Fiberwise building and stratification in tensor triangular geometry
Juan Omar Gómez
TL;DR
The paper develops a general fiberwise criterion for controlling localizing tensor-ideals in a big tt-category via a family of coproduct-preserving tt-functors with adjoints and projection formulas. Under suitable unit-and-adjunction hypotheses, it proves Detection and Building statements and deduces that global stratification follows from stratification of the fiber categories, provided Balmer-spectral conditions hold. Using Neeman’s stratification results over Noetherian bases and field-case stratification, the authors establish stratification for the big derived category of permutation modules $\mathcal{T}(G,R)$ and for the category $\mathrm{Rep}(\mathbb{G},R)$ of representations of a finite group scheme, together with Noetherianity of the small Balmer spectrum $\mathrm{Spc}(\mathcal{K}(G,R))$. They introduce triangular fixed points to prove Noetherianity in general and extend the framework to non-big tt-categories, such as stable module categories for infinite groups, thereby broadening the reach of fiberwise stratification in tensor-triangular geometry.
Abstract
We give 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 built from those objects whose image under $f_i$ lies in the localizing tensor-ideal generated by $f_i(x)$ for all $i$. This allows us to provide a criterion for stratification on a fiberwise level when restricted to big tt-categories. As an application, we show that the big derived category of permutation modules for a finite group over an arbitrary Noetherian base is indeed stratified. Furthermore, our approach extends to the category of representations of a finite group scheme over a Noetherian base, thereby recovering a recent result from the literature.