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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 | -20x+40y 6x-19y   -39x+18y 27x+38y  -21x+26y -3x+30y  2x-13y   -27x-11y |
              | -42x+18y -20x+21y -7x-31y  16x+40y  40x+17y  16x-20y  36x-39y  -25x+3y  |
              | -34x+25y x-12y    5x-18y   5x+48y   2x-15y   33x-y    -11x-19y -10x+50y |
              | 32x-10y  24x+19y  -17x+17y -35x+20y -33x-12y -26x+42y -30x-38y -x-7y    |
              | -44x-16y -20x+21y 31x-41y  -39x+21y 44x-30y  -x-40y   -8x-44y  -14x-14y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 18  -35 11  -34 15  |)
               | 0 0 x 0 y 0 0 0 |  | 20  41  -38 30  -44 |
               | 0 0 0 y x 0 0 0 |  | 7   -2  -18 30  -21 |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | -27 14  12  0   34  |

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 :