On localizing subcategories of Lie superalgebra representations

We state and prove a stratification result that allows us to classify the tensor ideal localizing subcategories for the stable module category $\text{Stab}(\mathcal{C}_{(\mathfrak{g}, \mathfrak{g}_{\bar 0})})$ of Lie superalgbera representations which are semisimple as representations of $\mathfrak{g}_{\bar 0}$ under the hypotheses that $\mathfrak{g}$ is a classical Lie superalgebra with a splitting detecting subalgebra $\mathfrak{z} \leq \mathfrak{g}$, as well as a natural hypothesis on realization of supports. This extends the work of the author and Nakano where a similar classification was obtained for the stable category of modules over a detecting subalgebra employing stratification in the sense of Benson, Iyengar, and Krause. Our new result involves making use of a more general stratification framework in weakly Noetherian contexts developed by Barthel, Heard, and Sanders using the Balmer-Favi notion of support for big objects in tensor triangulated categories, as well as the recently developed homological stratification of Barthel, Heard, Sanders, and Zou in using the homological spectrum.