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Kronecker :: decomposeModule

decomposeModule -- decompose a module into a direct sum of simple modules

Synopsis

Description

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 | 20x+40y  -39x-26y 6x-34y   34x+47y  -x-21y   -31x+14y -19x+40y -9x+28y |
              | -40x+35y -33x-22y -19x-46y 14x-39y  -20x+8y  -13x-48y -46x-17y 8x+9y   |
              | -44x-49y 23x+50y  -22x+50y -2x+11y  3x-13y   13x-3y   33x+32y  30x-49y |
              | 42x+35y  -5x+29y  48x+41y  -30x-36y -27x-35y 18x+25y  -46x+10y -46x+4y |
              | 11x+50y  34x+23y  29x+25y  -47x-33y 19x-19y  -47x+36y 45y      3x+30y  |

                            5
o3 : R-module, quotient of R
i4 : (N,f) = decomposeModule M

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 46  -39 26 17 -40 |)
               | 0 0 x 0 y 0 0 0 |  | 2   1   1  24 50  |
               | 0 0 0 y x 0 0 0 |  | -24 -49 0  3  3   |
               | 0 0 0 0 0 x 0 y |  | 1   0   0  0  0   |
               | 0 0 0 0 0 0 y x |  | -50 7   8  -4 -6  |

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

Ways to use decomposeModule :