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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 | -31x+14y 50x-16y  39x-28y  14x+23y  6x+50y   12x+50y -28x+3y  -35x-26y |
              | 8x-13y   17x-14y  -4x+46y  27x-23y  20x+24y  13x+17y 48x-17y  27x-25y  |
              | 14x-30y  9x-27y   -47x+43y -38x-21y -2x-37y  20x-31y 14x-25y  -12x+38y |
              | 39x-33y  6x-39y   31x+9y   10x-15y  -16x+39y 28x-38y -11x-4y  -36x+14y |
              | 5x-38y   -49x+17y -25x+7y  30x-42y  5x-50y   42x+20y -42x-37y -32x+36y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -7  -20 -13 -1 2   |)
               | 0 0 x 0 y 0 0 0 |  | 24  22  9   -2 2   |
               | 0 0 0 y x 0 0 0 |  | 27  33  -17 21 -42 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0  0   |
               | 0 0 0 0 0 0 y x |  | -21 32  40  38 25  |

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 :