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

Synopsis

Usage:

f = icMap R
Inputs:
- R, a ring, an affine domain
Outputs:
- f, a ring map, from R to its integral closure

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 : f = 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

i3 : isWellDefined f

o3 = true

i4 : source f === R

o4 = true

i5 : describe target f

               QQ[w   , x..y]
                   0,0
o5 = ---------------------------------
               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 \in R such that d R' \subset R).

i6 : S = QQ[a,b,c]/ideal(a^6-c^6-b^2*c^4);

i7 : F = icMap S;

                                QQ[w   , w   , a..c]
                                    4,0   3,0
o7 : 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

i8 : conductor F

             3     2   3    4
o8 = ideal (c , a*c , a c, a )

o8 : 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.

Ways to use icMap :

icMap(Ring)

For the programmer