Let
be a field and let
be the polynomial ring in two variables. We consider the ideal
and the element
. This element belongs to the radical of this ideal, hence the forcing morphism
-
is surjective. We claim that it is not a submersion. For this we look at the reduction modulo
. In
the ideal
becomes
which does not contain
. Hence by the valuative criterion for integral closure,
does not belong to the integral closure of the ideal. One can also say that the chain
in the affine plane does not have a lift
(as a chain)
to the spectrum of the forcing algebra.
For the ideal
-
![{\displaystyle {}I={\left(X^{2},Y^{2}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e39f319ff8fcdae48cdd7c87d7ca17c7294ce5)
and the element
the situation looks different. Let
-
be a ring homomorphism to a discrete valuation domain
. If
or
is mapped to
, then also
is mapped to
and hence belongs to the extended ideal. So assume that
and
,
where
is a local parameter of
and
and
are units. Then
and the exponent is at least the minimum of
and
,
hence
-
![{\displaystyle {}\theta (XY)\in {\left(\pi ^{2r},\pi ^{2s}\right)}={\left(\theta {\left(X^{2}\right)},\theta {\left(Y^{2}\right)}\right)}D\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a395e89ab507547dc2f7ec57cc9606000af174e6)
So
belongs to the integral closure of
and the forcing morphism
-
is a universal submersion.