingredients, the ambient saturation lemma, and the faithful-forgetful transfer is the FPRD Lab contribution. It has no external review.

8. Publication decision

The result answers a recognized open question, so it meets the Lab’s publication threshold. The public statement must retain all of the following boundaries:

• algebraically closed base field;
• characteristic zero;
• finite algebra object $A\in\mathscr C$;
• no external specialist review; and
• no asserted priority beyond a negative bounded search.