Test ideals

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

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

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

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

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,

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

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

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

• the normalization $\overline{A}$ of $A$, that is, we represent $\overline{A}$ as affine ring and describe the map $A \hookrightarrow \overline{A}$,
• generators for an ideal $I_{\text{\it NN}} \subset A$ describing the non-normal locus, that is, $V(I_{\text{\it NN}}) = NN(A) = \bigl\{P \in SpecA \:\big\vert\, A_P \text{ is not normal}\bigr\}\,,$
• for any ideal $I \subset A$, generators for the integral closure $\overline{I}$ of $I$ in $A$.