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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 | 11x-10y  7x-29y   -45x+4y  -44x+15y 4x+45y   -15x-21y 24x+31y -36x-9y  |
              | -31x-21y -4x-32y  42x+23y  37x+16y  -17x-23y -20x-20y 45x-38y 38x+17y  |
              | 22x-13y  31x-22y  47x-24y  4x-28y   -25x-33y 48x+15y  34x+2y  32x+32y  |
              | 11x-23y  -18x+38y -36x-14y 3x+40y   11x-4y   4x+34y   5x-24y  -40x-29y |
              | -50x+4y  17x-42y  -9x-30y  -48x-42y 37x-28y  26x-4y   2x-31y  -x-50y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 1   6   -11 -43 23  |)
               | 0 0 x 0 y 0 0 0 |  | -2  -5  -7  -45 -36 |
               | 0 0 0 y x 0 0 0 |  | -16 -32 -27 7   -37 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | 48  32  4   -44 -37 |

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 :