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)
-- 0.154094 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)
-- 0.0279498 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)
-- 0.0286034 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
|
The object kernelOfLocalization is a method function.