conductor -- the conductor of a finite ring map

Synopsis

Usage:
conductor F
conductor R
Inputs:
- F, a ring map, R →S, a finite map with R an affine reduced ring
- R, a ring, an affine domain. In this case, F : R →S is the inclusion map of R into the integral closure S
Outputs:
- an ideal, of R consisting of all d ∈R such that dS ⊂F(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. 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.

Caveat

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

Ways to use `conductor` :

conductor(Ring)
conductor(RingMap)

conductor -- the conductor of a finite ring map

Synopsis

Description

Caveat

See also

Ways to use conductor :

Ways to use `conductor` :