Matthew Hamil: Homological residue fields 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 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.