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Support varieties and modules of finite projective dimension for modular Lie superalgebras
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-06-30, DOI: 10.2140/ant.2021.15.1157
Christopher M. Drupieski, Jonathan R. Kujawa

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of Bøgvad, showing that if a finite-dimensional Lie superalgebra over a field of odd characteristic is absolutely torsion free, then its enveloping algebra is of finite global dimension.