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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 | -45x-y  -2x-11y  -48x+18y -32x+49y -23x-8y  -7x-6y  27x-22y  13x+38y  |
              | 49x+20y -4x-45y  -10x-22y -x+20y   47x-26y  -7x+32y -37x-31y 50x+21y  |
              | 31x+20y -30x-50y -11x+6y  -21x-6y  17x-48y  -16x-4y -21x+18y -30x-36y |
              | 38x+3y  39x+15y  -4x+26y  11x-30y  40x+y    15x-29y -8x+24y  9x-27y   |
              | 13x-29y -35x-y   -13x+6y  19x-25y  -43x+10y 37x+9y  5x+41y   -29x-24y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 25  -19 15 35 -10 |)
               | 0 0 x 0 y 0 0 0 |  | -3  -33 24 9  4   |
               | 0 0 0 y x 0 0 0 |  | -39 10  35 50 8   |
               | 0 0 0 0 0 x 0 y |  | 5   44  48 32 -6  |
               | 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 :