Next: Examples Up: Integral closure of rings Previous: Computing the normalization

## Integral closure algorithm

Let be a ring, an ideal. We propose an algorithm to compute is integral over , the integral closure of . is called integrally closed if and only if . It is called normal if for all . Note that . We are mainly interested in the case .

In the following, we describe an algorithm to compute for all , simultaneously. Consider the Rees algebra , and let denote the integral closure of in . Then

If is normal, then is normal and hence, the normalization of , that is, the integral closure of in , satisfies

Hence, computing the normalization of provides the integral closure of for all .

To be specific, let , with a perfect field. Then

where maps , . can be computed by eliminating from

that is, . For the integral closure of we need to compute

This means that we compute as an affine ring and, in each inductive step during the computation of , we also compute the map from the intermediate ring to .

The algorithm then reads as follows:

Input
, an integer, .
Output
Generators for .

1. Compute the Rees algebra .
2. Compute the normalization , together with maps , so that

commutes.
3. Determine , so that compute (indeed, we find a universal denominator for all ).
4. Determine generators of the -ideal which is mapped to the component of -degree of the subalgebra .
5. Return .

The algorithms described above are implemented in SINGULAR and contained in the libraries normal.lib [16] and reesclos.lib [23] contained in the distribution of SINGULAR 2.0 [17]. Similar procedures can be used to compute the conductor ideal of in . An implementation will be available soon.

Next: Examples Up: Integral closure of rings Previous: Computing the normalization
Christoph Lossen
2001-03-21