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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 | 15x-48y  48x+19y  15x+16y  -49x-44y 9x+48y   -39x-37y 8x+42y   23x-31y  |
              | -48x-29y 46x-33y  37x-42y  43x+23y  4x-31y   -26x+45y -43x+46y -28x-44y |
              | 42x-46y  20x-19y  23x+48y  -33x-38y -15x+24y 40x+8y   -50x-25y -40x-48y |
              | 36x-10y  -44x-45y -27x+45y -24x-39y -21x+21y 41x+48y  -48x-48y -50x-50y |
              | -10x+10y 9x+5y    -45x+43y 43x+29y  39x-41y  44x+11y  -6x-33y  -44x-45y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 16 35 30 -35 -27 |)
               | 0 0 x 0 y 0 0 0 |  | 40 28 24 -18 18  |
               | 0 0 0 y x 0 0 0 |  | -2 -2 46 9   -38 |
               | 0 0 0 0 0 x 0 y |  | 39 39 2  -9  -46 |
               | 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 :