Suppose that the
(primary)
ideal
(
is local and Cohen-Macaulay)
has finite projective dimension. Then we have a finite free resolution
-
and the length of this resolution is
due to the Auslander-Buchsbaum formula, since the depth of
is
. Then the top-dimensional syzygy module is free, just because
-
![{\displaystyle {}\operatorname {Syz} _{d-1}=F_{d}=R^{\beta _{d}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/468543b511354f568a40f356a9f7f4d22e65d136)
The cohomology class is then described as
-
and it is
if and only if all components are
. These components lie in the
(often)
well understood cohomology module
-
![{\displaystyle {}H^{d-1}(U,{\mathcal {O}}_{U})=H_{\mathfrak {m}}^{d}(R)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de209005e91c29bcc65eb54b790c8c0e40b91c53)
If
is even regular, then every ideal has a finite free resolution and the ideal containment problem
reduces to the computation of
(the components of)
and deciding whether they are
or not.