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conductor -- compute the conductor of a finite ring map

Synopsis

Usage:
conductor F
Inputs:
- F, a ring map from a ring R to a ring S. The map must be a finite
Outputs:
- an ideal that is the conductor of S into R.

Description

Suppose that the ring map F : R --> S is finite: i.e. S is a finitely generated R-module. The conductor of F is defined to be {g ∈R | g S ⊂f(R)}. One way to think about this is that the conductor is the set of universal denominators of S over R, or as the largest ideal of R which is also an ideal in S.

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

i2 : F = ICmap R

                      QQ [w , w , x, y, z]
                           7   6
o2 = map(---------------------------------------------,R,{x, y, z})
           2                    2         2    2    2
         (x  - w z, w x - w z, w  - w x, w  - y  - z )
                6    6     7    6    7    7

                          QQ [w , w , x, y, z]
                               7   6
o2 : RingMap --------------------------------------------- <--- R
               2                    2         2    2    2
             (x  - w z, w x - w z, w  - w x, w  - y  - z )
                    6    6     7    6    7    7

i3 : conductor F

             3     2   3    4
o3 = ideal (z , x*z , x z, x )

o3 : Ideal of R

The command conductor calls the command pushForward. Currently, the command pushForward does not work if the source of the map F is multgraded or inhomogeneous. If the source of the map F is multigraded or in homogeneous conductor returns the message -- No conductor for F.

See also

pushForward
integralClosure -- compute the integral closure of a ring

Ways to use `conductor` :

conductor(RingMap)