The Homological Spectrum and Nilpotence Theorems for Lie Superalgebra Representations

Abstract

Balmer recently showed that there is a general notion of a nilpotence theorem for tensor triangulated categories through the use of homological residue fields and the connection with the homological spectrum. The homological spectrum (like the theory of $\pi$-points) can be viewed as a topological space that provides an important realization of the Balmer spectrum. Let ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$ be a classical Lie superalgebra over ${\mathbb C}$. In this paper, the authors consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: $\text{stab}({\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})})$, The localizing subcategories for the detecting subalgebra ${\mathfrak f}$ are classified which answers a question of Boe, Kujawa, and Nakano. As a consequence of these results, the authors prove a nilpotence theorem and determine the homological spectrum for the stable module category of ${\mathcal F}_{({\mathfrak f},{\mathfrak f}_{\bar{0}})}$. The authors verify Balmer's ``Nerves of Steel'' Conjecture for ${\mathcal F}_{({\mathfrak f},{\mathfrak f}_{\bar{0}})}$. Let $F$ (resp. $G$) be the associated supergroup (scheme) for ${\mathfrak f}$ (resp. ${\mathfrak g}$). Under the condition that $F$ is a splitting subgroup for $G$, the results for the detecting subalgebra can be used to prove a nilpotence theorem for $\text{stab}({\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})})$, and to determine the homological spectrum in this case. Now using natural assumptions in terms of realization of supports, the authors provide a method to explicitly realize the Balmer spectrum of $\text{stab}({\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})})$, and prove the Nerves of Steel Conjecture in this case.