# conductor -- the conductor of a finite ring map

## Synopsis

• Usage:
conductor F
conductor R
• Inputs:
• F, , $R \rightarrow S$, a finite map with $R$ an affine reduced ring
• R, a ring, an affine domain. In this case, $F : R \rightarrow S$ is the inclusion map of $R$ into the integral closure $S$
• Outputs:
• an ideal, of $R$ consisting of all $d \in R$ such that $dS \subset F(R)$

## Description

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.

## Caveat

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