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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

NF    is not -free

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