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IntegralClosure :: integralClosure(Ring,Ring)

integralClosure(Ring,Ring) -- compute the integral closure (normalization) of an affine reduced ring over a base ring

Synopsis

Description

This function packages the output integral closure in the desired way. For more details about integral closure, see integralClosure(Ring).

In the following example, there are three possible coefficient rings for $R$: $R$, $A$ and ${\mathbb Q}$.

i1 : A = QQ[x,y]/(x^3-y^2)

o1 = A

o1 : QuotientRing
i2 : R = reesAlgebra(ideal(x*y,y^2), Variable => z)

o2 = R

o2 : QuotientRing
i3 : coefficientRing R

o3 = A

o3 : QuotientRing
i4 : describe R

                   A[z ..z ]
                      0   1
o4 = -------------------------------------
                           2     2      2
     (x*z  - y*z , y*z  - x z , z  - x*z )
         0      1     0      1   0      1
i5 : R' = integralClosure(R, R)

o5 = R'

o5 : QuotientRing
i6 : describe R'

                                 R[w   ]
                                    0,0
o6 = ---------------------------------------------------------------
                2               2
     (y*w    - x , x*w    - y, w    - x, z w    - z , z w    - x*z )
         0,0          0,0       0,0       1 0,0    0   0 0,0      1
i7 : icMap R

o7 = map(R',R,{z , z , x, y})
                0   1

o7 : RingMap R' <--- R
i8 : fracs1 = icFractions R

      z
       0
o8 = {--, z , z , x, y}
      z    0   1
       1

o8 : List
i9 : R'' = integralClosure(R, A)

o9 = R''

o9 : QuotientRing
i10 : describe R''

                                     A[w   , z ..z ]
                                        0,0   0   1
o10 = ----------------------------------------------------------------------------
                              2                                           2
      (x*z  - y*z , y*w    - x , x*w    - y, w   z  - z , w   z  - x*z , w    - x)
          0      1     0,0          0,0       0,0 1    0   0,0 0      1   0,0
i11 : icMap R

o11 = map(R'',R,{z , z , x, y})
                  0   1

o11 : RingMap R'' <--- R
i12 : fracs2 = icFractions R

       z
        0
o12 = {--, z , z , x, y}
       z    0   1
        1

o12 : List
i13 : assert(fracs1 == fracs2)
i14 : R''' = integralClosure(R, QQ)

o14 = R'''

o14 : QuotientRing
i15 : describe R'''

                               QQ[w   , z ..z , x..y]
                                   0,0   0   1
o15 = -----------------------------------------------------------------------
        2                                                            2
      (x  - w   y, z x - z y, w   x - y, w   z  - z , w   z  - z x, w    - x)
             0,0    0     1    0,0        0,0 1    0   0,0 0    1    0,0
i16 : icMap R

o16 = map(R''',R,{z , z , x, y})
                   0   1

o16 : RingMap R''' <--- R
i17 : fracs3 = icFractions R

       z
        0
o17 = {--, z , z , x, y}
       z    0   1
        1

o17 : List
i18 : assert(fracs1 == fracs3)

Note that the second and third calls to integralClosure changes the output of icMap but the fractions are the same.

Caveat

All the caveats of integralClosure(Ring) are in effect and the output of icMap changes upon each call to this function.

See also