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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 | 19x+40y  -19x-45y -43x-42y 40x+46y -25x+35y -46x+28y -14x+27y -33x-36y |
              | 27x-39y  10x-4y   46x-28y  -41x+8y -5x-17y  -26x-49y -41x-10y -32x+37y |
              | -45x+16y -44x-15y 37x-16y  11x-2y  -44x+37y -18x+18y 47x-37y  -7x+10y  |
              | -14x     45x+4y   45x-7y   13x-29y 47x+6y   9x+30y   -26x-8y  -12x-41y |
              | 4x-37y   44x+22y  41x-20y  19x-y   -28x-20y -38x-13y -8x+33y  30x+23y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | -34 22  -42 -24 0   |)
               | 0 0 x 0 y 0 0 0 |  | -35 -20 -17 0   -34 |
               | 0 0 0 y x 0 0 0 |  | 20  -26 20  -3  29  |
               | 0 0 0 0 0 x 0 y |  | 36  15  -21 -33 39  |
               | 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 :