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icMap -- natural map from an affine domain into its integral closure.

Synopsis

Description

If an integrally closed ring is given as input, the identity map from the ring to itself is returned.
i1 : R = QQ[x,y]/ideal(x+2);
i2 : icMap R

o2 = map(R,R,{-2, y})

o2 : RingMap R <--- R
This finite map is needed to compute the conductor of the integral closure into the original ring.
i3 : S = QQ[a,b,c]/ideal(a^6-c^6-b^2*c^4);
i4 : conductor(icMap S)
            +----------------+
choices are |(1, 1, a)       |
            +----------------+
            |        2     3 |
            |(3, 2, b c + c )|
            +----------------+
            +----------------+
choices are |(1, 1, a)       |
            +----------------+
            |(1, 1, w )      |
            |        0       |
            +----------------+
            |        2     3 |
            |(3, 2, b c + c )|
            +----------------+
            +----------------+
choices are |(1, 1, a)       |
            +----------------+
            |(1, 1, w )      |
            |        2       |
            +----------------+
            |(1, 1, w )      |
            |        1       |
            +----------------+
            |        2     3 |
            |(3, 2, b c + c )|
            +----------------+
            +----------+
choices are |(1, 1, a) |
            +----------+
            |(1, 1, w )|
            |        5 |
            +----------+
            |(1, 1, w )|
            |        4 |
            +----------+
            |(1, 1, w )|
            |        3 |
            +----------+
            +----------+
choices are |(1, 1, a) |
            +----------+
            |(1, 1, w )|
            |        3 |
            +----------+
            |(1, 1, w )|
            |        6 |
            +----------+
            +----------+
choices are |(1, 1, a) |
            +----------+
            |(1, 1, w )|
            |        6 |
            +----------+

             3     2   3    4
o4 = ideal (c , a*c , a c, a )

o4 : Ideal of S

If the user has already run the computation integralClosure R then this map can also be obtained by typing R.icMap.

i5 : integralClosure S;
            +----------------+
choices are |(1, 1, a)       |
            +----------------+
            |        2     3 |
            |(3, 2, b c + c )|
            +----------------+
            +----------------+
choices are |(1, 1, a)       |
            +----------------+
            |(1, 1, w )      |
            |        0       |
            +----------------+
            |        2     3 |
            |(3, 2, b c + c )|
            +----------------+
            +----------------+
choices are |(1, 1, a)       |
            +----------------+
            |(1, 1, w )      |
            |        2       |
            +----------------+
            |(1, 1, w )      |
            |        1       |
            +----------------+
            |        2     3 |
            |(3, 2, b c + c )|
            +----------------+
            +----------+
choices are |(1, 1, a) |
            +----------+
            |(1, 1, w )|
            |        5 |
            +----------+
            |(1, 1, w )|
            |        4 |
            +----------+
            |(1, 1, w )|
            |        3 |
            +----------+
            +----------+
choices are |(1, 1, a) |
            +----------+
            |(1, 1, w )|
            |        3 |
            +----------+
            |(1, 1, w )|
            |        6 |
            +----------+
            +----------+
choices are |(1, 1, a) |
            +----------+
            |(1, 1, w )|
            |        6 |
            +----------+
i6 : S.icMap

                      QQ[w , w , a, b, c]
                          7   6
o6 = map(---------------------------------------------,S,{a, b, c})
                 2                    2   2    2    2
         (w c - a , w c - w a, w a - w , w  - b  - c )
           6         7     6    7     6   7

                          QQ[w , w , a, b, c]
                              7   6
o6 : RingMap --------------------------------------------- <--- S
                     2                    2   2    2    2
             (w c - a , w c - w a, w a - w , w  - b  - c )
               6         7     6    7     6   7

See also

Ways to use icMap :