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IntegralClosure :: icMap

icMap -- natural map from an affine domain into its integral closure

Synopsis

Description

If the integral closure of R has not yet been computed, that computation is performed first. No extra computation is involved. If R is integrally closed, then the identity map is returned.

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

o1 = R

o1 : QuotientRing
i2 : icMap R

                   QQ[w   , x, y]
                       0,0
o2 = map(---------------------------------,R,{x, y})
                   2              2
         (w   y - x , w   x - y, w    - x)
           0,0         0,0        0,0

                       QQ[w   , x, y]
                           0,0
o2 : RingMap --------------------------------- <--- R
                       2              2
             (w   y - x , w   x - y, w    - x)
               0,0         0,0        0,0

This finite ring map can be used to compute the conductor, that is, the ideal of elements of R which are universal denominators for the integral closure (i.e. those d ∈R such that d R’ ⊂R).

i3 : S = QQ[a,b,c]/ideal(a^6-c^6-b^2*c^4);
i4 : F = icMap S;

                              QQ[w   , w   , a, b, c]
                                  4,0   3,0
o4 : RingMap --------------------------------------------------------- <--- S
                       2                          2     2      2    2
             (w   c - a , w   c - w   a, w   a - w   , w    - b  - c )
               3,0         4,0     3,0    4,0     3,0   4,0
i5 : conductor F

             3     2   3    4
o5 = ideal (c , a*c , a c, a )

o5 : Ideal of S

Caveat

If you want to control the computation of the integral closure via optional arguments, then make sure you call integralClosure(Ring) first, since icMap does not have optional arguments.

See also

Ways to use icMap :