# kernelOfLocalization -- the kernel of the localization map

## Synopsis

• Usage:
kernelOfLocalization(M, P)
• Inputs:
• M, ,
• P, an ideal, the prime ideal to localize at
• Outputs:
• , 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) -- 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