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`. An important case is the conductor of the map from a ring to its integral closure.

i1 : R = QQ[x,y,z]/ideal(x^7-z^7-y^2*z^5); |

i2 : icFractions R 3 2 x x o2 = {--, --, x, y, z} 2 z z o2 : List |

i3 : F = icMap R QQ[w , w , x, y, z] 5,0 4,0 o3 = map(--------------------------------------------------------------------------------------------------------,R,{x, y, z}) 2 2 2 2 3 2 2 2 3 2 2 (w z - x , w z - w x, w x - w , w x - y z - z , w w - x*y - x*z , w - w y - x z) 4,0 5,0 4,0 5,0 4,0 5,0 5,0 4,0 5,0 4,0 QQ[w , w , x, y, z] 5,0 4,0 o3 : RingMap -------------------------------------------------------------------------------------------------------- <--- R 2 2 2 2 3 2 2 2 3 2 2 (w z - x , w z - w x, w x - w , w x - y z - z , w w - x*y - x*z , w - w y - x z) 4,0 5,0 4,0 5,0 4,0 5,0 5,0 4,0 5,0 4,0 |

i4 : conductor F 4 3 2 2 4 5 o4 = ideal (z , x*z , x z , x z, x ) o4 : Ideal of R |

If an affine domain (a ring finitely generated over a field) is given as input, then the conductor of *R* in its integral closure is returned.

i5 : conductor R 4 3 2 2 4 5 o5 = ideal (z , x*z , x z , x z, x ) o5 : Ideal of R |

If the map is not `icFractions(R)`, then pushForward is called to compute the conductor.

Currently this function only works if `F` comes from a integral closure computation, or is homogeneous

- integralClosure -- integral closure of an ideal or a domain
- icFractions -- fractions integral over an affine domain
- icMap -- natural map from an affine domain into its integral closure
- pushForward

- conductor(Ring)
- conductor(RingMap)