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. On natural use is the conductor of the map from a ring to its integral closure.

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

i2 : S = integralClosure R
            +----------------+
choices are |(1, 1, x)       |
            +----------------+
            |        2     3 |
            |(3, 2, y z + z )|
            +----------------+
            +----------------+
choices are |(1, 1, x)       |
            +----------------+
            |(1, 1, w )      |
            |        0       |
            +----------------+
            |        2     3 |
            |(3, 2, y z + z )|
            +----------------+
            +----------------+
choices are |(1, 1, x)       |
            +----------------+
            |(1, 1, w )      |
            |        2       |
            +----------------+
            |(1, 1, w )      |
            |        1       |
            +----------------+
            |        2     3 |
            |(3, 2, y z + z )|
            +----------------+
            +----------+
choices are |(1, 1, x) |
            +----------+
            |(1, 1, w )|
            |        5 |
            +----------+
            |(1, 1, w )|
            |        4 |
            +----------+
            |(1, 1, w )|
            |        3 |
            +----------+
            +----------+
choices are |(1, 1, x) |
            +----------+
            |(1, 1, w )|
            |        3 |
            +----------+
            |(1, 1, w )|
            |        6 |
            +----------+
            +----------+
choices are |(1, 1, x) |
            +----------+
            |(1, 1, w )|
            |        6 |
            +----------+

o2 = S

o2 : QuotientRing

i3 : F = R.icMap

o3 = map(S,R,{x, y, z})

o3 : RingMap S <--- R

i4 : conductor F

             3     2   3    4
o4 = ideal (z , x*z , x z, x )

o4 : 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 inhomogeneous. If the source of the map F is not homogeneous conductor returns the message -- No conductor for F.

Ways to use `conductor` :

conductor(RingMap)

conductor -- compute the conductor of a finite ring map

Synopsis

Description

See also

Ways to use conductor :

Ways to use `conductor` :