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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 | -43x+25y -45x-48y -35x-40y -35x+43y -40x+33y 30x+9y   19x-23y  -20x+49y |
              | -5x+40y  -33x-y   11x+11y  -19x-4y  -44x-26y -10x-31y 7x-44y   -12x+8y  |
              | -20x-41y 23x-21y  17x+48y  -19x-22y -44x-3y  44x+47y  -19x-16y -46x+24y |
              | -20x+17y -45x-50y -20x-25y -35x-10y 11x+28y  -25x-14y -19x+38y -15x+40y |
              | 19x-18y  35x+48y  6x+2y    35x+39y  -7x+3y   -x-35y   x+25y    -35x+39y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 11  40 -2  49  -46 |)
               | 0 0 x 0 y 0 0 0 |  | -31 -3 -47 -36 -28 |
               | 0 0 0 y x 0 0 0 |  | 13  21 -23 -33 -9  |
               | 0 0 0 0 0 x 0 y |  | -14 -3 -20 47  -42 |
               | 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 :