Representations and cohomology of a family of finite supergroup schemes
| Autor: | Julia Pevtsova, Dave Benson |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Group cohomology 010102 general mathematics Mathematics - Rings and Algebras Unipotent Local cohomology 01 natural sciences Cohomology Cohomology ring Rings and Algebras (math.RA) Mathematics::K-Theory and Homology 0103 physical sciences Spectral sequence FOS: Mathematics 010307 mathematical physics Representation Theory (math.RT) 0101 mathematics Mathematics::Representation Theory Supergroup Mathematics - Representation Theory Quotient Mathematics |
| Zdroj: | Journal of Algebra. 561:84-110 |
| ISSN: | 0021-8693 |
| Popis: | We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes. 19 pages |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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