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##

Test ideals

It remains to find a test ideal. For this we consider the singular
locus
is not regular

Since every regular local ring is normal,
. For general Noetherian rings, however, Sing may
not be closed in the Zariski topology. Therefore, we pass to more
special rings.
Let
and be an affine ring where
is a perfect field. If is equidimensional of codimension ,
that is, all minimal primes of have the same height , then
the *Jacobian ideal* of defines Sing. That is, if
and

is the Jacobian ideal of , then Sing
. If, on the
other hand, is not equidimensional, then may be strictly
contained in Sing, if we define as above with the minimal
height of minimal primes of . In this case we need another ideal.
There are several alternatives to compute an ideal with
. Either we compute an equidimensional
decomposition
,
[8,15], of , compute the Jacobian ideal
for each equidimensional ideal and compute the ideal describing
the intersection of any two equidimensional parts. The same works for
a primary decomposition, [11,8,15])
instead of an equidimensional decomposition.
We can avoid an equidimensional, resp. primary, decomposition if we
compute the ideal of the non-free locus of the module of *Kähler
differentials*,

where
.
is isomorphic to modulo the submodule generated
by the rows of the Jacobian matrix of
, hence it is
finitely presented by the transpose of the Jacobian matrix.
For any finitely presented -module we can compute the non-free
locus

just by Gröbner basis and syzygy computations, cf. [15].
Let be an ideal defining the singular locus of . Since is reduced, contains a non-zerodivisor
of . Indeed, a general linear combination of the generators
of will be a non-zerodivisor. Hence,
any radical ideal of which contains and will be a test
ideal for normality. Two extreme choices for test ideals are
or
.

Since the radical of an ideal in an affine ring can be computed,
[8,25,15], we have all ingredients to compute the normalization
of .

In the remaining part of this section, we describe algorithms to
compute
- the
* normalization*
of , that is, we
represent
as affine ring and describe the map
,
- generators for an ideal
describing the
* non-normal locus*, that is,
- for any ideal
, generators for the
* integral
closure*
of in .

** Next:** Normalization algorithm
** Up:** Integral closure of rings
** Previous:** Ring normalization
*Christoph Lossen*

*2001-03-21*