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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 | 9x+29y  -42x-5y  21x+43y  12x-y    26x-48y  -20x+12y 44x-24y -33x+37y |
              | 49x-28y 25x-5y   -37x-16y 43x-2y   10y      -30x-44y -4x+10y -39x-6y  |
              | 12x+28y -12x+45y 48x+45y  -20x-14y -5x+39y  -46x+4y  50x-39y -9x+47y  |
              | -33x+9y 14y      -30x-3y  -2x-12y  -31x-40y -28x-37y 26x+37y 30x+19y  |
              | 4x-10y  -44x-21y 49x+28y  -26x+28y 32x-22y  -32x-19y 43x+14y -3x+44y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 37  18  27 -9  7  |)
               | 0 0 x 0 y 0 0 0 |  | -44 -17 47 20  34 |
               | 0 0 0 y x 0 0 0 |  | -49 -45 42 -45 13 |
               | 0 0 0 0 0 x 0 y |  | 29  41  6  -46 32 |
               | 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 :