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.
Highlights
The definition of the Koszul Complex, whose first introduction as an example of a complex of free modules over a commutative ring A dates back to Hilbert, marks the advent of homological methods in commutative algebra in the early 1950s
In Remark 3.6 at the end of section three, we address the differences with the classical Koszul homology and we briefly discuss the interesting case of characteristic p in the superalgebraic setting by means on an example
In section four we introduce the dual of the super Koszul complex and we briefly discuss the functoriality of the construction
Summary
The definition of the Koszul Complex, whose first introduction as an example of a complex of free modules over a commutative ring A dates back to Hilbert, marks the advent of homological methods in commutative algebra in the early 1950s. Xr) and calculate its homology, c⨁omputing Exti(A, S) This produces the Berezinian of the free A-module F = i Axi. it is fair to say that results regarding the Koszul complex in a “super setting” have previously appeared—see [8] -, we are not aware of a complete and detailed treatment of this fundamental construction in the existing superalgebra or supergeometry literature. The result is built upon Lemma 3.3 and Lemma 3.4, which compute the homotopy operator of the differential of the super Koszul complex In this respect, in Remark 3.6 at the end of section three, we address the differences with the classical Koszul homology and we briefly discuss the interesting case of characteristic p in the superalgebraic setting by means on an example. We would like to thank him for pointing out this reference to us
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