A Note on Super Koszul Complex and the Berezinian

TL;DR

This paper constructs and analyzes the super Koszul complex for free supermodules, demonstrating its homology properties and linking the Berezinian of automorphisms to the complex's dual homology class.

Contribution

It introduces the super Koszul complex for supermodules, proves its homology is concentrated in one degree, and connects the Berezinian to automorphisms via homology transformations.

Findings

Homology of super Koszul complex is concentrated in a single degree.

Dual complex's homology is isomorphic to a shifted version of the base algebra.

Automorphisms induce transformations given by the Berezinian on homology.

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 $V$.