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IntegralClosure :: IntegralClosure

IntegralClosure -- routines for integral closure of affine domains and ideals


This package contains several algorithms for computing the integral closure (i.e. normalization) of an affine domain, and also of an ideal.

The basic use of this package is shown in the following example.

i1 : R = QQ[x,y,z]/(x^3-x^2*z^5-z^2*y^5)

o1 = R

o1 : QuotientRing
i2 : S = integralClosure R

o2 = S

o2 : QuotientRing

Use icFractions to see what fractions have been added.

i3 : icFractions R

      y z
o3 = {---, x, y, z}

o3 : List

Look at the ideal of S or the generators of S to see the structure of the integral closure.

i4 : gens S

o4 = {w   , x, y, z}

o4 : List
i5 : trim ideal S

                     3        2       5    2   2        5
o5 = ideal (w   x - y z, w   y z + x*z  - x , w    + y*z  - x*y)
             1,0          1,0                  1,0

o5 : Ideal of QQ[w   , x, y, z]

If the characteristic is positive, yet small compared to the degree, but the fraction ring is still separable over a subring, then use icFracP, which is an implementation of an algorithm due to Leonard-Pellikaan, and modified by Singh-Swanson (see arXiv:0901.0871). However, the interface to this routine is likely to change in future releases to more closely match the functions above.

The method used by integralClosure is a modification of the basic algorithm explained in Theo De Jong’s paper An Algorithm for Computing the Integral Closure, J. Symbolic Computation, (1998) 26, 273-277.



This documentation describes version 1.0 of IntegralClosure.

Source code

The source code from which this documentation is derived is in the file IntegralClosure.m2. The auxiliary files accompanying it are in the directory IntegralClosure/.


  • Functions and commands
    • conductor -- the conductor of a finite ring map
    • icFracP -- compute the integral closure in prime characteristic
    • icFractions -- fractions integral over an affine domain
    • icMap -- natural map from an affine domain into its integral closure
    • icPIdeal -- compute the integral closure in prime characteristic of a principal ideal
    • idealizer -- compute Hom(I,I) as a quotient ring
    • integralClosure -- integral closure of an ideal or a domain
    • integralClosures (missing documentation)
    • isNormal -- determine if a reduced ring is normal
    • makeS2 -- compute the S2ification of a reduced ring
    • ringFromFractions -- find presentation for f.g. ring
  • Symbols
    • AllCodimensions (missing documentation)
    • ConductorElement -- Specifies a particular non-zerodivisor in the conductor.
    • Index -- Optional input for idealizer
    • Keep -- an optional argument for various functions
    • RadicalCodim1 -- a symbol denoting a strategy element usable with integralClosure(...,Strategy=>...)
    • Verbosity -- optional argument describing how verbose the output should be