Cohen-Macaulay/Parameter/Koszul-Auflösung/Kohomologie-Transport/Cech/Beispiel/en

Let ${}R$ be a Cohen-Macaulay local ring of dimension ${}d\geq 2$ and let ${}I={\left(f_{1},\ldots ,f_{d}\right)}$ be a (full) parameter ideal. We consider the Koszul-resolution of these parameters. This is a finite resolution and the top-dimensional syzygy sheaf is just the structure sheaf. A direct computation using Čech-cohomology shows that for an element ${}f\in R$ we get

${}c_{d-1}=\delta ^{d-1}(f)={\frac {f}{f_{1}\cdots f_{d}}}\in H_{\mathfrak {m}}^{d}(R)\,.$