Computing the normalization

The idea for computing the normalization of $A = S/I$ is as follows:

• We construct an increasing sequence of rings
  $$A \subset A_1 \subset A_2 \subset \dots \subset A_k \subset A_{k+1} \subset \dots \subset Q(A)$$
  with $J_0\subset A_0:=A$ a test ideal for $A$, $A_i:=Hom_{A_{i-1}}(J_{i-1},J_{i-1})$, and $J_i\subset A_i$ a test ideal for $A_i$, $i \ge 1$.

• If $A_k = A_{k+1}$ then $\overline{A} = A_k$.

For performance reasons we do not look for a non-zerodivisor in $J_i$ but choose any non-zero element $u$. If $u$ is a zerodivisor then it gives a splitting of the ring which makes the subsequent computations easier.

We obtain the following (highly recursive) algorithm for computing the normalization:

Input

$f_1, \dots, f_k \in S=K[x_1, \dots, x_n]$, $I = \langle f_1, \dots, f_k\rangle$
We assume that $I$ is a radical ideal.

Output

Polynomial rings $S_1, \dots, S_\ell$, ideals $I_j \subset S_j$, and maps $\pi_j : S \to S_j$ such that $S/I \to S_1/I_1 \oplus \dots \oplus S_\ell/I_\ell$, induced by $(\pi_1,\dots,\pi_\ell)$ is the normalization map of $S/I$.

1. Compute an ideal $I_{Sing}$ describing the singular locus.
2. Compute the radical $J=\sqrt{I_{Sing}}$.
3. Choose $u \in J\setminus\{0\}$ and compute $I : u$. If $I : u = I$ go to 4 (then $u$ is a non-zerodivisor of $S/I$). Otherwise, set $R = A/\langle u \rangle \oplus A/(I : u)$ and go to 6.
4. Compute $R := Hom_A(J,J)$ and $\pi : A \to R$.
5. If $A = R$ then return $(R, \pi)$, otherwise go to 1.
6. Suppose $R = R_1/I_1 \oplus \dots \oplus R_\ell/I_\ell$. Then, for each $i=1,\dots,\ell$, set $A=R_i/I_i$ and go to 1.