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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 | 24x-3y   -34x-y   -36x+y   -10x+32y 50x-30y  43x-39y 2x-41y   46x-17y |
              | 28x+31y  -x-43y   -34x+13y 46x+15y  -31x+37y 5x-12y  -23x+26y 35x-19y |
              | -14x+44y -24x+26y -35x+26y 4x+30y   37x+33y  26x-9y  -21x+28y 35x-17y |
              | -x-37y   42x-4y   12x-25y  -6x+37y  -9x+10y  19x+41y -26x-32y 43x-49y |
              | 40x-39y  37x-7y   -15x-47y 32x-27y  -45x-39y 5x+45y  42x+9y   26x+11y |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -11 31  -1  35  18  |)
               | 0 0 x 0 y 0 0 0 |  | 50  39  -9  -36 -42 |
               | 0 0 0 y x 0 0 0 |  | 17  -30 35  -36 3   |
               | 0 0 0 0 0 x 0 y |  | 1   0   0   0   0   |
               | 0 0 0 0 0 0 y x |  | 50  -48 -24 -34 -23 |

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 :