Normalization of rings

Abstract

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. A modification, applicable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper has been implemented in Singular, for localizations of affine rings with respect to arbitrary monomial orderings. Benchmark tests show that it is in general much faster than any other implementation of normalization algorithms known to us.turns out to be very fast for small p. However the computation of the Frobenius makes it impracticable when p is large.There are also very efficient methods for computing the normalization in some special cases. For example, for toric rings, one can apply fast combinatorial techniques, as explained in Bruns and Koch (2001).The algorithm we propose in this paper is a general algorithm and it is based on de Jong (1998) and Decker et al. (1999). In their algorithm, as we mentioned before, they construct an increasing chain of affine rings. They enlarge the rings by computing the endomorphism ring of a test ideal (see below), adding new variables for each module generator of the endomorphism ring and dividing out the relations among them. Then the algorithm is applied recursively to this new affine ring. Due to the increasing number of variables and relations this can produce a big slowdown in the performance of the algorithm already when the number of intermediate rings is 2 or 3. For a larger number, it usually makes the algorithm unusable, as the Groebner bases of the ideals of relations grow extensively. Our approach avoids the increasing complexity when enlarging the rings, benefiting from the finitely generated A-module structure of the normalization. We are able to do most computations over the original ring without adding new variables or relations.The main new results of this paper are presented in Section 3. In Section 4 we describe the algorithm and show, as an application, how the δ-invariant of the ring can be computed. Section 5 contains several benchmark examples and a comparison with previously known algorithms, while Section 6 is devoted to an extension of the algorithm to non-global monomial orderings. Basic definitions and toolsLet A be a reduced Noetherian ring. 1 The normalizationĀ of A is the integral closure of A in the total ring of fractions Q(A), which is the localization of A with respect to the non-zerodivisors on A. A is called normal if A =Ā.The conductor of A inĀ is C = {a ∈ Q(A) | aĀ ⊂ A} = Ann A (Ā/A).Lemma 2.1.Ā is a finitely generated A-module if and only if C contains a nonzerodivisor on A.Proof. If p ∈ C is a non-zerodivisor thenĀ ∼ = pĀ ⊂ A is module-finite over A, since A is Noetherian. Conversely, ifĀ is module-finite over A then any common multiple of the denominators of a finite set of generators is a non-zerodivisor on A contained in C.We recall the Grauert and Remmert criterion of normality.Proposition 2.2. Let A be a Noetherian reduced ring and J ⊂ A an idea...

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