HUNT packet: semisimplicity in supertannakian categories

Date: 6 September 2026
Mode: HUNT
Selected target: the unresolved characteristic-zero supertannakian case of Coulembier–Sherman’s equivalence theorem for finite algebra objects
Next mode: ATTACK
Status: selected research direction; no theorem or counterexample is claimed

Exact question

Let $\mathscr C$ be a supertannakian category over an algebraically closed field of characteristic zero, and let $A\in\mathscr C$ be a finite, not necessarily commutative, algebra object. Are the following conditions equivalent?

1. $A$ is absolutely flat: $\operatorname{Tor}_1^A\equiv0$.
2. $\underline{\operatorname{Ext}}_A^1(M,-)=0$ for every finitely presented left $A$-module $M$.
3. $A$ is a finite product of simple algebra objects.
4. $A$ is semisimple, in the internal Jacobson-radical sense.
5. $A$ has no nonzero nilpotent two-sided ideal object.
6. The category of finitely presented $A$-bimodules is rigid.

This is Question 5.16 of Kevin Coulembier and Alexander Sherman, Absolutely flat algebras in tensor categories, submitted 13 August 2026. Their Theorem 3.17 supplies several implications, so the decisive missing directions can be organized as

$$\text{internal semisimplicity}\Longrightarrow\text{absolute flatness} \quad\text{and}\quad \operatorname{Ext}^1\text{-vanishing}\Longrightarrow \text{internal semisimplicity}.$$

The commutative case is already covered by their broader theorem. The target is the finite noncommutative case.

Known frontier

Coulembier–Sherman prove the equivalence for:

• well-fibered categories in positive characteristic;
• tannakian categories in characteristic zero; and
• characteristic-zero supertannakian representation categories $\operatorname{Rep}^f G$ when $(\operatorname{Lie}G)_1$ is an ideal.

Their proof in the last situation restricts to a normal finite subgroup scheme $N$ with $G=NG_{\mathrm{red}}$. It transfers radicals, simple modules, and Ext groups to a finite tensor category, where the equivalence is known. The finite-category theorem is due to Coulembier, Stroiński, and Zorman, Simple algebras and exact module categories.

The gap is therefore not “all super representation theory.” It is the failure of that finite-normal-subgroup restriction when odd tangent directions have a nonzero square:

$$[\mathfrak g_1,\mathfrak g_1]\ne0.$$

Because parity forces $[\mathfrak g_1,\mathfrak g_1]\subseteq\mathfrak g_0$, the condition that $\mathfrak g_1$ itself be an ideal is precisely the square-zero boundary $[\mathfrak g_1,\mathfrak g_1]=0$. Question 5.16 asks what happens beyond that boundary.

Why this target passes the Lab’s selection filter

The target has one intrinsic object and one missing transfer mechanism. It does not ask us to enumerate presentations of an algebra or classify a growing list of supergroups.

The proposed bottleneck is rank-one odd-square detection. For an odd tangent vector $q$, set

$$z=q^2=\tfrac12[q,q].$$

The super Jacobi identity gives $[q,z]=0$. Thus every odd direction carries a canonical rank-one Lie subsuperalgebra

$$\mathfrak h_q=\langle z\rangle_{\bar0}\oplus \langle q\rangle_{\bar1},\qquad q^2=z,\quad [q,z]=0.$$

The first attack asks whether failure of the desired equivalence is detected after restriction to one such $\mathfrak h_q$, or to the corresponding rank-one distribution algebra/closed subsupergroup when it integrates. A positive detection theorem would replace a global supergroup by one canonical odd direction. A counterexample already in the rank-one category would answer Question 5.16 negatively. Either outcome is theorem-scale.

This is analogous to the successful Origami calibration in the relevant sense: a large ambient description is tested through one intrinsic local carrier, and the translation has a clean success or failure condition.

