A note on super Koszul complex and the Berezinian

• Verfasst von: Noja, Simone [VerfasserIn]
• Verfasst von: Re, Riccardo [VerfasserIn]
• Titel: A note on super Koszul complex and the Berezinian
• Verf.angabe: Simone Noja, Riccardo Re
• Jahr: 2022
• Umfang: 19 S.
• Fussnoten: Online veröffentlicht am 27. Mai 2021 ; Gesehen am 04.01.2023
• Titel Quelle: Enthalten in: Annali di matematica pura ed applicata
• Ort Quelle: Berlin : Springer, 1858
• Jahr Quelle: 2022
• Band/Heft Quelle: 201(2022), 1, Seite 403-421
• ISSN Quelle: 1618-1891
• DOI: doi:10.1007/s10231-021-01121-6
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 16E30
• Sach-SW: 16E40
• Sach-SW: 17A70
• Sach-SW: 58A50
• Sach-SW: Berezinian
• Sach-SW: Koszul Complex
• Sach-SW: Superalgebra
• Sach-SW: Supergeometry
• K10plus-PPN: 1830367048
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

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.