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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 | -x+37y   18x+8y  -42x+49y 25x-13y  19x-47y  8x       -26x+40y 48x+20y |
              | -2x+47y  -9x+32y 15x-36y  42x-12y  -40x-4y  14x-29y  21x-33y  -3x-41y |
              | -25x+11y -7x-44y 24x+12y  25x-6y   25x-8y   -26x-12y 43x+42y  28x-37y |
              | -13x-47y 38x+32y 26x+19y  -46x+29y -13x+13y 45x+31y  35x-y    5x-12y  |
              | 13x-36y  12x-37y -43x-11y 15x-19y  -43x-14y -9x-14y  48x-46y  -22x-5y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -10 -20 29  -38 32  |)
               | 0 0 x 0 y 0 0 0 |  | -41 19  -34 16  22  |
               | 0 0 0 y x 0 0 0 |  | -36 -36 9   4   -30 |
               | 0 0 0 0 0 x 0 y |  | 13  21  31  -10 30  |
               | 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 :