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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 | -36x-19y 30x+7y  47x-12y  -49x+18y -3x+6y   3x+33y   x-23y   41x-2y   |
              | 5x+35y   40x-44y x+26y    29y      31x      -23x+25y 24x-3y  17x+17y  |
              | -31x-39y 42x+19y 4x+30y   -23x-14y -27x+31y -42x-27y 41x-42y -45x+20y |
              | 16x+36y  4x+47y  -6x+19y  -4x+29y  48x+19y  35x+36y  49x-46y -21x+3y  |
              | 30x+18y  -4x-13y -10x+39y 9x-17y   13x      7x-18y   -6y     -17x+33y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 27 3   22  -1  40 |)
               | 0 0 x 0 y 0 0 0 |  | 30 -34 45  -26 21 |
               | 0 0 0 y x 0 0 0 |  | 48 -31 -7  34  -1 |
               | 0 0 0 0 0 x 0 y |  | 1  0   0   0   0  |
               | 0 0 0 0 0 0 y x |  | 20 -3  -28 -35 -6 |

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 :