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);
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i2 : S = integralClosure R
o2 = S
o2 : QuotientRing
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i3 : F = R.icMap
o3 = map(S,R,{x, y, z})
o3 : RingMap S <--- R
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i4 : conductor F
3 2 3 4
o4 = ideal (z , x*z , x z, x )
o4 : Ideal of R
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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.