Fiberwise building and stratification in tensor triangular geometry
Pith reviewed 2026-05-23 05:37 UTC · model grok-4.3
The pith
Conditions on a family of coproduct-preserving functors determine the localizing tensor ideal generated by an object from its images under those functors.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Under the stated conditions on the family of coproduct-preserving tt-functors f_i : T → T_i, an object y lies in the localizing tensor ideal generated by x in T if and only if f_i(y) lies in the localizing tensor ideal generated by f_i(x) in each T_i. This equivalence yields an explicit fiberwise test for stratification of rigidly-compactly generated tt-categories.
What carries the argument
The family of coproduct-preserving tt-functors that detect membership in localizing tensor ideals fiberwise.
If this is right
- The big derived category of permutation modules for any finite group over an arbitrary Noetherian base ring is stratified.
- The category of representations of any finite group scheme over a Noetherian base is stratified.
- Stratification of a rigidly-compactly generated tt-category can be verified by checking the images under each functor in the family separately.
- The same fiberwise test applies whenever a suitable detecting family of coproduct-preserving functors exists.
Where Pith is reading between the lines
- The criterion may be used to establish stratification for additional derived categories in representation theory by constructing appropriate detecting functors.
- Stratification behaves like a local property that can be checked after base change or restriction along the functors.
- The result suggests that similar fiberwise arguments could be developed for non-Noetherian bases if the coproduct-preserving conditions can be relaxed.
Load-bearing premise
The tt-categories are rigidly-compactly generated and the functors satisfy the coproduct-preserving and technical conditions needed for the fiberwise detection of ideals to hold.
What would settle it
A rigidly-compactly generated tt-category equipped with such a family of functors together with an object x and an object y not in the ideal generated by x whose images f_i(y) nevertheless lie in the ideal generated by f_i(x) for every i.
read the original abstract
We establish conditions on a family of coproduct-preserving tt-functors $f_i\colon \mathcal{T}\to \mathcal{T}_i$ between tt-categories with small coproducts, ensuring that the localizing tensor ideal generated by an object $x \in \mathcal{T}$ is determined by those objects whose image under each $f_i$ lies in the localizing tensor ideal generated by $f_i(x)$ for all $i$. This leads to a fiberwise criterion for stratification in the setting of rigidly-compactly generated tt-categories. As an application, we prove that the big derived category of permutation modules for a finite group over an arbitrary Noetherian base is stratified. Moreover, our methods extend to the category of representations of a finite group scheme over a Noetherian base, thereby recovering a recent result from the literature.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript establishes conditions on a family of coproduct-preserving tt-functors f_i : T → T_i between tt-categories with small coproducts such that the localizing tensor ideal generated by an object x ∈ T is determined by those objects whose images under each f_i lie in the localizing tensor ideal generated by f_i(x). This yields a fiberwise criterion for stratification in the setting of rigidly-compactly generated tt-categories. As an application, the paper proves that the big derived category of permutation modules for a finite group over an arbitrary Noetherian base is stratified and extends the methods to the category of representations of a finite group scheme over a Noetherian base, recovering a recent result from the literature.
Significance. If the derivations hold, the work supplies a practical fiberwise criterion for detecting localizing tensor ideals and stratification in tensor triangular geometry. The application to permutation modules over general Noetherian bases is a concrete advance in the representation theory of finite groups, and the extension to group schemes demonstrates the criterion's flexibility while aligning with existing literature. The precise formulation of the functor conditions and the use of standard tt-category axioms are strengths that make the result potentially reusable in other rigidly-compactly generated settings.
minor comments (3)
- [§2.2] §2.2: the definition of 'fiberwise building' is introduced via the family of functors but the subsequent stratification criterion in Theorem 3.4 would benefit from an explicit statement of how the Noetherian hypothesis on the base enters the argument, even if only for the application.
- [§4] Notation in §4: the symbol for the localizing tensor ideal generated by an object is overloaded between the source category T and the target categories T_i; a subscript or prime on the ideal notation would reduce ambiguity when comparing the fiberwise condition to the global one.
- [§5] The proof of the permutation-module application (around Proposition 5.3) invokes coproduct preservation of the restriction functors but does not explicitly verify that the family satisfies the full list of hypotheses from the main theorem; adding a short verification paragraph would strengthen the application section.
Simulated Author's Rebuttal
We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments appear in the report, so we have no points requiring point-by-point rebuttal or revision at this stage. Any minor editorial matters will be addressed in the revised manuscript.
Circularity Check
No significant circularity; derivation rests on standard tt-category axioms
full rationale
The paper derives a fiberwise criterion for stratification from conditions on coproduct-preserving tt-functors between rigidly-compactly generated tt-categories, using the standard axioms of tensor-triangular geometry. The central result on localizing tensor ideals generated by objects and their images under the functors follows directly from the functor properties and the definition of localizing tensor ideals, without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations. The application to permutation modules over a Noetherian base invokes the stated hypotheses on the functors and base ring as external conditions under which the result holds, and the recovery of a prior result is presented as an extension rather than a foundational premise. No step equates a prediction or uniqueness claim to its own inputs by construction.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard axioms and definitions of tensor-triangular categories and localizing tensor ideals
- domain assumption Rigidly-compactly generated tt-categories admit the stratification notion used
discussion (0)
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