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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 | 35x-39y -12x+6y 44x-25y  -10x-26y 36x-20y  -29x+40y -45x+27y 25x+30y  |
              | 13x-13y 29x-26y -37x+49y -28x     3x+14y   -13x-14y 26x+26y  21x+46y  |
              | 42x+31y 13x-43y 16x-y    -14x-20y -35x+26y 23x+48y  -27x-y   -7x+32y  |
              | 30x+12y -3x-46y 24x+8y   27x-37y  -30x+38y -8x+14y  31x+14y  -31x-47y |
              | 32x+42y -8x-37y 6x-39y   35x+8y   -46x+36y -25x+9y  38x-33y  -33x+6y  |

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

o4 = (cokernel | y x 0 0 0 0 0 0 |, | 25  45  50  2   -13 |)
               | 0 0 x 0 y 0 0 0 |  | -4  16  -17 -15 8   |
               | 0 0 0 y x 0 0 0 |  | -26 -39 6   2   1   |
               | 0 0 0 0 0 x 0 y |  | -11 22  47  49  15  |
               | 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 :