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Test ideals

It remains to find a test ideal. For this we consider the singular locus

$\displaystyle Sing(A) = \{P\in SpecA \mid A_P$    is not regular$\displaystyle \}\,.

Since every regular local ring is normal, $ NN(A) \subset
Sing(A)$. For general Noetherian rings, however, Sing$ (A)$ may not be closed in the Zariski topology. Therefore, we pass to more special rings.

Let $ S = K[x_1, \dots, x_n]$ and $ A = S/I$ be an affine ring where $ K$ is a perfect field. If $ A$ is equidimensional of codimension $ c$, that is, all minimal primes $ P$ of $ I$ have the same height $ c$, then the Jacobian ideal of $ I$ defines Sing$ (A)$. That is, if $ I =
\langle f_1, \dots, f_k\rangle$ and

$\displaystyle J = \left\langle c\text{-minors of } \left(\dfrac{\partial f_i}{\partial
x_j}\right),\; f_1, \dots, f_k\right\rangle \subset S

is the Jacobian ideal of $ I$, then Sing $ (A) = V(J)$. If, on the other hand, $ A$ is not equidimensional, then $ V(J)$ may be strictly contained in Sing$ (A)$, if we define $ J$ as above with $ c$ the minimal height of minimal primes of $ I$. In this case we need another ideal. There are several alternatives to compute an ideal $ I_{Sing}$ with $ V(I_{Sing}) = Sing(A)$. Either we compute an equidimensional decomposition $ I=\bigcap_i I_i$, [8,15], of $ I$, compute the Jacobian ideal $ J_i$ for each equidimensional ideal $ I_i$ 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,

$\displaystyle \Omega^1_{A/K} = \Omega^1_{S/K} \Big/\left(\sum^k_{i=1} f_i
\Omega^1_{S/K} + \sum^k_{i=1} Sdf_i\right)\,,

where $ \Omega^1_{S/K} = {\bigoplus}_{i=1}^n Sdx_i$. $ \Omega^1_{A/K}$ is isomorphic to $ A^n$ modulo the submodule generated by the rows of the Jacobian matrix of $ (f_1, \dots, f_k)$, hence it is finitely presented by the transpose of the Jacobian matrix.

For any finitely presented $ A$-module $ M$ we can compute the non-free locus

   NF$\displaystyle (M) = \{P \in SpecA \mid M_P$    is not $\displaystyle A_P$-free$\displaystyle \}

just by Gröbner basis and syzygy computations, cf. [15].

Let $ I_{Sing}$ be an ideal defining the singular locus of $ A = S/I$. Since $ A$ is reduced, $ I_{Sing}$ contains a non-zerodivisor of $ A$. Indeed, a general linear combination $ u$ of the generators of $ I_{Sing}$ will be a non-zerodivisor. Hence, any radical ideal of $ S$ which contains $ I$ and $ u$ will be a test ideal for normality. Two extreme choices for test ideals are $ \sqrt{I_{Sing}}$ or $ \sqrt{\langle I,u\rangle}$.

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 $ A$.

In the remaining part of this section, we describe algorithms to compute

next up previous
Next: Normalization algorithm Up: Integral closure of rings Previous: Ring normalization
Christoph Lossen