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 can be viewed as a topological space that realizes the Balmer spectrum in a concrete way.
Let 𝔤=𝔤0 ⊕ 𝔤1 be a Type I classical Lie superalgebra with an ample detecting subalgebra. In this talk, the speaker will consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: stab(ℱ(𝔤,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 we prove a nilpotence theorem and determine the homological spectrum for the stable module category of ℱ(𝔣,𝔣0).
The orbit structure of the reductive group G0 on 𝔤1 where Lie G0=𝔤0 along with the results for the detecting subalgebra is used to prove a nilpotence theorem for stab(ℱ(𝔤,𝔤0)) and to determine the homological spectrum in this case.
This is joint work with Dan Nakano.
This video is part of the University of Georgia‘s Algebra seminar.
