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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-27y  6x-40y   15x-12y  4x-27y   21x+21y 10x-35y  -41x-2y  -32x-12y |
              | -18x-17y 6x+7y    15x-y    -36x-43y 16x+16y -35x+46y -21x-5y  -15x-39y |
              | -32x-22y -16x+34y -30x-15y 3x+15y   -36x-9y -2x+36y  -17x-29y 30x-27y  |
              | -6x+35y  26x-44y  -47x-17y -10x-21y 4x-29y  16x-43y  -23x-3y  19x-37y  |
              | -46x-11y 15x-38y  x-34y    20x+33y  38x-27y 42x-3y   49x+27y  50x+37y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -12 13  -14 -37 17 |)
               | 0 0 x 0 y 0 0 0 |  | -42 -49 29  -36 13 |
               | 0 0 0 y x 0 0 0 |  | 48  9   -1  48  5  |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0  |
               | 0 0 0 0 0 0 y x |  | 44  -30 -17 18  9  |

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 :