Calibration: the obvious toy model is not the open case

The exterior algebra $\Lambda(\theta)$ with odd translation $D(\theta)=1$ is tempting: its ordinary radical $(\theta)$ is not stable under the odd action. But $D^2=0$. It lives on the square-zero side already handled by the finite/odd-ideal machinery, so it cannot test the missing phenomenon.

The smallest relevant local relation must retain

$$q^2=z\ne0.$$

This calibration is useful because it prevents an attack from mistaking a familiar non-stable ordinary radical for evidence about Question 5.16. The new information is carried by the even infinitesimal motion generated by the square of the odd one.

The rank-one algebra has a second canonical split, through the Jordan type of the even operator $z$:

• a toral/local-semisimple part, where $z$ supplies weights and $q$ preserves them because $[q,z]=0$;
• a unipotent part, where the $z$-adic filtration is finite on every finite-dimensional rational representation.

This is a structural dichotomy, not a list of groups. The attack must formulate it functorially for algebra objects and their relative tensor products.

The first ATTACK

Gate A: settle the rank-one category

Work with the minimal Harish–Chandra/distribution pair generated by $q$ and $z=q^2$. Prove one of the following:

1. every internally semisimple finite algebra object is absolutely flat; or
2. exhibit a finite algebra object $A$ with $\mathcal J_{\mathscr C}(A)=0$ but $\operatorname{Tor}_1^A(M,N)\ne0$ for explicit equivariant modules.

For a positive proof, the concrete route is to filter by the even operator $z$, compare the associated graded category with the square-zero odd case, and identify the exact obstruction to lifting rigidity or flatness through one $z$-extension. The attack should produce a short spectral-sequence or extension lemma, not a catalogue of algebra structures.

Gate B: prove or kill rank-one detection

Formulate the candidate detection statement:

If the equivalence fails for a finite algebra object in $\operatorname{Rep}^fG$, then some odd direction $q\in\mathfrak g_1$ detects the failure after the appropriate rank-one restriction.

The exact detector may be nonzero relative Tor, failure of a bimodule to be rigid, or a support class attached to a simple $A$-module. The Duflo–Serganova functor is an important precedent for reducing super-representation questions along square-zero odd directions; Alexander Sherman’s work on its symmetries influences this approach. It is not a formal dependency here because the present directions may satisfy $q^2\ne0$.

The distribution-algebra interpretation and integration obligations should be checked against Benson–Pevtsova, Group schemes and their Lie algebras over a symmetric tensor category.

Decisive stopping rule

Stop this route if there is an explicit failure of rank-one detection whose support vanishes on every $\mathfrak h_q$, and no replacement by a bounded family of intrinsic odd subspaces is available. Do not respond by enumerating higher-rank supergroups or low-dimensional algebra laws.

What would count as success

• A counterexample in the rank-one category would settle Question 5.16 negatively.
• A rank-one theorem plus a global detection theorem would settle it positively.
• A proof for the rank-one non-square-zero category alone would be a useful structural advance, but not the full answer.
• A finite-dimensional census, a list of examples with no detection theorem, or a proof only after forgetting equivariance would not count.

Candidates considered but not selected

The quantum-counting and min-plus problems remain worthwhile future candidates. They were not rejected as unimportant; they were rejected because this hunt found a smaller and more falsifiable structural gate.

Source and priority boundary

The exact open question, the known implications, and the solved cases belong to Coulembier–Sherman. The finite tensor-category theorem belongs to Coulembier–Stroiński–Zorman. Rank-one odd reduction is a standard organizing idea in super representation theory, with Duflo–Serganova theory as a major precedent. The specific proposed odd-square detection lemma and the calibration above are a research plan, not a novelty claim. No literature search can establish priority for a theorem that has not yet been proved.

Durable next question

In the rank-one non-square-zero supergroup generated by $q$ and $z=q^2$, does internal semisimplicity of every finite algebra object force absolute flatness, and is any global failure detected by such a rank-one restriction?