An example of a non acyclic Koszul complex of a module

Abstract

In his paper "Residues of a Pfaff system relative to an invariant subscheme" in Trans. Amer. Math. Soc. 352, 2000, 4019-4035, F. Sancho de Salas defines the universal Koszul complex of a module $M$ over a sheaf of rings $\mathcal{O}$ as ${\rm Kos}(M)=Λ(M)\otimes_{\mathcal{O}}S(M)$, where $Λ(M)$ and $S(M)$ stand for the exterior and symmetric algebras of $M$, endowed with the usual differential, and he conjectures (Conjecture 2.3.) that ${\rm Kos}(M)$ is always acyclic. We give here an example of a non acyclic Koszul complex ${\rm Kos}(M)$.