A note on super Koszul complex and the Berezinian
We construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single deg...
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| Main Authors: | , |
|---|---|
| Format: | Article (Journal) |
| Language: | English |
| Published: |
2022
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| In: |
Annali di matematica pura ed applicata
Year: 2022, Volume: 201, Issue: 1, Pages: 403-421 |
| ISSN: | 1618-1891 |
| DOI: | 10.1007/s10231-021-01121-6 |
| Online Access: | Resolving-System, kostenfrei, Volltext: https://doi.org/10.1007/s10231-021-01121-6
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| Author Notes: | Simone Noja, Riccardo Re |
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| 520 | |a We construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $$\Pi ^{p+q} A$$, with $$\Pi $$the parity changing functor. Finally, we show that, given an automorphism of V, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of V. | ||
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