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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 | 33x+47y  -7x-38y  50x+23y  -9x+38y  6x+45y   -45x+37y 13x+y    -8x+2y   |
              | 46x-44y  39x-35y  48x-29y  33x+39y  -19x+16y 18x+25y  29x-34y  -20x-14y |
              | -32x+37y 19x-21y  6x-13y   -19x-40y 9x+40y   32x-9y   -38x+17y -20x+20y |
              | -45x+34y -14x-29y 36x-47y  30x+23y  -19x+5y  -35x-22y 24x+y    35x-43y  |
              | 29x-32y  5x+13y   -26x-15y -x+5y    43x-4y   -21x+26y 28x-9y   -12x-46y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 12 -49 42  -23 34 |)
               | 0 0 x 0 y 0 0 0 |  | 9  -21 16  16  36 |
               | 0 0 0 y x 0 0 0 |  | 15 -22 -46 37  1  |
               | 0 0 0 0 0 x 0 y |  | -1 1   37  -19 24 |
               | 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

Ways to use decomposeModule :