Vol. 15, No. 5, 2021

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Support varieties and modules of finite projective dimension for modular Lie superalgebras

Christopher M. Drupieski and Jonathan R. Kujawa

Appendix: Luchezar L. Avramov and Srikanth B. Iyengar

Vol. 15 (2021), No. 5, 1157–1180

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.

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Lie superalgebras, cohomology, support varieties, rank varieties, projective dimension
Mathematical Subject Classification
Primary: 20G10
Secondary: 17B56
Received: 15 December 2019
Revised: 19 June 2020
Accepted: 21 November 2020
Published: 30 June 2021
Christopher M. Drupieski
Department of Mathematical Sciences
DePaul University
Chicago, IL
United States
Jonathan R. Kujawa
Department of Mathematics
University of Oklahoma
Norman, OK
United States
Luchezar L. Avramov
Department of Mathematics
University of Nebraska
Lincoln, NE
United States
Srikanth B. Iyengar
Department of Mathematics
University of Utah
Salt Lake City, Ut
United States