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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 | -31x+22y -37x+y   39x-50y  -20x+26y -36x+39y -29x+9y  -16x-28y 42x-42y  |
              | 30x-23y  13x-50y  -9x-16y  14x+50y  -7x+43y  -33x-43y -14x+46y -18x-32y |
              | 50x-23y  -46y     -21x+20y 3x-22y   -32x-6y  48x-36y  39y      5x+28y   |
              | -28x+23y 33x-6y   -22x     13x+39y  -15x-41y 21x-44y  44x+13y  -15x-30y |
              | -26x-29y -16x-38y 41x-y    44x-13y  31x+32y  27x+29y  -x-2y    11x-9y   |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -48 17  -12 29  -19 |)
               | 0 0 x 0 y 0 0 0 |  | -11 -50 14  -17 -50 |
               | 0 0 0 y x 0 0 0 |  | -41 12  -5  -11 -24 |
               | 0 0 0 0 0 x 0 y |  | 22  46  13  -27 -19 |
               | 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 :