The Homological Spectrum and Nilpotence Theorems for Lie Superalgebra Representations

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 -points) can be viewed as a topological space that provides an important realization of the Balmer spectrum. Let g= g₀ g₁ 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: stab (F (₆, { g₀) }), 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 (₅, { f₀) }. The authors verify Balmer's ``Nerves of Steel'' Conjecture for F (₅, { f₀) }. 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 stab (F (₆, { g₀) }), 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 stab (F (₆, { g₀) }), and prove the Nerves of Steel Conjecture in this case.