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IntegralClosure :: integralClosure(Ring)

integralClosure(Ring) -- compute the integral closure (normalization) of an affine domain

Synopsis

Description

The integral closure of a domain is the subring of the fraction field consisting of all fractions integral over the domain. For example,

i1 : R = QQ[x,y,z]/ideal(x^6-z^6-y^2*z^4-z^3);
i2 : R' = integralClosure R

o2 = R'

o2 : QuotientRing
i3 : gens R'

o3 = {w   , x, y, z}
       3,0

o3 : List
i4 : icFractions R

       2
      x
o4 = {--, x, y, z}
       z

o4 : List
i5 : icMap R

o5 = map (R', R, {x, y, z})

o5 : RingMap R' <--- R
i6 : I = trim ideal R'

                     2   3      2     3
o6 = ideal (w   z - x , w    - y z - z  - 1)
             3,0         3,0

o6 : Ideal of QQ[w   , x..z]
                  3,0

Sometimes using trim provides a cleaner set of generators.

If $R$ is not a domain, first decompose it, and collect all of the integral closures.

i7 : S = ZZ/101[a..d]/ideal(a*(b-c),c*(b-d),b*(c-d));
i8 : C = decompose ideal S

o8 = {ideal (c, b), ideal (c - d, b - d), ideal (d, c, a), ideal (d, b, a)}

o8 : List
i9 : Rs = apply(C, I -> (ring I)/I);
i10 : Rs/integralClosure

        ZZ            ZZ           ZZ         ZZ
       ---[a..d]     ---[a..d]    ---[a..d]  ---[a..d]
       101           101          101        101
o10 = {---------, --------------, ---------, ---------}
         (c, b)   (c - d, b - d)  (d, c, a)  (d, b, a)

o10 : List
i11 : oo/prune

        ZZ         ZZ         ZZ      ZZ
o11 = {---[a, d], ---[a, d], ---[b], ---[c]}
       101        101        101     101

o11 : List

This function is roughly based on Theo De Jong's paper, An Algorithm for Computing the Integral Closure, J. Symbolic Computation, (1998) 26, 273-277. This algorithm is similar to the round two algorithm of Zassenhaus in algebraic number theory.

There are several optional parameters which allows the user to control the way the integral closure is computed. These options may change in the future.

Caveat

This function requires that the degree of the field extension (over a pure transcendental subfield) be greater than the characteristic of the base field. If not, use icFracP. This function requires that the ring be finitely generated over a ring. If not (e.g. if it is f.g. over the integers), then the result is integral, but not necessarily the entire integral closure. Finally, if the ring is not a domain, then the answers will often be incorrect, or an obscure error will occur.

See also