Proof
We assume
,
lower dimensions may be treated directly. Because of
,
we can also reduce to the case of a primary ideal
. Suppose that
,
and let
be the corresponding non-zero class arising from a finite free resolution. At least one component, say
,
is then also non-zero, and we can write it in terms of Čech-cohomology as
-
![{\displaystyle {}c'={\frac {h}{x_{1}^{n_{1}}\cdots x_{d}^{n_{d}}}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c93b56627581e0584612a159a98a9f3d0e746839)
where
is a regular system of parameters of
and
.
We have to show that there is no
such that
for all
.
Multiplying the class with some element of
we may assume that
is a unit.
We have
(with
)
-
![{\displaystyle {}F^{e*}(c')={\frac {h^{q}}{x_{1}^{qn_{1}}\cdots x_{d}^{qn_{d}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73393305a5f833dc271959f6869c4505edce43bb)
and its annihilator is
. But then
-
![{\displaystyle {}\bigcap _{e\in \mathbb {N} }{\left(x_{1}^{qn_{1}},\ldots ,x_{d}^{qn_{d}}\right)}\subseteq \bigcap _{e\in \mathbb {N} }{\left(x_{1}^{n_{1}},\ldots ,x_{d}^{n_{d}}\right)}^{q}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba62931f94d6177258c9af2db93ed959997209f8)