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# kernelOfLocalization -- the kernel of the localization map

- Usage:
    `kernelOfLocalization(M, P)`
- Inputs:
  - M, a module,
  - P, an ideal, the prime ideal to localize at
- Outputs:
  - a module, the kernel of the localization map M -> M_P

## 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)
     -- .103958s 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)
     -- .0178636s 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)
     -- .0662661s 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

- associatedPrimes(Module) -- find associated primes
- primaryDecomposition(Module) -- irredundant primary decomposition of a module

## Ways to use kernelOfLocalization:

- `kernelOfLocalization(Module,Ideal)`

## For the programmer

The object kernelOfLocalization is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PrimaryDecomposition/doc.m2:750:0.