Suppose that the ring map $F : R \rightarrow S$ is finite: i.e. $S$ is a finitely generated $R$-module. The conductor of $F$ is defined to be $\{ g \in R \mid g S \subset F(R) \}$. 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..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..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
The object conductor is a method function.