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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 | 8x+25y   19x-36y -42x+11y 8x+7y    23x-19y  43x-8y   44x-10y  -38x-47y |
              | 26y      8x-42y  -11x+6y  20x-38y  -31x-34y -27x-14y 47x-7y   -15x-38y |
              | -46x+50y 21x-35y 20x-7y   4x-3y    -6x-36y  48x+5y   -43x+21y 38x-40y  |
              | 49x+24y  35x     -13x+y   -16x+42y -9x+27y  22x+12y  -25x-8y  -34x+10y |
              | -43x+21y -8x+26y -18x+19y 7x+37y   46x+40y  -8x+21y  44x+5y   -47x+6y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -5  -26 -44 1  -21 |)
               | 0 0 x 0 y 0 0 0 |  | 19  19  13  -5 -2  |
               | 0 0 0 y x 0 0 0 |  | 0   31  -21 19 31  |
               | 0 0 0 0 0 x 0 y |  | -37 -14 24  31 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 :