Suppose that
in the situation of
fact.
Then the generic elements
are parameters. In the polynomial ring
we have for parameters of degree
the inclusion
-
![{\displaystyle {}P_{\geq \sum _{i=0}^{d}a_{i}-d}\subseteq {\left(f_{1},\ldots ,f_{d+1}\right)}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a14d6fa998ba549569faeabd327267e0f667a06)
because the graded Koszul resolution ends in
and
-
So the theorem implies for a graded ring
finite over
that
holds for generic elements. But by the graded Briançon-Skoda Theorem (see
fact)
this holds for parameters even without the generic assumption.