On localizing subcategories of Lie superalgebra representations
Matthew H. Hamil
TL;DR
The paper develops a comprehensive stratification program for the stable category of Lie superalgebra representations with semisimple even part, leveraging splitting detecting subalgebras and cohomological supports. It integrates three stratification paradigms—BIK, Balmer–Favi, and homological stratification—and proves that for classical Lie superalgebras with a splitting detecting subalgebra, the stable category Stab(C_{(g,g0)}) is tt-stratified by Spc(stab(F_{(g,g0)})), with tt-stratification equivalent to h-stratification. This yields a bijection between tensor-ideal localizing subcategories and subsets of the Balmer spectrum, reduced to the cohomological spectrum of a splitting subalgebra, and it applies descent to Type A via Nerves-of-Steel. The work extends prior results for detecting subalgebras and provides a robust framework for classifying subcategories in Lie superalgebra representations, with explicit implications for cohomological support varieties and Nerves-of-Steel conjectures. The methods combine Balmer–Favi and homological approaches, enabling stratification without requiring a global ring action, and offer practical reductions to elementary blocks such as exterior algebras for key substructures.
Abstract
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.