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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-50y   -24x-12y 13x+27y  47x+12y 23x+40y -40x-34y -25x-6y  -25x-41y |
              | 43x+9y   42x-38y  -47x+41y 11x-45y -2x-3y  49x+25y  -30x-21y -10x+35y |
              | 4x-50y   -18x-28y -20x-25y -4x-29y 17x-26y -45x+4y  -8x+16y  17x+38y  |
              | 39x-33y  x-33y    12x-44y  6x-31y  16x+26y 45x+47y  -13x+47y 11x-48y  |
              | -13x-20y -42x+25y 15x-47y  17x-15y -31x-4y 6x       49x-19y  -32x+41y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -38 -10 16  -21 -24 |)
               | 0 0 x 0 y 0 0 0 |  | 35  -11 -36 35  40  |
               | 0 0 0 y x 0 0 0 |  | -24 37  18  4   0   |
               | 0 0 0 0 0 x 0 y |  | -2  -42 31  -36 23  |
               | 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 :