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 $\mathrm{Kos}(M)=\Lambda (M){\otimes }_{\mathcal{O}}S(M)$, where $\Lambda (M)$ and $S(M)$ stand for the exterior and symmetric algebras of $M$, endowed with the usual differential, and he conjectures (Conjecture 2.3.) that $\mathrm{Kos}(M)$ is always acyclic. We give here an example of a non acyclic Koszul complex $\mathrm{Kos}(M)$.