A note on super Koszul complex and the Berezinian
| Autor: | Riccardo Re, Simone Noja |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
Pure mathematics
Rank (linear algebra) Berezinian FOS: Physical sciences Koszul complex Homology (mathematics) Superalgebra 01 natural sciences Mathematics::Algebraic Topology Mathematics - Algebraic Geometry Koszul Complex Supergeometry Mathematics::K-Theory and Homology 0103 physical sciences FOS: Mathematics Algebraic Geometry (math.AG) Mathematical Physics Mathematics Functor Degree (graph theory) 010308 nuclear & particles physics Applied Mathematics Mathematical Physics (math-ph) Automorphism 010307 mathematical physics Resolution (algebra) |
| Popis: | 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 $V$. Comment: 13 pages, reference added |
| Databáze: | OpenAIRE |
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