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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 | -35x+26y -34x     29x+50y  6x-17y   -23x+47y 32x+40y 21x+5y  8x+41y   |
              | -33x+6y  -15x+41y -13x+15y 28x-33y  -42x-11y 49x-48y 7x+14y  43x+45y  |
              | -30y     15x-31y  11x+27y  -38x+34y 47x+2y   -38y    29x+27y 32x-42y  |
              | -37x-32y 27x+34y  -23x-8y  -28x+11y 50x+10y  -x-7y   -8x+11y 38x-43y  |
              | 23x-12y  -7x-10y  29x+29y  -7x+42y  -43x-8y  -45x+7y -14x-5y -41x+42y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -21 -47 -31 26  15  |)
               | 0 0 x 0 y 0 0 0 |  | 25  -20 -5  -1  30  |
               | 0 0 0 y x 0 0 0 |  | 48  23  48  33  -10 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | -47 46  15  -13 35  |

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 :