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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 | 45x+21y  -5x-50y  -38x+14y -44x-39y -41x+12y 14x+14y  10x+45y -7x-25y  |
              | 19x-5y   37x-26y  -2x+42y  45x+40y  5x-19y   32x-25y  11x-8y  -22x+2y  |
              | -13x+40y -37x-16y -37x+28y 13x-9y   35x-16y  -13x-39y 3x-6y   42x+2y   |
              | 8x+23y   33x-20y  -32x-9y  21x+34y  x-8y     8x-43y   4x-50y  38x-7y   |
              | 21x-32y  17x+34y  9x+40y   -7x-32y  8x+31y   25x-44y  7x+3y   -32x-24y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -23 9   -37 27  34 |)
               | 0 0 x 0 y 0 0 0 |  | 40  -32 29  -18 48 |
               | 0 0 0 y x 0 0 0 |  | 46  -10 6   47  45 |
               | 0 0 0 0 0 x 0 y |  | -7  25  45  -18 49 |
               | 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 :