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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 | 5x-32y   48x-29y  25x-46y  22x-2y   35x+17y 18x-35y 6x-32y   -28x+23y |
              | -47x+45y 33x-37y  -36x+27y 24x-33y  39x-20y -5x-17y 2x-15y   -31x-11y |
              | 2x-33y   -9x+20y  -x+22y   -4x+35y  35x+18y 47x-42y 46x-29y  33x-28y  |
              | 42x-6y   -x-8y    29x-49y  -x-27y   9x-46y  4x+19y  -14x-16y 47x-23y  |
              | -42x-42y -38x+38y 38x-19y  -24x-50y 14x+30y 13x-33y 5x+43y   11x+2y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 4   -23 -11 -19 -18 |)
               | 0 0 x 0 y 0 0 0 |  | 30  15  48  12  -14 |
               | 0 0 0 y x 0 0 0 |  | -50 4   -1  48  11  |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | -33 11  30  39  27  |

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 :