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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 | 22x+40y  -18x+20y -12x+17y -35x-39y -16x+43y -15x-26y -46x-25y -49x-49y |
              | 42x-16y  10x-19y  -17x+36y 17x-47y  18x-38y  33x+6y   43x-38y  6x+46y   |
              | 33x-30y  -35x+22y 42x+28y  -41x+26y -7x+23y  -16x-14y -44x+22y 23x-44y  |
              | -41x-12y 36x-42y  -40x-23y 45x+18y  -7x+17y  -27x+9y  28x+46y  46x-34y  |
              | -8x+14y  4x       -37x-5y  46x-47y  -2x+36y  3x+16y   38x+13y  5x-19y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -33 -49 4   -13 -27 |)
               | 0 0 x 0 y 0 0 0 |  | -33 26  -15 -22 -2  |
               | 0 0 0 y x 0 0 0 |  | -8  40  -31 28  42  |
               | 0 0 0 0 0 x 0 y |  | 10  8   -9  -13 -48 |
               | 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 :