This function decomposes a module into a direct sum of simple modules, given some fairly strong assumptions on the ring which acts on the ring which acts on the module. This ring must only have two variables, and the square of each of those variables must kill the module.
i1 : Q = ZZ/101[x,y]
o1 = Q
o1 : PolynomialRing
|
i2 : R = Q/(x^2,y^2)
o2 = R
o2 : QuotientRing
|
i3 : M = coker random(R^5, R^8 ** R^{-1})
o3 = cokernel | 22x+40y -18x+20y -12x+17y -35x-39y -16x+43y -15x-26y -46x-25y -49x-49y |
| 42x-16y 10x-19y -17x+36y 17x-47y 18x-38y 33x+6y 43x-38y 6x+46y |
| 33x-30y -35x+22y 42x+28y -41x+26y -7x+23y -16x-14y -44x+22y 23x-44y |
| -41x-12y 36x-42y -40x-23y 45x+18y -7x+17y -27x+9y 28x+46y 46x-34y |
| -8x+14y 4x -37x-5y 46x-47y -2x+36y 3x+16y 38x+13y 5x-19y |
5
o3 : R-module, quotient of R
|
i4 : (N,f) = decomposeModule M
o4 = (cokernel | y x 0 0 0 0 0 0 |, | -33 -49 4 -13 -27 |)
| 0 0 x 0 y 0 0 0 | | -33 26 -15 -22 -2 |
| 0 0 0 y x 0 0 0 | | -8 40 -31 28 42 |
| 0 0 0 0 0 x 0 y | | 10 8 -9 -13 -48 |
| 0 0 0 0 0 0 y x | | 1 0 0 0 0 |
o4 : Sequence
|
i5 : components N
o5 = {cokernel | y x |, cokernel | x 0 y |, cokernel | x 0 y |}
| 0 y x | | 0 y x |
o5 : List
|
i6 : ker f == 0
o6 = true
|
i7 : coker f == 0
o7 = true
|