The homological spectrum and nilpotence theorems for Lie superalgebra representations

In the study of cohomology of finite group schemes it is well known that nilpotence theorems play a key role in determining the spectrum of the cohomology ring. Balmer recently showed that there is a more 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 π-points) can be viewed as a topological space that provides an important realization of the Balmer spectrum.

Let $g=g_{\overline{0}}\oplus g_{\overline{1}}$ be a classical Lie superalgebra over $C$. In this paper, the authors consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: $\text{stab}\left(F_{(g,g_{\overline{0}})}\right)$. The localizing subcategories for the detecting subalgebra $f$ 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 $F_{(f,f_{\overline{0}})}$. The authors also verify Balmer's “Nerves-of-Steel” Conjecture for $\mathrm{stab}\left(F_{(f,f_{\overline{0}})}\right)$ where $f$ is a detecting subalgebra.

Let F (resp. G) be the associated supergroup (scheme) for $f$ (resp. $g$). 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 $\text{stab}\left(F_{(g,g_{\overline{0}})}\right)$, and to determine the homological spectrum in this case. Then using natural assumptions in terms of realization of supports, the authors provide a method to explicitly realize the Balmer spectrum of $\text{stab}\left(F_{(g,g_{\overline{0}})}\right)$, and prove the Nerves-of-Steel Conjecture in this case which includes all classical Lie superalgebras of Type A.