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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 | -49x+23y x-6y     -28x-46y 38x+16y  x+4y     5x+17y  39x-14y  -36x+8y  |
              | -40x-43y 7x-34y   -29x+9y  -27x+16y -4x-34y  14x-23y -50x-22y -13x-45y |
              | 30x+32y  -11x-9y  -3x+6y   4x-28y   -15x-20y 33x-45y -9x-11y  21x-40y  |
              | -24x+36y 43x+19y  -7x+47y  -28x-35y 39x+33y  24x-7y  13x+21y  -29x-15y |
              | -8x-11y  -14x-10y 20x-41y  42x-48y  16x+23y  19x-36y 2x-4y    23x+32y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 20 -24 2   46  14  |)
               | 0 0 x 0 y 0 0 0 |  | -9 40  -19 34  -11 |
               | 0 0 0 y x 0 0 0 |  | 6  23  -16 -36 15  |
               | 0 0 0 0 0 x 0 y |  | 44 15  -32 -18 27  |
               | 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 :