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 π€=π€0 β π€1 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(β±(π€,π€0)).
The localizing subcategories for the detecting subalgebra π£ 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 β±(π£,π£0).
The orbit structure of the reductive group G0 on π€1 where Lie G0=π€0 along with the results for the detecting subalgebra is used to prove a nilpotence theorem for stab(β±(π€,π€0)) 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.
