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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 | 10x-31y  39x-y    -33x-8y  -36x+22y 11x+2y   32x-40y  6x-37y   7x-48y   |
              | 24x+17y  19x+3y   27x-14y  13x-26y  -12x+19y 40x-49y  17x-6y   46x+11y  |
              | 28x-31y  -48x+16y 48x+13y  -5x+21y  19x+7y   23x+12y  -43x-44y 50x-2y   |
              | 15x-41y  48x-47y  -10x+26y -41x-28y 42x+25y  21x-27y  31x-39y  -45x+31y |
              | -15x+32y 33x+16y  27x-7y   26x-44y  -15x-4y  -15x-45y 6x-48y   -6x-32y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 27  -23 47  46 -45 |)
               | 0 0 x 0 y 0 0 0 |  | 16  8   15  -3 20  |
               | 0 0 0 y x 0 0 0 |  | 40  -19 -23 30 3   |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0  0   |
               | 0 0 0 0 0 0 y x |  | -13 -4  8   29 -38 |

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 :