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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 | -15x+15y 13x-18y  -21x+29y x-28y    25x+26y  35x+39y  -36x+19y 5x+8y    |
              | -41x-3y  -18x-48y 25x+26y  27x-9y   -12x+30y 47x+37y  -35x+7y  41x+3y   |
              | -40x+31y 5x-45y   -32x+15y -49x+5y  -15x+3y  -50x-23y -20x+28y -48x-26y |
              | 39x+12y  -18x+41y -38x-3y  -37x+25y -35x+12y 45x-28y  -31x+17y 22x-10y  |
              | 50x+44y  -12x-22y -39x+21y -40x-49y -34x+43y -4x-27y  6x+5y    27x-17y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -14 -8  6   -4  -31 |)
               | 0 0 x 0 y 0 0 0 |  | 39  29  -29 -45 -43 |
               | 0 0 0 y x 0 0 0 |  | -47 -28 50  -47 -4  |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | 27  -40 42  50  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 :