Semisimplicity under odd symmetry

The question

For a finite algebra object $A$ in a characteristic-zero supertannakian category, Coulembier and Sherman ask whether six familiar conditions remain equivalent: absolute flatness, internal $\operatorname{Ext}^1$ vanishing, product decomposition into simple algebra objects, internal semisimplicity, absence of nilpotent invariant ideals, and rigidity of finitely presented bimodules.

Their general theorem supplies most arrows. The decisive missing implication is

$$\text{no nonzero nilpotent invariant ideal} \Longrightarrow \text{absolute flatness}.$$

The structural proof

1. Connected reduction

Write $H=G^\circ$. If a nilpotent $H$-stable ideal existed, the sum of its finitely many component-group translates would lie in the ordinary Jacobson radical. It would therefore be a nonzero nilpotent $G$-stable ideal. Thus the hypothesis remains true after passing to the connected component.

2. Minimal ideals acquire identities

A minimal nonzero $H$-stable ideal $I$ has $I^2=I$. If a nonzero derivation-stable ideal $J\triangleleft I$ is not itself an ambient ideal, its saturation satisfies

$$AJA=I, \qquad I=I(AJA)I=IJI\subseteq J.$$

Thus $I$ is differentiably simple. This saturation is the proof repair found by the hostile audit. Kac’s Proposition 5.1.1, whose method adapts Block’s differentiably-simple-ring theorem, gives

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

with $S$ simple. This gives $I$ an identity. That identity is a central invariant idempotent in the ambient algebra, so induction splits the whole algebra into connected simple factors. The component group merely permutes those factors.

3. The Grassmann algebra is canonical

The apparent presentation factor is actually intrinsic:

$$Z_s\!\left(S\widehat\otimes\Lambda(V)\right) =\Lambda(V).$$

Any invariant ideal of this supercenter would generate an invariant ideal of the full simple factor. Hence the Grassmann supercenter is a simple commutative algebra object.

4. Use the solved commutative theorem

Coulembier–Sherman already prove that a simple commutative algebra in this setting is absolutely flat. Applying that theorem to $\Lambda(V)$ controls every module over the canonical center.

5. Lift through the simple factor

Finite simple associative superalgebras are separable. Therefore the extension

$$\Lambda(V)\longrightarrow S\widehat\otimes\Lambda(V)$$

is separable after forgetting symmetry. On equivariant module sequences, the faithful-forgetful image of tensor product over the full algebra is a natural retract of tensor product over the center. The latter is exact, so the retract is exact; faithful forgetting reflects the conclusion. This gives the missing implication for arbitrary ind-modules without asserting ordinary flatness against nonequivariant sequences.

The remaining arrow is elementary

If internal $\operatorname{Ext}^1$ vanishes and $R$ is a nilpotent invariant ideal, internal Hom applied to

$$0\to R\to A\to A/R\to0$$

lifts the identity of $A/R$ to an underlying $A$-linear section. Its value at one is an idempotent congruent to one modulo $R$ and annihilated by $R$. A nilpotent ideal contains no nonzero idempotent, forcing $R=0$.

Why this is not another presentation census

The initial attack settled the non-square-zero rank-one carrier, but it also found that restriction to an odd line can create a spurious radical. The global proof does not enumerate or compare those line restrictions. Kac’s theorem packages all odd directions simultaneously, and the supercenter identifies the Grassmann factor without choosing coordinates. The alternative presentation has become an exact structural reduction.

What the hostile audit established

1. Connected integration follows from the super Harish–Chandra pair and the connected even group; see Masuoka.
2. Kac’s proposition applies to the possibly nonunital minimal ideal once $I^2\ne0$ is proved.
3. The coefficientwise supercenter proof covers matrix and queer types.
4. Explicit fixed-column, averaged-matrix, and queer separability sections pass exact sign checks; the classification context is due to Wall.
5. The faithful-forgetful retract handles all ind-modules and does not require an equivariant separability section.
6. The source remains arXiv version 1, dated 13 August 2026. No competing resolution was found through 6 September 2026; this negative search is not a priority proof.

The candidate shortcut from a differential ideal to ambient minimality was rejected and repaired by the displayed$AJA$ argument. No fatal gap remained in that internal audit. The candidate remains in research and review as claim FPRD-ST-T01, with the no-external-review boundary retained.

Proofs, controls, and sources

• Main claim record
• Hostile internal audit
• Superseded candidate packet with corrections
• Rank-one odd-slice precursor and warning
• Publication audit, exact controls, and source ledger
• Primary-source ledger

The target and commutative theorem are due to Coulembier and Sherman. The normal form is due to Kac, following Block’s method. The optional exterior-module control uses Bergh–Erdmann rank varieties. The application of these ingredients to Question 5.16 has no external review. The bounded search did not locate an earlier resolution, but no priority claim is made.