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kernelOfLocalization -- the kernel of the localization map

Synopsis

Description

This method computes the kernel of the natural map from a module to its localization at a given prime ideal. The efficiency of this method is intimately tied to the efficiency of computation of associated primes for the module - if the associated primes of M have previously been computed, then this method should finish quickly.

i1 : R = QQ[x_0..x_3]

o1 = R

o1 : PolynomialRing
 
i2 : (I1,I2,I3) = monomialCurveIdeal_R \ ({1,2,3},{2,3},{4,5})

              2                       2                 3      2          5  
o2 = (ideal (x  - x x , x x  - x x , x  - x x ), ideal(x  - x x ), ideal(x  -
              2    1 3   1 2    0 3   1    0 2          1    0 2          1  
     ------------------------------------------------------------------------
        4
     x x ))
      0 2

o2 : Sequence
 
i3 : M = comodule I1 ++ comodule I2 ++ comodule I3

o3 = cokernel | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |
              | 0            0             0            x_1^3-x_0x_2^2 0              |
              | 0            0             0            0              x_1^5-x_0x_2^4 |

                            3
o3 : R-module, quotient of R
 
i4 : elapsedTime kernelOfLocalization(M, I1)
 -- 0.0854081 seconds elapsed

o4 = subquotient (| 0 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
                  | 1 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
                  | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |

                               3
o4 : R-module, subquotient of R
 
i5 : elapsedTime kernelOfLocalization(M, I2)
 -- 0.0128964 seconds elapsed

o5 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
                  | 0 0 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
                  | 0 1 |  | 0            0             0            0              x_1^5-x_0x_2^4 |

                               3
o5 : R-module, subquotient of R
 
i6 : elapsedTime kernelOfLocalization(M, I3)
 -- 0.0227855 seconds elapsed

o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0              0              |)
                  | 0 1 |  | 0            0             0            x_1^3-x_0x_2^2 0              |
                  | 0 0 |  | 0            0             0            0              x_1^5-x_0x_2^4 |

                               3
o6 : R-module, subquotient of R

See also

Ways to use kernelOfLocalization :

For the programmer

The object kernelOfLocalization is a method function.