Cohen-Macaulay/Finite projective dimension/Transport of cohomology/Regular/Example

Suppose that the (primary) ideal ${}I={\left(f_{1},\ldots ,f_{n}\right)}$ (${}R$ is local and Cohen-Macaulay) has finite projective dimension. Then we have a finite free resolution

$0\longrightarrow F_{d}\longrightarrow F_{d-1}\longrightarrow \cdots \longrightarrow F_{1}{\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}F_{0}=R\longrightarrow R/I\longrightarrow 0$

and the length of this resolution is ${}d$ due to the Auslander-Buchsbaum formula, since the depth of ${}R/I$ is ${}0$. Then the top-dimensional syzygy module is free, just because

${}\operatorname {Syz} _{d-1}=F_{d}=R^{\beta _{d}}\,.$

The cohomology class is then described as

$c_{d-1}=(c_{d-1,j}){\text{ where }}c_{d-1,j}\in H^{d-1}(U,{\mathcal {O}}_{U}),$

and it is ${}0$ if and only if all components are ${}0$. These components lie in the (often) well understood cohomology module

${}H^{d-1}(U,{\mathcal {O}}_{U})=H_{\mathfrak {m}}^{d}(R)\,.$

If ${}R$ is even regular, then every ideal has a finite free resolution and the ideal containment problem ${}f\in I$ reduces to the computation of (the components of) ${}c_{d-1}$ and deciding whether they are ${}0$ or not.